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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv_0.7.4.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" Uncertainty, Event Detection, Efficiency, Microcontrollers " lang="en" name="keywords"/> <base href="/html/2402.09264v3/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S1" title="I Introduction ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2" title="II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Related Works</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S3" title="III UR2M System overview ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">UR2M System overview</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4" title="IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Efficient Uncertainty Quantification</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.SS1" title="IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">Evidential Deep Learning</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.SS2" title="IV-B Efficient Evidential Modeling for Event Detection on MCUs ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Efficient Evidential Modeling for Event Detection on MCUs</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.SS3" title="IV-C Uncertainty-aware training and optimization ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span> </span><span class="ltx_text ltx_font_italic">Uncertainty-aware training and optimization</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5" title="V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Cascade learning</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5.SS1" title="V-A Single-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-A</span> </span><span class="ltx_text ltx_font_italic">Single-event Sharing</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5.SS2" title="V-B Multiple-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">V-B</span> </span><span class="ltx_text ltx_font_italic">Multiple-event Sharing</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6" title="VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span><span class="ltx_text ltx_font_smallcaps">Implementation</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6.SS1" title="VI-A System Implementation ‣ VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VI-A</span> </span><span class="ltx_text ltx_font_italic">System Implementation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6.SS2" title="VI-B Uncertainty Operator Implementation ‣ VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VI-B</span> </span><span class="ltx_text ltx_font_italic">Uncertainty Operator Implementation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6.SS3" title="VI-C MCU Library Optimization ‣ VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VI-C</span> </span><span class="ltx_text ltx_font_italic">MCU Library Optimization</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7" title="VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VII </span><span class="ltx_text ltx_font_smallcaps">Evaluation settings</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.SS1" title="VII-A Evaluated Datasets ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VII-A</span> </span><span class="ltx_text ltx_font_italic">Evaluated Datasets</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.SS2" title="VII-B Uncertainty Metrics ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VII-B</span> </span><span class="ltx_text ltx_font_italic">Uncertainty Metrics</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.SS3" title="VII-C Uncertainty Quantification Baselines ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VII-C</span> </span><span class="ltx_text ltx_font_italic">Uncertainty Quantification Baselines</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8" title="VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VIII </span><span class="ltx_text ltx_font_smallcaps">Results</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS1" title="VIII-A Performance of Event Detection ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VIII-A</span> </span><span class="ltx_text ltx_font_italic">Performance of Event Detection</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS2" title="VIII-B Impact of different Uncertainty Thresholds ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VIII-B</span> </span><span class="ltx_text ltx_font_italic">Impact of different Uncertainty Thresholds</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS3" title="VIII-C End-to-end System Efficiency ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VIII-C</span> </span><span class="ltx_text ltx_font_italic">End-to-end System Efficiency</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS4" title="VIII-D Robustness Against Signal Uncertainties ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VIII-D</span> </span><span class="ltx_text ltx_font_italic">Robustness Against Signal Uncertainties</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S9" title="IX Discussion ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IX </span><span class="ltx_text ltx_font_smallcaps">Discussion</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S10" title="X Conclusion ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">X </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S11" title="XI Acknowledgment ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">XI </span><span class="ltx_text ltx_font_smallcaps">Acknowledgment</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <div aria-label="Conversion errors have been found" class="package-alerts ltx_document" role="status"> <button aria-label="Dismiss 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Feedback on these issues are not necessary; they are known and are being worked on.</p> <ul arial-label="Unsupported packages used in this paper"> <li>failed: oplotsymbl</li> <li>failed: nth</li> </ul> <p>Authors: achieve the best HTML results from your LaTeX submissions by following these <a href="https://info.arxiv.org/help/submit_latex_best_practices.html" target="_blank">best practices</a>.</p> </div><div class="section" id="target-section"><div id="license-tr">License: arXiv.org perpetual non-exclusive license</div><div id="watermark-tr">arXiv:2402.09264v3 [cs.LG] 12 Mar 2024</div></div> <script> function closePopup() { document.querySelector('.package-alerts').style.display = 'none'; } </script> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Hong Jia1, Young D. Kwon1, Dong Ma2, Nhat Pham3, Lorena Qendro4, Tam Vu5 and Cecilia Mascolo1 </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">1University of Cambridge, Cambridge, UK 2Singapore Management University, Singapore <br class="ltx_break"/>3Cardiff University, Cardiff, UK 4Nokia Bell Labs, Cambridge, UK 5University of Colorado Boulder, Colorado, US <br class="ltx_break"/>{hj359, ydk21}@cam.ac.uk, dongma@smu.edu.sg, phamn@cardiff.ac.uk, <br class="ltx_break"/>lorena.qendro@nokia-bell-labs.com, tam.vu@colorado.edu, cm542@cam.ac.uk </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Traditional machine learning techniques are prone to generating inaccurate predictions when confronted with shifts in the distribution of data between the training and testing phases. This vulnerability can lead to severe consequences, especially in applications such as mobile healthcare. Uncertainty estimation has the potential to mitigate this issue by assessing the reliability of a model’s output. However, existing uncertainty estimation techniques often require substantial computational resources and memory, making them impractical for implementation on microcontrollers (MCUs). This limitation hinders the feasibility of many important on-device wearable event detection (WED) applications, such as heart attack detection.</p> <p class="ltx_p" id="id2.id2">In this paper, we present <span class="ltx_text ltx_font_bold" id="id2.id2.1">UR2M</span>, a novel <span class="ltx_text ltx_font_bold" id="id2.id2.2">Uncertainty</span> and <span class="ltx_text ltx_font_bold" id="id2.id2.3">Resource-aware</span> event detection framework for <span class="ltx_text ltx_font_bold" id="id2.id2.4">MCUs</span>. Specifically, we (i) develop an uncertainty-aware WED based on evidential theory for accurate event detection and reliable uncertainty estimation; (ii) introduce a cascade ML framework to achieve efficient model inference via early exits, by sharing shallower model layers among different event models; (iii) optimize the deployment of the model and MCU library for system efficiency. We conducted extensive experiments and compared UR2M to traditional uncertainty baselines using three wearable datasets. Our results demonstrate that UR2M achieves up to 864% faster inference speed, 857% energy-saving for uncertainty estimation, 55% memory saving on two popular MCUs, and a 22% improvement in uncertainty quantification performance. UR2M can be deployed on a wide range of MCUs, significantly expanding real-time and reliable WED applications.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Uncertainty, Event Detection, Efficiency, Microcontrollers </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">With advancements in pervasive, low-power, and embedded sensors, a range of human physiological signals can be collected and continuously analyzed. Empowered by machine learning (ML), especially deep learning (DL), these sensors provide great opportunities for a plethora of wearable event detection (WED) applications, such as the detection of stress levels <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib1" title="">1</a>]</cite>, blood pressure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib2" title="">2</a>]</cite>, or respiratory illnesses <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib3" title="">3</a>]</cite>. Recently, deploying ML models directly on microcontrollers (MCUs) has attracted tremendous attention due to their potential to improve user privacy and computational latency in WED, especially under unstable network conditions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib4" title="">4</a>]</cite>. However, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S1.F1" title="Figure 1 ‣ I Introduction ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a>, designing and deploying efficient WED models on MCUs is challenging due to their limited memory space and battery life, especially in comparison to mobile phones <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib4" title="">4</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Furthermore, many existing WED models prioritize enhancing classification accuracy while overlooking the importance of prediction reliability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib5" title="">5</a>]</cite>, which is crucial in fields like health. Reliability is quantified as <span class="ltx_text ltx_font_italic" id="S1.p2.1.1">uncertainty</span>, indicating the trustworthiness of the classification results <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib6" title="">6</a>]</cite>. Factors such as hardware differences, environmental variations, data collection methods, and sensor degradation can lead to distribution shifts between training and testing data (data uncertainty) or unseen data (model uncertainty <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib7" title="">7</a>]</cite>), reducing the reliability of WED models.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="222" id="S1.F1.g1" src="x1.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Memory and power comparison between a typical mobile phone and microcontrollers.</figcaption> </figure> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Several methods for quantifying uncertainty have been investigated. Bayesian Neural Networks (BNNs), a prominent approach for uncertainty estimation, quantify uncertainty by estimating posteriors over model weights <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib8" title="">8</a>]</cite>. However, BNNs entail substantial computational expenses <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib9" title="">9</a>]</cite>. Although approximation techniques such as Monte Carlo dropout (MCDP) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib10" title="">10</a>]</cite> and deep ensembles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib7" title="">7</a>]</cite> have been proposed, these methods still require ensembling multiple models and various inference steps, which introduce intensive computational and memory demands, as well as increased latency. Recent research has also introduced deterministic models that require only one forward pass, making them more efficient but at the cost of lower accuracy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib11" title="">11</a>]</cite>. As a result, integrating reliable uncertainty could pose additional complexities in the design and deployment of trustworthy WED models on MCUs.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Lastly, existing works demonstrate inefficiency in supporting multi-event detection on MCUs, as they typically employ individual models for each event to ensure reusability across different applications or use cases and to optimize efficiency for each model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib12" title="">12</a>]</cite>. However, wearable devices often require the simultaneous detection of multiple events. For instance, a single electroencephalography (EEG) input might be utilized to concurrently detect the brain’s alpha wave (event 1) for a guided-meditation application, and beta wave (event 2) for a focus monitoring application. Additionally, executing multiple inferences (encompassing both prediction and uncertainty estimation) for varied events can be resource-intensive, potentially rendering WED deployment on MCUs impracticable due to memory constraints.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">To address the aforementioned challenges, we propose an efficient uncertainty estimation approach based on evidential deep learning (EDL) and cascade learning. Specifically, (i) EDL is designed to predict a distribution, parameterized by a vector, instead of providing a point prediction through a single DL model, which allows for the direct prediction of event detection and its associated uncertainty via a single inference. (ii) For each event (intra-event), we consider three models of varied depths (i.e., shallow, medium, and deep); herein, deeper models are stacked upon shallower ones, meaning the lower layers are shared. A classifier layer (termed a “head”) is appended to each model. This design adheres to the observation that some testing samples, particularly those near the center of the training sample distribution, do not require a full pass through the deep model to ensure a reliable prediction <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib13" title="">13</a>]</cite>. Consequently, early exits can be employed to enhance computational cost-effectiveness and inference speed, with uncertainty chosen as the criterion for an early exit to ensure the reliability of the prediction. (iii) For multiple events (inter-event) using the same input, we propose the sharing of all layers for feature extraction and the training of individual classification layers (referred to as “multi-heads”). As a result, our framework can be effortlessly scaled to multiple events with minimal memory overhead, since only the heads need to be added. Additionally, reusing shared layers for different events reduces computation time and cost.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">We further apply three techniques to improve the efficiency of our approach during implementation. First, we implement an architecture search to find the optimal model structure automatically (e.g., number of model layers and size of channels) for specific WED tasks based on recent success models designed for MCUs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib14" title="">14</a>]</cite>. Second, we conduct scalar quantization of the model weights into 8-bit integers to decrease the model size and further save memory. Third, to reduce the memory consumption of the deep learning library, we remove unnecessary components that are not utilized in our models. Finally, we conduct comprehensive experiments with two MCU platforms to demonstrate the effectiveness of the proposed approach.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">To summarize, we make the following contributions:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">We propose a cascade model architecture with intra-event and inter-event layer sharing to enable efficient multi-event detection. We also conduct efficient architecture search, model compression, and library optimization to improve system efficiency (§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5" title="V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">V</span></a>-§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6" title="VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">VI</span></a>).</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">We propose a novel uncertainty-aware learning paradigm based on evidential theory for efficient and reliable WED uncertainty estimation on MCUs (§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4" title="IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">IV</span></a>).</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">We conduct extensive experiments on three popular wearable datasets and implement our framework on two off-the-shelf MCUs, including STM32F446ZE and STM32H747F7, with limited SRAM memory (128KB and 512KB, respectively). Our evaluation shows that the proposed framework performs up to 864% better inference speed and 857% energy saving compared to uncertainty baselines. The approach also saves 55% of memory compared with existing uncertainty estimation baselines (§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7" title="VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">VII</span></a>-§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8" title="VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">VIII</span></a>), enabling the deployment of WED models on MCUs with limited memory (e.g., STM32F205VB with 64KB SRAM).</p> </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Related Works</span> </h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">This section briefly discusses the literature on machine learning on MCUs, event detection on resource-constrained devices, and efficient methods for uncertainty estimation.</p> </div> <figure class="ltx_figure" id="S2.F2"> <figure class="ltx_figure ltx_align_center" id="S2.F2.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="227" id="S2.F2.sf1.g1" src="x2.png" width="688"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(a) </span></figcaption> </figure> <br class="ltx_break ltx_centering"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>System overview.</figcaption> </figure> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.p2.1.1">Tiny machine learning on MCUs.</span> Tiny Machine Learning <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib14" title="">14</a>]</cite> (TinyML) aims to execute deep learning models locally on extremely resource-constrained devices such as MCUs. Recent studies have concentrated on optimizing network architectures considering constraints such as limited memory, energy, FLOPs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib4" title="">4</a>]</cite>, and processor speed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib15" title="">15</a>]</cite>. However, these approaches focus solely on classification accuracy, treating them as single-point predictions without considering uncertainty estimation. In contrast, we further include uncertainty estimation of the desired predictions to enable a more reliable WED.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.p3.1.1">Event detection on resource-constrained devices.</span> Recent years have seen a surge in research focused on event detection using wearables, exploring various sensing modalities including image <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib16" title="">16</a>]</cite>, audio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib17" title="">17</a>]</cite>, electrocardiogram (ECG) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib18" title="">18</a>]</cite>, and others. However, most existing WED approaches only utilize wearables for data collection, offloading processing tasks like pre-processing, feature extraction, and ML modelling to cloud-based GPUs (through WiFi) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib19" title="">19</a>]</cite>, desktop GPUs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib20" title="">20</a>]</cite>, mobile devices <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib1" title="">1</a>]</cite> or IoT devices <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib21" title="">21</a>]</cite>. This category of approaches can lead to high latency during signal transmission or raise privacy concerns. To address these challenges, our focus is on comprehensive WED for on-MCU computation, developing efficient and lightweight ML models suitable for limited-resource environments.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.p4.1.1">Efficient uncertainty estimation.</span> Some effort has been devoted to achieving efficient uncertainty estimation, such as regulating the neural network weights to simulate BNNs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib22" title="">22</a>]</cite>. Another stream of studies focuses on expensive and not deployable operations on MCUs like flow <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib23" title="">23</a>]</cite>, spectral normalization <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib24" title="">24</a>]</cite>, and stochastic Convolutional layers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib9" title="">9</a>]</cite>. Despite their success in improving computation efficiency, their accuracy still either performs four times worse than the state-of-the-art (SOTA) method of deep ensembles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib7" title="">7</a>]</cite> or require customized operators and libraries that are currently unavailable on MCUs. As an alternative to using ensembles, knowledge distillation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib25" title="">25</a>]</cite> has been proposed as a means of training a single model. However, knowledge distillation typically requires out-of-distribution (OOD) data, which is often difficult to obtain for real-world applications. Compared to existing work, our study is the first to propose an efficient model for uncertainty quantification on MCUs.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">UR2M System overview</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.3">UR2M includes two stages: <span class="ltx_text ltx_font_bold" id="S3.p1.3.1">model training</span> (<math alttext="\lx@sectionsign" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi id="S3.p1.1.m1.1.1" mathvariant="normal" xref="S3.p1.1.m1.1.1.cmml">§</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\lx@sectionsign</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">§</annotation></semantics></math><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4" title="IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">IV</span></a>-<math alttext="\lx@sectionsign" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi id="S3.p1.2.m2.1.1" mathvariant="normal" xref="S3.p1.2.m2.1.1.cmml">§</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\lx@sectionsign</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">§</annotation></semantics></math><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5" title="V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">V</span></a>) and <span class="ltx_text ltx_font_bold" id="S3.p1.3.2">deployment</span> (<math alttext="\lx@sectionsign" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi id="S3.p1.3.m3.1.1" mathvariant="normal" xref="S3.p1.3.m3.1.1.cmml">§</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">\lx@sectionsign</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">§</annotation></semantics></math><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6" title="VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">VI</span></a>) as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2.F2" title="Figure 2 ‣ II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>. During the <span class="ltx_text ltx_font_bold" id="S3.p1.3.3">training stage</span>, there are three objectives: (1) EDL for efficient uncertainty quantification, (2) Cascade ML learning which includes single-event (intra-event) detection via early exits, and multi-event (inter-event) detection via feature sharing and multi-heads. During the <span class="ltx_text ltx_font_bold" id="S3.p1.3.4">deployment stage</span>, we first carry out (1) multi-tenancy deployment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib26" title="">26</a>]</cite>, allowing multiple ML models (referred to as “tenants”) to efficiently and dynamically share the same memory space among intra-event models. We then further focus on (2) optimizing the model and the MCU library.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">In detail, wearable sensors first capture event streaming signals. Features are then extracted for different signals, such as Mel-frequency cepstral coefficients (MFCC) for the audio signals. Following this, evidential modeling via EDL and one-vs-all training <span class="ltx_text ltx_font_bold" id="S3.p2.1.1">(§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4" title="IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">IV</span></a>)</span> are applied to obtain reliable WED predictions and estimate uncertainty. Within the EDL framework, we specifically designed a cascade learning architecture <span class="ltx_text ltx_font_bold" id="S3.p2.1.2">(§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5" title="V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">V</span></a>)</span> for single-event detection, which divides the network layers into shallow, medium, and deep levels to enable intra-event sharing (sharing shallower layers and inferring with early exits within an event model) and process samples at different levels of recognition difficulty. Further, we propose inter-event sharing (sharing entire layers for feature extraction) for multi-event detection. In addition to the modeling, we further carry out efficiency improvements <span class="ltx_text ltx_font_bold" id="S3.p2.1.3">(§<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S6" title="VI Implementation ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">VI</span></a>)</span> via model architecture search (during model training), quantization, uncertainty operator wrap-up, and MCU library optimizations. </p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Efficient Uncertainty Quantification</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this Section, we propose a highly efficient EDL model tailored for event detection on MCUs. This model is optimized to adhere to the constraints of MCUs, employing distributions to achieve accurate uncertainty quantification in real-time scenarios through a single forward pass.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.5.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.6.2">Evidential Deep Learning</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.8">For a given input <math alttext="x^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><msup id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">x</mi><mi id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" 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xref="S4.SS1.p1.2.m2.1.1.4">𝑖</ci><ci id="S4.SS1.p1.2.m2.1.1.5.cmml" xref="S4.SS1.p1.2.m2.1.1.5">𝑟</ci><apply id="S4.SS1.p1.2.m2.1.1.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1.1.1">superscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.1.1.1.2">𝜶</ci><ci id="S4.SS1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">Dir(\bm{\alpha}^{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_D italic_i italic_r ( bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math>, where <math alttext="\bm{\alpha}^{i}=[\alpha_{1}^{i}" class="ltx_math_unparsed" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><mrow id="S4.SS1.p1.3.m3.1b"><msup id="S4.SS1.p1.3.m3.1.1"><mi id="S4.SS1.p1.3.m3.1.1.2">𝜶</mi><mi id="S4.SS1.p1.3.m3.1.1.3">i</mi></msup><mo id="S4.SS1.p1.3.m3.1.2">=</mo><mrow id="S4.SS1.p1.3.m3.1.3"><mo id="S4.SS1.p1.3.m3.1.3.1" stretchy="false">[</mo><msubsup id="S4.SS1.p1.3.m3.1.3.2"><mi id="S4.SS1.p1.3.m3.1.3.2.2.2">α</mi><mn id="S4.SS1.p1.3.m3.1.3.2.2.3">1</mn><mi id="S4.SS1.p1.3.m3.1.3.2.3">i</mi></msubsup></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">\bm{\alpha}^{i}=[\alpha_{1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = [ italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>,<math alttext="\alpha_{2}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><msubsup id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2.2" xref="S4.SS1.p1.4.m4.1.1.2.2.cmml">α</mi><mn id="S4.SS1.p1.4.m4.1.1.2.3" xref="S4.SS1.p1.4.m4.1.1.2.3.cmml">2</mn><mi id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">superscript</csymbol><apply id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.2.1.cmml" xref="S4.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2.2">𝛼</ci><cn id="S4.SS1.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS1.p1.4.m4.1.1.2.3">2</cn></apply><ci id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">\alpha_{2}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>,…,<math alttext="\alpha_{C}^{i}]" class="ltx_math_unparsed" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mrow id="S4.SS1.p1.5.m5.1b"><msubsup id="S4.SS1.p1.5.m5.1.1"><mi id="S4.SS1.p1.5.m5.1.1.2.2">α</mi><mi id="S4.SS1.p1.5.m5.1.1.2.3">C</mi><mi id="S4.SS1.p1.5.m5.1.1.3">i</mi></msubsup><mo id="S4.SS1.p1.5.m5.1.2" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">\alpha_{C}^{i}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">italic_α start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ]</annotation></semantics></math> denotes the concentration parameters of the distribution (dense distribution means high evidence and low uncertainty) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib22" title="">22</a>]</cite>. Being a conjugate prior to the categorical distribution, the Dirichlet distribution enables EDL to determine the belief mass <math alttext="\bm{b}^{i}=[b_{1}^{i}" class="ltx_math_unparsed" display="inline" id="S4.SS1.p1.6.m6.1"><semantics id="S4.SS1.p1.6.m6.1a"><mrow id="S4.SS1.p1.6.m6.1b"><msup id="S4.SS1.p1.6.m6.1.1"><mi id="S4.SS1.p1.6.m6.1.1.2">𝒃</mi><mi id="S4.SS1.p1.6.m6.1.1.3">i</mi></msup><mo id="S4.SS1.p1.6.m6.1.2">=</mo><mrow id="S4.SS1.p1.6.m6.1.3"><mo id="S4.SS1.p1.6.m6.1.3.1" stretchy="false">[</mo><msubsup id="S4.SS1.p1.6.m6.1.3.2"><mi id="S4.SS1.p1.6.m6.1.3.2.2.2">b</mi><mn id="S4.SS1.p1.6.m6.1.3.2.2.3">1</mn><mi id="S4.SS1.p1.6.m6.1.3.2.3">i</mi></msubsup></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS1.p1.6.m6.1c">\bm{b}^{i}=[b_{1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.6.m6.1d">bold_italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = [ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>,<math alttext="b_{2}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.7.m7.1"><semantics id="S4.SS1.p1.7.m7.1a"><msubsup id="S4.SS1.p1.7.m7.1.1" xref="S4.SS1.p1.7.m7.1.1.cmml"><mi id="S4.SS1.p1.7.m7.1.1.2.2" xref="S4.SS1.p1.7.m7.1.1.2.2.cmml">b</mi><mn id="S4.SS1.p1.7.m7.1.1.2.3" xref="S4.SS1.p1.7.m7.1.1.2.3.cmml">2</mn><mi id="S4.SS1.p1.7.m7.1.1.3" xref="S4.SS1.p1.7.m7.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.7.m7.1b"><apply id="S4.SS1.p1.7.m7.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.1.1.1.cmml" xref="S4.SS1.p1.7.m7.1.1">superscript</csymbol><apply id="S4.SS1.p1.7.m7.1.1.2.cmml" xref="S4.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.7.m7.1.1.2.1.cmml" xref="S4.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p1.7.m7.1.1.2.2.cmml" xref="S4.SS1.p1.7.m7.1.1.2.2">𝑏</ci><cn id="S4.SS1.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S4.SS1.p1.7.m7.1.1.2.3">2</cn></apply><ci id="S4.SS1.p1.7.m7.1.1.3.cmml" xref="S4.SS1.p1.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.7.m7.1c">b_{2}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.7.m7.1d">italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>,…,<math alttext="b_{C}^{i}]" class="ltx_math_unparsed" display="inline" id="S4.SS1.p1.8.m8.1"><semantics id="S4.SS1.p1.8.m8.1a"><mrow id="S4.SS1.p1.8.m8.1b"><msubsup id="S4.SS1.p1.8.m8.1.1"><mi id="S4.SS1.p1.8.m8.1.1.2.2">b</mi><mi id="S4.SS1.p1.8.m8.1.1.2.3">C</mi><mi id="S4.SS1.p1.8.m8.1.1.3">i</mi></msubsup><mo id="S4.SS1.p1.8.m8.1.2" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="S4.SS1.p1.8.m8.1c">b_{C}^{i}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.8.m8.1d">italic_b start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ]</annotation></semantics></math> correlating directly with uncertainty. A higher belief mass indicates a higher confidence in the prediction, whereas a lower belief mass suggests the presence of uncertainty. Formally,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bm{b}^{i}=(\bm{\alpha}^{i}-1)/S^{i}," class="ltx_Math" display="block" id="S4.E1.m1.1"><semantics id="S4.E1.m1.1a"><mrow id="S4.E1.m1.1.1.1" xref="S4.E1.m1.1.1.1.1.cmml"><mrow id="S4.E1.m1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.cmml"><msup id="S4.E1.m1.1.1.1.1.3" xref="S4.E1.m1.1.1.1.1.3.cmml"><mi id="S4.E1.m1.1.1.1.1.3.2" xref="S4.E1.m1.1.1.1.1.3.2.cmml">𝒃</mi><mi id="S4.E1.m1.1.1.1.1.3.3" xref="S4.E1.m1.1.1.1.1.3.3.cmml">i</mi></msup><mo id="S4.E1.m1.1.1.1.1.2" xref="S4.E1.m1.1.1.1.1.2.cmml">=</mo><mrow id="S4.E1.m1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.cmml"><mrow id="S4.E1.m1.1.1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.E1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E1.m1.1.1.1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.1.1.1.cmml"><msup id="S4.E1.m1.1.1.1.1.1.1.1.1.2" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.E1.m1.1.1.1.1.1.1.1.1.2.2" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝜶</mi><mi id="S4.E1.m1.1.1.1.1.1.1.1.1.2.3" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msup><mo id="S4.E1.m1.1.1.1.1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.E1.m1.1.1.1.1.1.1.1.1.3" xref="S4.E1.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.E1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E1.m1.1.1.1.1.1.2" xref="S4.E1.m1.1.1.1.1.1.2.cmml">/</mo><msup id="S4.E1.m1.1.1.1.1.1.3" xref="S4.E1.m1.1.1.1.1.1.3.cmml"><mi id="S4.E1.m1.1.1.1.1.1.3.2" xref="S4.E1.m1.1.1.1.1.1.3.2.cmml">S</mi><mi id="S4.E1.m1.1.1.1.1.1.3.3" xref="S4.E1.m1.1.1.1.1.1.3.3.cmml">i</mi></msup></mrow></mrow><mo id="S4.E1.m1.1.1.1.2" xref="S4.E1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E1.m1.1b"><apply id="S4.E1.m1.1.1.1.1.cmml" xref="S4.E1.m1.1.1.1"><eq id="S4.E1.m1.1.1.1.1.2.cmml" xref="S4.E1.m1.1.1.1.1.2"></eq><apply id="S4.E1.m1.1.1.1.1.3.cmml" xref="S4.E1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E1.m1.1.1.1.1.3.1.cmml" xref="S4.E1.m1.1.1.1.1.3">superscript</csymbol><ci id="S4.E1.m1.1.1.1.1.3.2.cmml" xref="S4.E1.m1.1.1.1.1.3.2">𝒃</ci><ci id="S4.E1.m1.1.1.1.1.3.3.cmml" xref="S4.E1.m1.1.1.1.1.3.3">𝑖</ci></apply><apply id="S4.E1.m1.1.1.1.1.1.cmml" xref="S4.E1.m1.1.1.1.1.1"><divide id="S4.E1.m1.1.1.1.1.1.2.cmml" xref="S4.E1.m1.1.1.1.1.1.2"></divide><apply id="S4.E1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1"><minus id="S4.E1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1.1.1"></minus><apply id="S4.E1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E1.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.E1.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2.2">𝜶</ci><ci id="S4.E1.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.E1.m1.1.1.1.1.1.1.1.1.2.3">𝑖</ci></apply><cn id="S4.E1.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.E1.m1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S4.E1.m1.1.1.1.1.1.3.cmml" xref="S4.E1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E1.m1.1.1.1.1.1.3.1.cmml" xref="S4.E1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.E1.m1.1.1.1.1.1.3.2.cmml" xref="S4.E1.m1.1.1.1.1.1.3.2">𝑆</ci><ci id="S4.E1.m1.1.1.1.1.1.3.3.cmml" xref="S4.E1.m1.1.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.1c">\bm{b}^{i}=(\bm{\alpha}^{i}-1)/S^{i},</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.1d">bold_italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = ( bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - 1 ) / italic_S start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p1.13">where <math alttext="S^{i}=\sum_{c=1}^{C}\alpha_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.9.m1.1"><semantics id="S4.SS1.p1.9.m1.1a"><mrow id="S4.SS1.p1.9.m1.1.1" xref="S4.SS1.p1.9.m1.1.1.cmml"><msup id="S4.SS1.p1.9.m1.1.1.2" xref="S4.SS1.p1.9.m1.1.1.2.cmml"><mi id="S4.SS1.p1.9.m1.1.1.2.2" xref="S4.SS1.p1.9.m1.1.1.2.2.cmml">S</mi><mi id="S4.SS1.p1.9.m1.1.1.2.3" xref="S4.SS1.p1.9.m1.1.1.2.3.cmml">i</mi></msup><mo id="S4.SS1.p1.9.m1.1.1.1" rspace="0.111em" xref="S4.SS1.p1.9.m1.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.9.m1.1.1.3" xref="S4.SS1.p1.9.m1.1.1.3.cmml"><msubsup id="S4.SS1.p1.9.m1.1.1.3.1" xref="S4.SS1.p1.9.m1.1.1.3.1.cmml"><mo id="S4.SS1.p1.9.m1.1.1.3.1.2.2" xref="S4.SS1.p1.9.m1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.SS1.p1.9.m1.1.1.3.1.2.3" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.cmml"><mi id="S4.SS1.p1.9.m1.1.1.3.1.2.3.2" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.2.cmml">c</mi><mo id="S4.SS1.p1.9.m1.1.1.3.1.2.3.1" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.SS1.p1.9.m1.1.1.3.1.2.3.3" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS1.p1.9.m1.1.1.3.1.3" xref="S4.SS1.p1.9.m1.1.1.3.1.3.cmml">C</mi></msubsup><msubsup id="S4.SS1.p1.9.m1.1.1.3.2" xref="S4.SS1.p1.9.m1.1.1.3.2.cmml"><mi id="S4.SS1.p1.9.m1.1.1.3.2.2.2" xref="S4.SS1.p1.9.m1.1.1.3.2.2.2.cmml">α</mi><mi id="S4.SS1.p1.9.m1.1.1.3.2.2.3" xref="S4.SS1.p1.9.m1.1.1.3.2.2.3.cmml">c</mi><mi id="S4.SS1.p1.9.m1.1.1.3.2.3" xref="S4.SS1.p1.9.m1.1.1.3.2.3.cmml">i</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.9.m1.1b"><apply id="S4.SS1.p1.9.m1.1.1.cmml" xref="S4.SS1.p1.9.m1.1.1"><eq id="S4.SS1.p1.9.m1.1.1.1.cmml" xref="S4.SS1.p1.9.m1.1.1.1"></eq><apply id="S4.SS1.p1.9.m1.1.1.2.cmml" xref="S4.SS1.p1.9.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m1.1.1.2.1.cmml" xref="S4.SS1.p1.9.m1.1.1.2">superscript</csymbol><ci id="S4.SS1.p1.9.m1.1.1.2.2.cmml" xref="S4.SS1.p1.9.m1.1.1.2.2">𝑆</ci><ci id="S4.SS1.p1.9.m1.1.1.2.3.cmml" xref="S4.SS1.p1.9.m1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS1.p1.9.m1.1.1.3.cmml" xref="S4.SS1.p1.9.m1.1.1.3"><apply id="S4.SS1.p1.9.m1.1.1.3.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m1.1.1.3.1.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1">superscript</csymbol><apply id="S4.SS1.p1.9.m1.1.1.3.1.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m1.1.1.3.1.2.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1">subscript</csymbol><sum id="S4.SS1.p1.9.m1.1.1.3.1.2.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1.2.2"></sum><apply id="S4.SS1.p1.9.m1.1.1.3.1.2.3.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3"><eq id="S4.SS1.p1.9.m1.1.1.3.1.2.3.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.1"></eq><ci id="S4.SS1.p1.9.m1.1.1.3.1.2.3.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.2">𝑐</ci><cn id="S4.SS1.p1.9.m1.1.1.3.1.2.3.3.cmml" type="integer" xref="S4.SS1.p1.9.m1.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S4.SS1.p1.9.m1.1.1.3.1.3.cmml" xref="S4.SS1.p1.9.m1.1.1.3.1.3">𝐶</ci></apply><apply id="S4.SS1.p1.9.m1.1.1.3.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m1.1.1.3.2.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2">superscript</csymbol><apply id="S4.SS1.p1.9.m1.1.1.3.2.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p1.9.m1.1.1.3.2.2.1.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2">subscript</csymbol><ci id="S4.SS1.p1.9.m1.1.1.3.2.2.2.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2.2.2">𝛼</ci><ci id="S4.SS1.p1.9.m1.1.1.3.2.2.3.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2.2.3">𝑐</ci></apply><ci id="S4.SS1.p1.9.m1.1.1.3.2.3.cmml" xref="S4.SS1.p1.9.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.9.m1.1c">S^{i}=\sum_{c=1}^{C}\alpha_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.9.m1.1d">italic_S start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_c = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is the Dirichlet strength. From <math alttext="\bm{\alpha}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.10.m2.1"><semantics id="S4.SS1.p1.10.m2.1a"><msup id="S4.SS1.p1.10.m2.1.1" xref="S4.SS1.p1.10.m2.1.1.cmml"><mi id="S4.SS1.p1.10.m2.1.1.2" xref="S4.SS1.p1.10.m2.1.1.2.cmml">𝜶</mi><mi id="S4.SS1.p1.10.m2.1.1.3" xref="S4.SS1.p1.10.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.10.m2.1b"><apply id="S4.SS1.p1.10.m2.1.1.cmml" xref="S4.SS1.p1.10.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.10.m2.1.1.1.cmml" xref="S4.SS1.p1.10.m2.1.1">superscript</csymbol><ci id="S4.SS1.p1.10.m2.1.1.2.cmml" xref="S4.SS1.p1.10.m2.1.1.2">𝜶</ci><ci id="S4.SS1.p1.10.m2.1.1.3.cmml" xref="S4.SS1.p1.10.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.10.m2.1c">\bm{\alpha}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.10.m2.1d">bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\bm{b}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.11.m3.1"><semantics id="S4.SS1.p1.11.m3.1a"><msup id="S4.SS1.p1.11.m3.1.1" xref="S4.SS1.p1.11.m3.1.1.cmml"><mi id="S4.SS1.p1.11.m3.1.1.2" xref="S4.SS1.p1.11.m3.1.1.2.cmml">𝒃</mi><mi id="S4.SS1.p1.11.m3.1.1.3" xref="S4.SS1.p1.11.m3.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.11.m3.1b"><apply id="S4.SS1.p1.11.m3.1.1.cmml" xref="S4.SS1.p1.11.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.11.m3.1.1.1.cmml" xref="S4.SS1.p1.11.m3.1.1">superscript</csymbol><ci id="S4.SS1.p1.11.m3.1.1.2.cmml" xref="S4.SS1.p1.11.m3.1.1.2">𝒃</ci><ci id="S4.SS1.p1.11.m3.1.1.3.cmml" xref="S4.SS1.p1.11.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.11.m3.1c">\bm{b}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.11.m3.1d">bold_italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, we can further infer the categorical prediction <math alttext="\hat{y}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.12.m4.1"><semantics id="S4.SS1.p1.12.m4.1a"><msup id="S4.SS1.p1.12.m4.1.1" xref="S4.SS1.p1.12.m4.1.1.cmml"><mover accent="true" id="S4.SS1.p1.12.m4.1.1.2" xref="S4.SS1.p1.12.m4.1.1.2.cmml"><mi id="S4.SS1.p1.12.m4.1.1.2.2" xref="S4.SS1.p1.12.m4.1.1.2.2.cmml">y</mi><mo id="S4.SS1.p1.12.m4.1.1.2.1" xref="S4.SS1.p1.12.m4.1.1.2.1.cmml">^</mo></mover><mi id="S4.SS1.p1.12.m4.1.1.3" xref="S4.SS1.p1.12.m4.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.12.m4.1b"><apply id="S4.SS1.p1.12.m4.1.1.cmml" xref="S4.SS1.p1.12.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.12.m4.1.1.1.cmml" xref="S4.SS1.p1.12.m4.1.1">superscript</csymbol><apply id="S4.SS1.p1.12.m4.1.1.2.cmml" xref="S4.SS1.p1.12.m4.1.1.2"><ci id="S4.SS1.p1.12.m4.1.1.2.1.cmml" xref="S4.SS1.p1.12.m4.1.1.2.1">^</ci><ci id="S4.SS1.p1.12.m4.1.1.2.2.cmml" xref="S4.SS1.p1.12.m4.1.1.2.2">𝑦</ci></apply><ci id="S4.SS1.p1.12.m4.1.1.3.cmml" xref="S4.SS1.p1.12.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.12.m4.1c">\hat{y}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.12.m4.1d">over^ start_ARG italic_y end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and the associated uncertainty <math alttext="u^{i}" class="ltx_Math" display="inline" id="S4.SS1.p1.13.m5.1"><semantics id="S4.SS1.p1.13.m5.1a"><msup id="S4.SS1.p1.13.m5.1.1" xref="S4.SS1.p1.13.m5.1.1.cmml"><mi id="S4.SS1.p1.13.m5.1.1.2" xref="S4.SS1.p1.13.m5.1.1.2.cmml">u</mi><mi id="S4.SS1.p1.13.m5.1.1.3" xref="S4.SS1.p1.13.m5.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.13.m5.1b"><apply id="S4.SS1.p1.13.m5.1.1.cmml" xref="S4.SS1.p1.13.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.13.m5.1.1.1.cmml" xref="S4.SS1.p1.13.m5.1.1">superscript</csymbol><ci id="S4.SS1.p1.13.m5.1.1.2.cmml" xref="S4.SS1.p1.13.m5.1.1.2">𝑢</ci><ci id="S4.SS1.p1.13.m5.1.1.3.cmml" xref="S4.SS1.p1.13.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.13.m5.1c">u^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.13.m5.1d">italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> as:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\hat{y}^{i}=\arg\max_{c}[{\alpha}^{i}/S^{i}],\quad u^{i}=1-\sum_{c=1}^{C}{b}_{% c}^{i}" class="ltx_Math" display="block" id="S4.E2.m1.2"><semantics id="S4.E2.m1.2a"><mrow id="S4.E2.m1.2.2.2" xref="S4.E2.m1.2.2.3.cmml"><mrow id="S4.E2.m1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.cmml"><msup 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xref="S4.E2.m1.2.2.2.2.3.3.2"><csymbol cd="ambiguous" id="S4.E2.m1.2.2.2.2.3.3.2.2.1.cmml" xref="S4.E2.m1.2.2.2.2.3.3.2">subscript</csymbol><ci id="S4.E2.m1.2.2.2.2.3.3.2.2.2.cmml" xref="S4.E2.m1.2.2.2.2.3.3.2.2.2">𝑏</ci><ci id="S4.E2.m1.2.2.2.2.3.3.2.2.3.cmml" xref="S4.E2.m1.2.2.2.2.3.3.2.2.3">𝑐</ci></apply><ci id="S4.E2.m1.2.2.2.2.3.3.2.3.cmml" xref="S4.E2.m1.2.2.2.2.3.3.2.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.2c">\hat{y}^{i}=\arg\max_{c}[{\alpha}^{i}/S^{i}],\quad u^{i}=1-\sum_{c=1}^{C}{b}_{% c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.2d">over^ start_ARG italic_y end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = roman_arg roman_max start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT [ italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_S start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ] , italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = 1 - ∑ start_POSTSUBSCRIPT italic_c = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.4">Before the training process, acknowledging our initial state of complete uncertainty about the outputs (i.e., uncertainty <math alttext="u^{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msup id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">u</mi><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑢</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">u^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_u start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is set to 1), we initialize <math alttext="\bm{\alpha}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><msup id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">𝜶</mi><mi id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">superscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝜶</ci><ci id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\bm{\alpha}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="[1,1,1]" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.3"><semantics id="S4.SS1.p2.3.m3.3a"><mrow id="S4.SS1.p2.3.m3.3.4.2" xref="S4.SS1.p2.3.m3.3.4.1.cmml"><mo id="S4.SS1.p2.3.m3.3.4.2.1" stretchy="false" xref="S4.SS1.p2.3.m3.3.4.1.cmml">[</mo><mn id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">1</mn><mo id="S4.SS1.p2.3.m3.3.4.2.2" xref="S4.SS1.p2.3.m3.3.4.1.cmml">,</mo><mn id="S4.SS1.p2.3.m3.2.2" xref="S4.SS1.p2.3.m3.2.2.cmml">1</mn><mo id="S4.SS1.p2.3.m3.3.4.2.3" xref="S4.SS1.p2.3.m3.3.4.1.cmml">,</mo><mn id="S4.SS1.p2.3.m3.3.3" xref="S4.SS1.p2.3.m3.3.3.cmml">1</mn><mo id="S4.SS1.p2.3.m3.3.4.2.4" stretchy="false" xref="S4.SS1.p2.3.m3.3.4.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.3b"><list id="S4.SS1.p2.3.m3.3.4.1.cmml" xref="S4.SS1.p2.3.m3.3.4.2"><cn id="S4.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p2.3.m3.1.1">1</cn><cn id="S4.SS1.p2.3.m3.2.2.cmml" type="integer" xref="S4.SS1.p2.3.m3.2.2">1</cn><cn id="S4.SS1.p2.3.m3.3.3.cmml" type="integer" xref="S4.SS1.p2.3.m3.3.3">1</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.3c">[1,1,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.3d">[ 1 , 1 , 1 ]</annotation></semantics></math>, corresponding to <math alttext="\bm{b}^{i}=[0,0,0]" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.3"><semantics id="S4.SS1.p2.4.m4.3a"><mrow id="S4.SS1.p2.4.m4.3.4" xref="S4.SS1.p2.4.m4.3.4.cmml"><msup id="S4.SS1.p2.4.m4.3.4.2" xref="S4.SS1.p2.4.m4.3.4.2.cmml"><mi id="S4.SS1.p2.4.m4.3.4.2.2" xref="S4.SS1.p2.4.m4.3.4.2.2.cmml">𝒃</mi><mi id="S4.SS1.p2.4.m4.3.4.2.3" xref="S4.SS1.p2.4.m4.3.4.2.3.cmml">i</mi></msup><mo id="S4.SS1.p2.4.m4.3.4.1" xref="S4.SS1.p2.4.m4.3.4.1.cmml">=</mo><mrow id="S4.SS1.p2.4.m4.3.4.3.2" xref="S4.SS1.p2.4.m4.3.4.3.1.cmml"><mo id="S4.SS1.p2.4.m4.3.4.3.2.1" stretchy="false" xref="S4.SS1.p2.4.m4.3.4.3.1.cmml">[</mo><mn id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">0</mn><mo id="S4.SS1.p2.4.m4.3.4.3.2.2" xref="S4.SS1.p2.4.m4.3.4.3.1.cmml">,</mo><mn id="S4.SS1.p2.4.m4.2.2" xref="S4.SS1.p2.4.m4.2.2.cmml">0</mn><mo id="S4.SS1.p2.4.m4.3.4.3.2.3" xref="S4.SS1.p2.4.m4.3.4.3.1.cmml">,</mo><mn id="S4.SS1.p2.4.m4.3.3" xref="S4.SS1.p2.4.m4.3.3.cmml">0</mn><mo id="S4.SS1.p2.4.m4.3.4.3.2.4" stretchy="false" xref="S4.SS1.p2.4.m4.3.4.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.3b"><apply id="S4.SS1.p2.4.m4.3.4.cmml" xref="S4.SS1.p2.4.m4.3.4"><eq id="S4.SS1.p2.4.m4.3.4.1.cmml" xref="S4.SS1.p2.4.m4.3.4.1"></eq><apply id="S4.SS1.p2.4.m4.3.4.2.cmml" xref="S4.SS1.p2.4.m4.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.3.4.2.1.cmml" xref="S4.SS1.p2.4.m4.3.4.2">superscript</csymbol><ci id="S4.SS1.p2.4.m4.3.4.2.2.cmml" xref="S4.SS1.p2.4.m4.3.4.2.2">𝒃</ci><ci id="S4.SS1.p2.4.m4.3.4.2.3.cmml" xref="S4.SS1.p2.4.m4.3.4.2.3">𝑖</ci></apply><list id="S4.SS1.p2.4.m4.3.4.3.1.cmml" xref="S4.SS1.p2.4.m4.3.4.3.2"><cn id="S4.SS1.p2.4.m4.1.1.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1">0</cn><cn id="S4.SS1.p2.4.m4.2.2.cmml" type="integer" xref="S4.SS1.p2.4.m4.2.2">0</cn><cn id="S4.SS1.p2.4.m4.3.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.3.3">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.3c">\bm{b}^{i}=[0,0,0]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.3d">bold_italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = [ 0 , 0 , 0 ]</annotation></semantics></math> according to Eq. <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.E1" title="1 ‣ IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a> and Eq. <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.E2" title="2 ‣ IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>. To refine the model, we employ a loss function defined as:</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\min_{\theta}\mathcal{L}=\frac{1}{N}\sum_{i}^{N}CE(\alpha_{c}^{i}/S^{i},y^{i})% -\lambda\cdot H(Dir(\bm{\alpha}^{i}))" class="ltx_Math" display="block" id="S4.E3.m1.3"><semantics id="S4.E3.m1.3a"><mrow id="S4.E3.m1.3.3" xref="S4.E3.m1.3.3.cmml"><mrow id="S4.E3.m1.3.3.5" xref="S4.E3.m1.3.3.5.cmml"><munder id="S4.E3.m1.3.3.5.1" xref="S4.E3.m1.3.3.5.1.cmml"><mi id="S4.E3.m1.3.3.5.1.2" xref="S4.E3.m1.3.3.5.1.2.cmml">min</mi><mi id="S4.E3.m1.3.3.5.1.3" xref="S4.E3.m1.3.3.5.1.3.cmml">θ</mi></munder><mo id="S4.E3.m1.3.3.5a" lspace="0.167em" xref="S4.E3.m1.3.3.5.cmml">⁡</mo><mi class="ltx_font_mathcaligraphic" id="S4.E3.m1.3.3.5.2" 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id="S4.E3.m1.3d">roman_min start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT caligraphic_L = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_C italic_E ( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_S start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) - italic_λ ⋅ italic_H ( italic_D italic_i italic_r ( bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.4">where <math alttext="CE" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mrow id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml"><mi id="S4.SS1.p4.1.m1.1.1.2" xref="S4.SS1.p4.1.m1.1.1.2.cmml">C</mi><mo id="S4.SS1.p4.1.m1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p4.1.m1.1.1.3" xref="S4.SS1.p4.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><apply id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1"><times id="S4.SS1.p4.1.m1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1"></times><ci id="S4.SS1.p4.1.m1.1.1.2.cmml" xref="S4.SS1.p4.1.m1.1.1.2">𝐶</ci><ci id="S4.SS1.p4.1.m1.1.1.3.cmml" xref="S4.SS1.p4.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">CE</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">italic_C italic_E</annotation></semantics></math> denotes the cross-entropy loss, and <math alttext="H" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_H</annotation></semantics></math> represents the entropy of a Dirichlet distribution parameterized by <math alttext="\bm{\alpha}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.1"><semantics id="S4.SS1.p4.3.m3.1a"><msup id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml"><mi id="S4.SS1.p4.3.m3.1.1.2" xref="S4.SS1.p4.3.m3.1.1.2.cmml">𝜶</mi><mi id="S4.SS1.p4.3.m3.1.1.3" xref="S4.SS1.p4.3.m3.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.1b"><apply id="S4.SS1.p4.3.m3.1.1.cmml" 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The first term of the loss function aims to maximize classification accuracy, while the second term controls the output distribution to avoid overconfidence. The hyperparameter <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m4.1"><semantics id="S4.SS1.p4.4.m4.1a"><mi id="S4.SS1.p4.4.m4.1.1" xref="S4.SS1.p4.4.m4.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m4.1b"><ci id="S4.SS1.p4.4.m4.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m4.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m4.1d">italic_λ</annotation></semantics></math> plays a crucial role in balancing these two terms.</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.2">Finally, this procedure will lead to a predicted <math alttext="\bm{\alpha}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><msup id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml"><mi id="S4.SS1.p5.1.m1.1.1.2" xref="S4.SS1.p5.1.m1.1.1.2.cmml">𝜶</mi><mi id="S4.SS1.p5.1.m1.1.1.3" xref="S4.SS1.p5.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><apply id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.1.m1.1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.p5.1.m1.1.1.2.cmml" xref="S4.SS1.p5.1.m1.1.1.2">𝜶</ci><ci id="S4.SS1.p5.1.m1.1.1.3.cmml" xref="S4.SS1.p5.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">\bm{\alpha}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">bold_italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> for each sample which is used to infer the categorical outcome and the associated uncertainty (e.g., <math alttext="u=1-\sum b^{i}" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.1"><semantics id="S4.SS1.p5.2.m2.1a"><mrow id="S4.SS1.p5.2.m2.1.1" xref="S4.SS1.p5.2.m2.1.1.cmml"><mi id="S4.SS1.p5.2.m2.1.1.2" xref="S4.SS1.p5.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS1.p5.2.m2.1.1.1" xref="S4.SS1.p5.2.m2.1.1.1.cmml">=</mo><mrow id="S4.SS1.p5.2.m2.1.1.3" xref="S4.SS1.p5.2.m2.1.1.3.cmml"><mn id="S4.SS1.p5.2.m2.1.1.3.2" xref="S4.SS1.p5.2.m2.1.1.3.2.cmml">1</mn><mo id="S4.SS1.p5.2.m2.1.1.3.1" rspace="0.055em" xref="S4.SS1.p5.2.m2.1.1.3.1.cmml">−</mo><mrow id="S4.SS1.p5.2.m2.1.1.3.3" xref="S4.SS1.p5.2.m2.1.1.3.3.cmml"><mo id="S4.SS1.p5.2.m2.1.1.3.3.1" xref="S4.SS1.p5.2.m2.1.1.3.3.1.cmml">∑</mo><msup id="S4.SS1.p5.2.m2.1.1.3.3.2" xref="S4.SS1.p5.2.m2.1.1.3.3.2.cmml"><mi id="S4.SS1.p5.2.m2.1.1.3.3.2.2" xref="S4.SS1.p5.2.m2.1.1.3.3.2.2.cmml">b</mi><mi id="S4.SS1.p5.2.m2.1.1.3.3.2.3" xref="S4.SS1.p5.2.m2.1.1.3.3.2.3.cmml">i</mi></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m2.1b"><apply id="S4.SS1.p5.2.m2.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1"><eq id="S4.SS1.p5.2.m2.1.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1.1"></eq><ci id="S4.SS1.p5.2.m2.1.1.2.cmml" xref="S4.SS1.p5.2.m2.1.1.2">𝑢</ci><apply id="S4.SS1.p5.2.m2.1.1.3.cmml" xref="S4.SS1.p5.2.m2.1.1.3"><minus id="S4.SS1.p5.2.m2.1.1.3.1.cmml" xref="S4.SS1.p5.2.m2.1.1.3.1"></minus><cn id="S4.SS1.p5.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS1.p5.2.m2.1.1.3.2">1</cn><apply id="S4.SS1.p5.2.m2.1.1.3.3.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3"><sum id="S4.SS1.p5.2.m2.1.1.3.3.1.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3.1"></sum><apply id="S4.SS1.p5.2.m2.1.1.3.3.2.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.p5.2.m2.1.1.3.3.2.1.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3.2">superscript</csymbol><ci id="S4.SS1.p5.2.m2.1.1.3.3.2.2.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3.2.2">𝑏</ci><ci id="S4.SS1.p5.2.m2.1.1.3.3.2.3.cmml" xref="S4.SS1.p5.2.m2.1.1.3.3.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.1c">u=1-\sum b^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.1d">italic_u = 1 - ∑ italic_b start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>).</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.5.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.6.2">Efficient Evidential Modeling for Event Detection on MCUs</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">Implementing the EDL discussed in §<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.SS1" title="IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-A</span></span></a> for WED requires deploying multiple models and performing a series of inferences to detect various events, which significantly challenges the limited computational resources of MCUs. To mitigate this, we propose an efficient EDL modeling for WED, along with related training and optimization techniques designed to infer multiple events concurrently.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S4.SS2.p2.1.1">Efficient EDL Modeling for WED<span class="ltx_text ltx_font_upright" id="S4.SS2.p2.1.1.1">.</span></span> WED is designed to identify an event signal coming from a wearable device. In ML/DL, this objective is defined as a binary classification task over a given duration/period of sensor data. For each binary classifier that detects classes of the event <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_c</annotation></semantics></math>, the outputs of EDL include the binomial belief mass, which can be used to infer the uncertainty of the WED prediction, i.e., how confident it is to be classified as positive (i.e., an event happening) or negative (i.e., an event not happening).</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.2">Given the binary nature of our EDL framework (positive vs negative), we adopt a Beta distribution (a special case of the Dirichlet distribution) to model the event probability. Specifically, a Beta distribution is characterized by two parameters <math alttext="\alpha_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><msubsup id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2.2" xref="S4.SS2.p3.1.m1.1.1.2.2.cmml">α</mi><mi id="S4.SS2.p3.1.m1.1.1.2.3" xref="S4.SS2.p3.1.m1.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">superscript</csymbol><apply id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.2.1.cmml" xref="S4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.1.2.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2.2">𝛼</ci><ci id="S4.SS2.p3.1.m1.1.1.2.3.cmml" xref="S4.SS2.p3.1.m1.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p3.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\alpha_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\beta_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><msubsup id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.2.2" xref="S4.SS2.p3.2.m2.1.1.2.2.cmml">β</mi><mi id="S4.SS2.p3.2.m2.1.1.2.3" xref="S4.SS2.p3.2.m2.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">superscript</csymbol><apply id="S4.SS2.p3.2.m2.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.2.1.cmml" xref="S4.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.2.2.cmml" xref="S4.SS2.p3.2.m2.1.1.2.2">𝛽</ci><ci id="S4.SS2.p3.2.m2.1.1.2.3.cmml" xref="S4.SS2.p3.2.m2.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p3.2.m2.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">\beta_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\mathrm{P}({p_{c}^{i}}\mid{x}^{i};{\theta_{c}})&=\operatorname{% Beta}(p_{c}^{i}\mid\alpha_{c}^{i},\beta_{c}^{i})\\ &=\frac{1}{B(\alpha_{c}^{i},\beta_{c}^{i})}p^{\alpha_{c}^{i}-1}(1-p)^{\beta_{c% }^{i}-1},\end{split}" class="ltx_Math" display="block" id="S4.E4.m1.42"><semantics id="S4.E4.m1.42a"><mtable columnspacing="0pt" displaystyle="true" id="S4.E4.m1.42.42.4" rowspacing="0pt"><mtr id="S4.E4.m1.42.42.4a"><mtd class="ltx_align_right" columnalign="right" id="S4.E4.m1.42.42.4b"><mrow id="S4.E4.m1.40.40.2.39.28.13"><mi id="S4.E4.m1.1.1.1.1.1.1" mathvariant="normal" xref="S4.E4.m1.1.1.1.1.1.1.cmml">P</mi><mo id="S4.E4.m1.40.40.2.39.28.13.14" xref="S4.E4.m1.39.39.1.1.1.cmml">⁢</mo><mrow id="S4.E4.m1.40.40.2.39.28.13.13.1"><mo id="S4.E4.m1.2.2.2.2.2.2" stretchy="false" xref="S4.E4.m1.39.39.1.1.1.cmml">(</mo><mrow id="S4.E4.m1.40.40.2.39.28.13.13.1.1"><msubsup 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xref="S4.E4.m1.37.37.37.10.10.10.1.2.2.3">𝑐</ci></apply><ci id="S4.E4.m1.37.37.37.10.10.10.1.2.3.cmml" xref="S4.E4.m1.37.37.37.10.10.10.1.2.3">𝑖</ci></apply><cn id="S4.E4.m1.37.37.37.10.10.10.1.3.cmml" type="integer" xref="S4.E4.m1.37.37.37.10.10.10.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.42c">\begin{split}\mathrm{P}({p_{c}^{i}}\mid{x}^{i};{\theta_{c}})&=\operatorname{% Beta}(p_{c}^{i}\mid\alpha_{c}^{i},\beta_{c}^{i})\\ &=\frac{1}{B(\alpha_{c}^{i},\beta_{c}^{i})}p^{\alpha_{c}^{i}-1}(1-p)^{\beta_{c% }^{i}-1},\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.42d">start_ROW start_CELL roman_P ( italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∣ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ; italic_θ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) end_CELL start_CELL = roman_Beta ( italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∣ italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG italic_B ( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) end_ARG italic_p start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.9">where <math alttext="\mathrm{P}\left({p_{c}^{i}}\mid{x}^{i};{\theta_{c}}\right)" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><mrow id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml"><mi id="S4.SS2.p5.1.m1.1.1.3" mathvariant="normal" xref="S4.SS2.p5.1.m1.1.1.3.cmml">P</mi><mo id="S4.SS2.p5.1.m1.1.1.2" xref="S4.SS2.p5.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p5.1.m1.1.1.1.1" xref="S4.SS2.p5.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.p5.1.m1.1.1.1.1.2" xref="S4.SS2.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p5.1.m1.1.1.1.1.1" xref="S4.SS2.p5.1.m1.1.1.1.1.1.cmml"><msubsup 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id="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2.1.cmml" xref="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2.2.cmml" xref="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2.2">𝜃</ci><ci id="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2.3.cmml" xref="S4.SS2.p5.1.m1.1.1.1.1.1.2.2.2.3">𝑐</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">\mathrm{P}\left({p_{c}^{i}}\mid{x}^{i};{\theta_{c}}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">roman_P ( italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ∣ italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ; italic_θ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math> denotes the probability distribution of the event given the sensor sample <math alttext="{x}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.1"><semantics id="S4.SS2.p5.2.m2.1a"><msup id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml"><mi id="S4.SS2.p5.2.m2.1.1.2" xref="S4.SS2.p5.2.m2.1.1.2.cmml">x</mi><mi id="S4.SS2.p5.2.m2.1.1.3" xref="S4.SS2.p5.2.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><apply id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.2.m2.1.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.p5.2.m2.1.1.2.cmml" xref="S4.SS2.p5.2.m2.1.1.2">𝑥</ci><ci id="S4.SS2.p5.2.m2.1.1.3.cmml" xref="S4.SS2.p5.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">{x}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, with both <math alttext="\alpha_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.1"><semantics id="S4.SS2.p5.3.m3.1a"><msubsup id="S4.SS2.p5.3.m3.1.1" xref="S4.SS2.p5.3.m3.1.1.cmml"><mi id="S4.SS2.p5.3.m3.1.1.2.2" xref="S4.SS2.p5.3.m3.1.1.2.2.cmml">α</mi><mi id="S4.SS2.p5.3.m3.1.1.2.3" xref="S4.SS2.p5.3.m3.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p5.3.m3.1.1.3" xref="S4.SS2.p5.3.m3.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.1b"><apply id="S4.SS2.p5.3.m3.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.1.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1">superscript</csymbol><apply id="S4.SS2.p5.3.m3.1.1.2.cmml" xref="S4.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.1.1.2.1.cmml" xref="S4.SS2.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p5.3.m3.1.1.2.2.cmml" xref="S4.SS2.p5.3.m3.1.1.2.2">𝛼</ci><ci id="S4.SS2.p5.3.m3.1.1.2.3.cmml" xref="S4.SS2.p5.3.m3.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p5.3.m3.1.1.3.cmml" xref="S4.SS2.p5.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.1c">\alpha_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\beta_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.1"><semantics id="S4.SS2.p5.4.m4.1a"><msubsup id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml"><mi id="S4.SS2.p5.4.m4.1.1.2.2" xref="S4.SS2.p5.4.m4.1.1.2.2.cmml">β</mi><mi id="S4.SS2.p5.4.m4.1.1.2.3" xref="S4.SS2.p5.4.m4.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p5.4.m4.1.1.3" xref="S4.SS2.p5.4.m4.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.1b"><apply id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1">superscript</csymbol><apply id="S4.SS2.p5.4.m4.1.1.2.cmml" xref="S4.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.2.1.cmml" xref="S4.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p5.4.m4.1.1.2.2.cmml" xref="S4.SS2.p5.4.m4.1.1.2.2">𝛽</ci><ci id="S4.SS2.p5.4.m4.1.1.2.3.cmml" xref="S4.SS2.p5.4.m4.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p5.4.m4.1.1.3.cmml" xref="S4.SS2.p5.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.1c">\beta_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.1d">italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> being greater than zero. <math alttext="B(\alpha_{c}^{i},\beta_{c}^{i})=\Gamma(\alpha_{c}^{i})\Gamma(\beta_{c}^{i})/% \Gamma(\alpha_{c}^{i}+\beta_{c}^{i})" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m5.5"><semantics id="S4.SS2.p5.5.m5.5a"><mrow id="S4.SS2.p5.5.m5.5.5" xref="S4.SS2.p5.5.m5.5.5.cmml"><mrow id="S4.SS2.p5.5.m5.2.2.2" xref="S4.SS2.p5.5.m5.2.2.2.cmml"><mi 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xref="S4.SS2.p5.5.m5.5.5.5.3.1"><plus id="S4.SS2.p5.5.m5.5.5.5.3.1.1.1.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.1"></plus><apply id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.1.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2">superscript</csymbol><apply id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.1.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2">subscript</csymbol><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.2.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.2">𝛼</ci><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.3.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.2.3">𝑐</ci></apply><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.3.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.2.3">𝑖</ci></apply><apply id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.1.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3">superscript</csymbol><apply id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.1.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3">subscript</csymbol><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.2.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.2">𝛽</ci><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.3.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.2.3">𝑐</ci></apply><ci id="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.3.cmml" xref="S4.SS2.p5.5.m5.5.5.5.3.1.1.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m5.5c">B(\alpha_{c}^{i},\beta_{c}^{i})=\Gamma(\alpha_{c}^{i})\Gamma(\beta_{c}^{i})/% \Gamma(\alpha_{c}^{i}+\beta_{c}^{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m5.5d">italic_B ( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) = roman_Γ ( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) roman_Γ ( italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) / roman_Γ ( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math> is the Beta function, <math alttext="\Gamma(\cdot)" class="ltx_Math" display="inline" id="S4.SS2.p5.6.m6.1"><semantics id="S4.SS2.p5.6.m6.1a"><mrow id="S4.SS2.p5.6.m6.1.2" xref="S4.SS2.p5.6.m6.1.2.cmml"><mi id="S4.SS2.p5.6.m6.1.2.2" mathvariant="normal" xref="S4.SS2.p5.6.m6.1.2.2.cmml">Γ</mi><mo id="S4.SS2.p5.6.m6.1.2.1" xref="S4.SS2.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.6.m6.1.2.3.2" xref="S4.SS2.p5.6.m6.1.2.cmml"><mo id="S4.SS2.p5.6.m6.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.6.m6.1.2.cmml">(</mo><mo id="S4.SS2.p5.6.m6.1.1" lspace="0em" rspace="0em" xref="S4.SS2.p5.6.m6.1.1.cmml">⋅</mo><mo id="S4.SS2.p5.6.m6.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.6.m6.1b"><apply id="S4.SS2.p5.6.m6.1.2.cmml" xref="S4.SS2.p5.6.m6.1.2"><times id="S4.SS2.p5.6.m6.1.2.1.cmml" xref="S4.SS2.p5.6.m6.1.2.1"></times><ci id="S4.SS2.p5.6.m6.1.2.2.cmml" xref="S4.SS2.p5.6.m6.1.2.2">Γ</ci><ci id="S4.SS2.p5.6.m6.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.6.m6.1c">\Gamma(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.6.m6.1d">roman_Γ ( ⋅ )</annotation></semantics></math> is the gamma function, and <math alttext="p_{c}^{i}\neq 0" class="ltx_Math" display="inline" id="S4.SS2.p5.7.m7.1"><semantics id="S4.SS2.p5.7.m7.1a"><mrow id="S4.SS2.p5.7.m7.1.1" xref="S4.SS2.p5.7.m7.1.1.cmml"><msubsup id="S4.SS2.p5.7.m7.1.1.2" xref="S4.SS2.p5.7.m7.1.1.2.cmml"><mi id="S4.SS2.p5.7.m7.1.1.2.2.2" xref="S4.SS2.p5.7.m7.1.1.2.2.2.cmml">p</mi><mi id="S4.SS2.p5.7.m7.1.1.2.2.3" xref="S4.SS2.p5.7.m7.1.1.2.2.3.cmml">c</mi><mi id="S4.SS2.p5.7.m7.1.1.2.3" xref="S4.SS2.p5.7.m7.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p5.7.m7.1.1.1" xref="S4.SS2.p5.7.m7.1.1.1.cmml">≠</mo><mn id="S4.SS2.p5.7.m7.1.1.3" xref="S4.SS2.p5.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.7.m7.1b"><apply id="S4.SS2.p5.7.m7.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1"><neq id="S4.SS2.p5.7.m7.1.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1.1"></neq><apply id="S4.SS2.p5.7.m7.1.1.2.cmml" xref="S4.SS2.p5.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.7.m7.1.1.2.1.cmml" xref="S4.SS2.p5.7.m7.1.1.2">superscript</csymbol><apply id="S4.SS2.p5.7.m7.1.1.2.2.cmml" xref="S4.SS2.p5.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.7.m7.1.1.2.2.1.cmml" xref="S4.SS2.p5.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.p5.7.m7.1.1.2.2.2.cmml" xref="S4.SS2.p5.7.m7.1.1.2.2.2">𝑝</ci><ci id="S4.SS2.p5.7.m7.1.1.2.2.3.cmml" xref="S4.SS2.p5.7.m7.1.1.2.2.3">𝑐</ci></apply><ci id="S4.SS2.p5.7.m7.1.1.2.3.cmml" xref="S4.SS2.p5.7.m7.1.1.2.3">𝑖</ci></apply><cn id="S4.SS2.p5.7.m7.1.1.3.cmml" type="integer" xref="S4.SS2.p5.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.7.m7.1c">p_{c}^{i}\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.7.m7.1d">italic_p start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ≠ 0</annotation></semantics></math>. Applying the mapping rule in Eq. <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.E2" title="2 ‣ IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>, the belief mass and uncertainty <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.p5.8.m8.1"><semantics id="S4.SS2.p5.8.m8.1a"><mi id="S4.SS2.p5.8.m8.1.1" xref="S4.SS2.p5.8.m8.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.8.m8.1b"><ci id="S4.SS2.p5.8.m8.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.8.m8.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.8.m8.1d">italic_u</annotation></semantics></math> for each sample <math alttext="i" class="ltx_Math" display="inline" id="S4.SS2.p5.9.m9.1"><semantics id="S4.SS2.p5.9.m9.1a"><mi id="S4.SS2.p5.9.m9.1.1" xref="S4.SS2.p5.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.9.m9.1b"><ci id="S4.SS2.p5.9.m9.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.9.m9.1d">italic_i</annotation></semantics></math> are derived via a NN:</p> </div> <div class="ltx_para" id="S4.SS2.p6"> <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{b}_{1}^{i}=\frac{\alpha_{c}^{i}-1}{\alpha_{c}^{i}+\beta_{c}^{i}},\hskip 5.0pt% {b}_{2}^{i}=\frac{\beta_{c}^{i}-1}{\alpha_{c}^{i}+\beta_{c}^{i}}" class="ltx_Math" display="block" id="S4.E5.m1.2"><semantics id="S4.E5.m1.2a"><mrow id="S4.E5.m1.2.2.2" xref="S4.E5.m1.2.2.3.cmml"><mrow id="S4.E5.m1.1.1.1.1" 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end_POSTSUPERSCRIPT end_ARG , italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = divide start_ARG italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - 1 end_ARG start_ARG italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT + italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.2">where <math alttext="{b}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p7.1.m1.1"><semantics id="S4.SS2.p7.1.m1.1a"><msub id="S4.SS2.p7.1.m1.1.1" xref="S4.SS2.p7.1.m1.1.1.cmml"><mi id="S4.SS2.p7.1.m1.1.1.2" xref="S4.SS2.p7.1.m1.1.1.2.cmml">b</mi><mn id="S4.SS2.p7.1.m1.1.1.3" xref="S4.SS2.p7.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.1.m1.1b"><apply id="S4.SS2.p7.1.m1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p7.1.m1.1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p7.1.m1.1.1.2.cmml" xref="S4.SS2.p7.1.m1.1.1.2">𝑏</ci><cn id="S4.SS2.p7.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.1.m1.1c">{b}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.1.m1.1d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> represents the belief mass of a positive prediction while <math alttext="{b}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p7.2.m2.1"><semantics id="S4.SS2.p7.2.m2.1a"><msub id="S4.SS2.p7.2.m2.1.1" xref="S4.SS2.p7.2.m2.1.1.cmml"><mi id="S4.SS2.p7.2.m2.1.1.2" xref="S4.SS2.p7.2.m2.1.1.2.cmml">b</mi><mn id="S4.SS2.p7.2.m2.1.1.3" xref="S4.SS2.p7.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.2.m2.1b"><apply id="S4.SS2.p7.2.m2.1.1.cmml" xref="S4.SS2.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p7.2.m2.1.1.1.cmml" xref="S4.SS2.p7.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p7.2.m2.1.1.2.cmml" xref="S4.SS2.p7.2.m2.1.1.2">𝑏</ci><cn id="S4.SS2.p7.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p7.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.2.m2.1c">{b}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.2.m2.1d">italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> denotes that of a negative prediction.</p> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.12"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S4.SS2.p8.12.1">One-versus-all classifiers</span>. To obtain the parameters of <math alttext="\alpha_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p8.1.m1.1"><semantics id="S4.SS2.p8.1.m1.1a"><msubsup id="S4.SS2.p8.1.m1.1.1" xref="S4.SS2.p8.1.m1.1.1.cmml"><mi id="S4.SS2.p8.1.m1.1.1.2.2" xref="S4.SS2.p8.1.m1.1.1.2.2.cmml">α</mi><mi id="S4.SS2.p8.1.m1.1.1.2.3" xref="S4.SS2.p8.1.m1.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p8.1.m1.1.1.3" xref="S4.SS2.p8.1.m1.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.1.m1.1b"><apply id="S4.SS2.p8.1.m1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.1.m1.1.1.1.cmml" xref="S4.SS2.p8.1.m1.1.1">superscript</csymbol><apply id="S4.SS2.p8.1.m1.1.1.2.cmml" xref="S4.SS2.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.1.m1.1.1.2.1.cmml" xref="S4.SS2.p8.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p8.1.m1.1.1.2.2.cmml" xref="S4.SS2.p8.1.m1.1.1.2.2">𝛼</ci><ci id="S4.SS2.p8.1.m1.1.1.2.3.cmml" xref="S4.SS2.p8.1.m1.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p8.1.m1.1.1.3.cmml" xref="S4.SS2.p8.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.1.m1.1c">\alpha_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\beta_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p8.2.m2.1"><semantics id="S4.SS2.p8.2.m2.1a"><msubsup id="S4.SS2.p8.2.m2.1.1" xref="S4.SS2.p8.2.m2.1.1.cmml"><mi id="S4.SS2.p8.2.m2.1.1.2.2" xref="S4.SS2.p8.2.m2.1.1.2.2.cmml">β</mi><mi id="S4.SS2.p8.2.m2.1.1.2.3" xref="S4.SS2.p8.2.m2.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p8.2.m2.1.1.3" xref="S4.SS2.p8.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.2.m2.1b"><apply id="S4.SS2.p8.2.m2.1.1.cmml" xref="S4.SS2.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.2.m2.1.1.1.cmml" xref="S4.SS2.p8.2.m2.1.1">superscript</csymbol><apply id="S4.SS2.p8.2.m2.1.1.2.cmml" xref="S4.SS2.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.2.m2.1.1.2.1.cmml" xref="S4.SS2.p8.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p8.2.m2.1.1.2.2.cmml" xref="S4.SS2.p8.2.m2.1.1.2.2">𝛽</ci><ci id="S4.SS2.p8.2.m2.1.1.2.3.cmml" xref="S4.SS2.p8.2.m2.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p8.2.m2.1.1.3.cmml" xref="S4.SS2.p8.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.2.m2.1c">\beta_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.2.m2.1d">italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> in EDL across multiple events, we adopt the one-versus-all (OVA) classifier, where each classifier distinguishes a specific event from all others, leading to <math alttext="C" class="ltx_Math" display="inline" id="S4.SS2.p8.3.m3.1"><semantics id="S4.SS2.p8.3.m3.1a"><mi id="S4.SS2.p8.3.m3.1.1" xref="S4.SS2.p8.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.3.m3.1b"><ci id="S4.SS2.p8.3.m3.1.1.cmml" xref="S4.SS2.p8.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.3.m3.1d">italic_C</annotation></semantics></math> binary classifiers (i.e., heads). Specifically, in multi-event WED, we split the entire training dataset into <math alttext="C" class="ltx_Math" display="inline" id="S4.SS2.p8.4.m4.1"><semantics id="S4.SS2.p8.4.m4.1a"><mi id="S4.SS2.p8.4.m4.1.1" xref="S4.SS2.p8.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.4.m4.1b"><ci id="S4.SS2.p8.4.m4.1.1.cmml" xref="S4.SS2.p8.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.4.m4.1d">italic_C</annotation></semantics></math> independent datasets with binary labels (i.e., event <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p8.5.m5.1"><semantics id="S4.SS2.p8.5.m5.1a"><mi id="S4.SS2.p8.5.m5.1.1" xref="S4.SS2.p8.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.5.m5.1b"><ci id="S4.SS2.p8.5.m5.1.1.cmml" xref="S4.SS2.p8.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.5.m5.1d">italic_c</annotation></semantics></math> vs. non-event <math alttext="c" class="ltx_Math" display="inline" id="S4.SS2.p8.6.m6.1"><semantics id="S4.SS2.p8.6.m6.1a"><mi id="S4.SS2.p8.6.m6.1.1" xref="S4.SS2.p8.6.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.6.m6.1b"><ci id="S4.SS2.p8.6.m6.1.1.cmml" xref="S4.SS2.p8.6.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.6.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.6.m6.1d">italic_c</annotation></semantics></math> for <math alttext="c\in[1,C]" class="ltx_Math" display="inline" id="S4.SS2.p8.7.m7.2"><semantics id="S4.SS2.p8.7.m7.2a"><mrow id="S4.SS2.p8.7.m7.2.3" xref="S4.SS2.p8.7.m7.2.3.cmml"><mi id="S4.SS2.p8.7.m7.2.3.2" xref="S4.SS2.p8.7.m7.2.3.2.cmml">c</mi><mo id="S4.SS2.p8.7.m7.2.3.1" xref="S4.SS2.p8.7.m7.2.3.1.cmml">∈</mo><mrow id="S4.SS2.p8.7.m7.2.3.3.2" xref="S4.SS2.p8.7.m7.2.3.3.1.cmml"><mo id="S4.SS2.p8.7.m7.2.3.3.2.1" stretchy="false" xref="S4.SS2.p8.7.m7.2.3.3.1.cmml">[</mo><mn id="S4.SS2.p8.7.m7.1.1" xref="S4.SS2.p8.7.m7.1.1.cmml">1</mn><mo id="S4.SS2.p8.7.m7.2.3.3.2.2" xref="S4.SS2.p8.7.m7.2.3.3.1.cmml">,</mo><mi id="S4.SS2.p8.7.m7.2.2" xref="S4.SS2.p8.7.m7.2.2.cmml">C</mi><mo id="S4.SS2.p8.7.m7.2.3.3.2.3" stretchy="false" xref="S4.SS2.p8.7.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.7.m7.2b"><apply id="S4.SS2.p8.7.m7.2.3.cmml" xref="S4.SS2.p8.7.m7.2.3"><in id="S4.SS2.p8.7.m7.2.3.1.cmml" xref="S4.SS2.p8.7.m7.2.3.1"></in><ci id="S4.SS2.p8.7.m7.2.3.2.cmml" xref="S4.SS2.p8.7.m7.2.3.2">𝑐</ci><interval closure="closed" id="S4.SS2.p8.7.m7.2.3.3.1.cmml" xref="S4.SS2.p8.7.m7.2.3.3.2"><cn id="S4.SS2.p8.7.m7.1.1.cmml" type="integer" xref="S4.SS2.p8.7.m7.1.1">1</cn><ci id="S4.SS2.p8.7.m7.2.2.cmml" xref="S4.SS2.p8.7.m7.2.2">𝐶</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.7.m7.2c">c\in[1,C]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.7.m7.2d">italic_c ∈ [ 1 , italic_C ]</annotation></semantics></math>). For each event, we then develop a model to learn a set of mapping functions <math alttext="h_{c}({x}^{i};\theta_{c})" class="ltx_Math" display="inline" id="S4.SS2.p8.8.m8.2"><semantics id="S4.SS2.p8.8.m8.2a"><mrow id="S4.SS2.p8.8.m8.2.2" xref="S4.SS2.p8.8.m8.2.2.cmml"><msub id="S4.SS2.p8.8.m8.2.2.4" xref="S4.SS2.p8.8.m8.2.2.4.cmml"><mi id="S4.SS2.p8.8.m8.2.2.4.2" xref="S4.SS2.p8.8.m8.2.2.4.2.cmml">h</mi><mi id="S4.SS2.p8.8.m8.2.2.4.3" xref="S4.SS2.p8.8.m8.2.2.4.3.cmml">c</mi></msub><mo id="S4.SS2.p8.8.m8.2.2.3" xref="S4.SS2.p8.8.m8.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.p8.8.m8.2.2.2.2" xref="S4.SS2.p8.8.m8.2.2.2.3.cmml"><mo id="S4.SS2.p8.8.m8.2.2.2.2.3" stretchy="false" xref="S4.SS2.p8.8.m8.2.2.2.3.cmml">(</mo><msup id="S4.SS2.p8.8.m8.1.1.1.1.1" xref="S4.SS2.p8.8.m8.1.1.1.1.1.cmml"><mi id="S4.SS2.p8.8.m8.1.1.1.1.1.2" xref="S4.SS2.p8.8.m8.1.1.1.1.1.2.cmml">x</mi><mi id="S4.SS2.p8.8.m8.1.1.1.1.1.3" xref="S4.SS2.p8.8.m8.1.1.1.1.1.3.cmml">i</mi></msup><mo id="S4.SS2.p8.8.m8.2.2.2.2.4" xref="S4.SS2.p8.8.m8.2.2.2.3.cmml">;</mo><msub id="S4.SS2.p8.8.m8.2.2.2.2.2" xref="S4.SS2.p8.8.m8.2.2.2.2.2.cmml"><mi id="S4.SS2.p8.8.m8.2.2.2.2.2.2" xref="S4.SS2.p8.8.m8.2.2.2.2.2.2.cmml">θ</mi><mi id="S4.SS2.p8.8.m8.2.2.2.2.2.3" xref="S4.SS2.p8.8.m8.2.2.2.2.2.3.cmml">c</mi></msub><mo id="S4.SS2.p8.8.m8.2.2.2.2.5" stretchy="false" xref="S4.SS2.p8.8.m8.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.8.m8.2b"><apply id="S4.SS2.p8.8.m8.2.2.cmml" xref="S4.SS2.p8.8.m8.2.2"><times id="S4.SS2.p8.8.m8.2.2.3.cmml" xref="S4.SS2.p8.8.m8.2.2.3"></times><apply id="S4.SS2.p8.8.m8.2.2.4.cmml" xref="S4.SS2.p8.8.m8.2.2.4"><csymbol cd="ambiguous" id="S4.SS2.p8.8.m8.2.2.4.1.cmml" xref="S4.SS2.p8.8.m8.2.2.4">subscript</csymbol><ci id="S4.SS2.p8.8.m8.2.2.4.2.cmml" xref="S4.SS2.p8.8.m8.2.2.4.2">ℎ</ci><ci id="S4.SS2.p8.8.m8.2.2.4.3.cmml" xref="S4.SS2.p8.8.m8.2.2.4.3">𝑐</ci></apply><list id="S4.SS2.p8.8.m8.2.2.2.3.cmml" xref="S4.SS2.p8.8.m8.2.2.2.2"><apply id="S4.SS2.p8.8.m8.1.1.1.1.1.cmml" xref="S4.SS2.p8.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS2.p8.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p8.8.m8.1.1.1.1.1.2.cmml" xref="S4.SS2.p8.8.m8.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS2.p8.8.m8.1.1.1.1.1.3.cmml" xref="S4.SS2.p8.8.m8.1.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS2.p8.8.m8.2.2.2.2.2.cmml" xref="S4.SS2.p8.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p8.8.m8.2.2.2.2.2.1.cmml" xref="S4.SS2.p8.8.m8.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p8.8.m8.2.2.2.2.2.2.cmml" xref="S4.SS2.p8.8.m8.2.2.2.2.2.2">𝜃</ci><ci id="S4.SS2.p8.8.m8.2.2.2.2.2.3.cmml" xref="S4.SS2.p8.8.m8.2.2.2.2.2.3">𝑐</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.8.m8.2c">h_{c}({x}^{i};\theta_{c})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.8.m8.2d">italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ; italic_θ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math>, where <math alttext="{x}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p8.9.m9.1"><semantics id="S4.SS2.p8.9.m9.1a"><msup id="S4.SS2.p8.9.m9.1.1" xref="S4.SS2.p8.9.m9.1.1.cmml"><mi id="S4.SS2.p8.9.m9.1.1.2" xref="S4.SS2.p8.9.m9.1.1.2.cmml">x</mi><mi id="S4.SS2.p8.9.m9.1.1.3" xref="S4.SS2.p8.9.m9.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.9.m9.1b"><apply id="S4.SS2.p8.9.m9.1.1.cmml" xref="S4.SS2.p8.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.9.m9.1.1.1.cmml" xref="S4.SS2.p8.9.m9.1.1">superscript</csymbol><ci id="S4.SS2.p8.9.m9.1.1.2.cmml" xref="S4.SS2.p8.9.m9.1.1.2">𝑥</ci><ci id="S4.SS2.p8.9.m9.1.1.3.cmml" xref="S4.SS2.p8.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.9.m9.1c">{x}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.9.m9.1d">italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> represents the input signal, and <math alttext="\theta_{c}" class="ltx_Math" display="inline" id="S4.SS2.p8.10.m10.1"><semantics id="S4.SS2.p8.10.m10.1a"><msub id="S4.SS2.p8.10.m10.1.1" xref="S4.SS2.p8.10.m10.1.1.cmml"><mi id="S4.SS2.p8.10.m10.1.1.2" xref="S4.SS2.p8.10.m10.1.1.2.cmml">θ</mi><mi id="S4.SS2.p8.10.m10.1.1.3" xref="S4.SS2.p8.10.m10.1.1.3.cmml">c</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.10.m10.1b"><apply id="S4.SS2.p8.10.m10.1.1.cmml" xref="S4.SS2.p8.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.10.m10.1.1.1.cmml" xref="S4.SS2.p8.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.p8.10.m10.1.1.2.cmml" xref="S4.SS2.p8.10.m10.1.1.2">𝜃</ci><ci id="S4.SS2.p8.10.m10.1.1.3.cmml" xref="S4.SS2.p8.10.m10.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.10.m10.1c">\theta_{c}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.10.m10.1d">italic_θ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math> are the model weights. The outputs of the mapping functions yield the parameters <math alttext="\alpha_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p8.11.m11.1"><semantics id="S4.SS2.p8.11.m11.1a"><msubsup id="S4.SS2.p8.11.m11.1.1" xref="S4.SS2.p8.11.m11.1.1.cmml"><mi id="S4.SS2.p8.11.m11.1.1.2.2" xref="S4.SS2.p8.11.m11.1.1.2.2.cmml">α</mi><mi id="S4.SS2.p8.11.m11.1.1.2.3" xref="S4.SS2.p8.11.m11.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p8.11.m11.1.1.3" xref="S4.SS2.p8.11.m11.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.11.m11.1b"><apply id="S4.SS2.p8.11.m11.1.1.cmml" xref="S4.SS2.p8.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.11.m11.1.1.1.cmml" xref="S4.SS2.p8.11.m11.1.1">superscript</csymbol><apply id="S4.SS2.p8.11.m11.1.1.2.cmml" xref="S4.SS2.p8.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.11.m11.1.1.2.1.cmml" xref="S4.SS2.p8.11.m11.1.1">subscript</csymbol><ci id="S4.SS2.p8.11.m11.1.1.2.2.cmml" xref="S4.SS2.p8.11.m11.1.1.2.2">𝛼</ci><ci id="S4.SS2.p8.11.m11.1.1.2.3.cmml" xref="S4.SS2.p8.11.m11.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p8.11.m11.1.1.3.cmml" xref="S4.SS2.p8.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.11.m11.1c">\alpha_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.11.m11.1d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\beta_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p8.12.m12.1"><semantics id="S4.SS2.p8.12.m12.1a"><msubsup id="S4.SS2.p8.12.m12.1.1" xref="S4.SS2.p8.12.m12.1.1.cmml"><mi id="S4.SS2.p8.12.m12.1.1.2.2" xref="S4.SS2.p8.12.m12.1.1.2.2.cmml">β</mi><mi id="S4.SS2.p8.12.m12.1.1.2.3" xref="S4.SS2.p8.12.m12.1.1.2.3.cmml">c</mi><mi id="S4.SS2.p8.12.m12.1.1.3" xref="S4.SS2.p8.12.m12.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p8.12.m12.1b"><apply id="S4.SS2.p8.12.m12.1.1.cmml" xref="S4.SS2.p8.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.12.m12.1.1.1.cmml" xref="S4.SS2.p8.12.m12.1.1">superscript</csymbol><apply id="S4.SS2.p8.12.m12.1.1.2.cmml" xref="S4.SS2.p8.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p8.12.m12.1.1.2.1.cmml" xref="S4.SS2.p8.12.m12.1.1">subscript</csymbol><ci id="S4.SS2.p8.12.m12.1.1.2.2.cmml" xref="S4.SS2.p8.12.m12.1.1.2.2">𝛽</ci><ci id="S4.SS2.p8.12.m12.1.1.2.3.cmml" xref="S4.SS2.p8.12.m12.1.1.2.3">𝑐</ci></apply><ci id="S4.SS2.p8.12.m12.1.1.3.cmml" xref="S4.SS2.p8.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p8.12.m12.1c">\beta_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p8.12.m12.1d">italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> in the Beta distribution, computed as:</p> </div> <div class="ltx_para" id="S4.SS2.p9"> <table 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encoding="application/x-llamapun" id="S4.E7.m1.4d">italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_h start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ; italic_θ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p10"> <p class="ltx_p" id="S4.SS2.p10.2">From this, we can deduce binomial decisions, with <math alttext="b_{1}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p10.1.m1.1"><semantics id="S4.SS2.p10.1.m1.1a"><msubsup id="S4.SS2.p10.1.m1.1.1" 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id="S4.SS2.p10.1.m1.1c">b_{1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p10.1.m1.1d">italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> denoting a positive prediction (i.e., event happening), and <math alttext="b_{2}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p10.2.m2.1"><semantics id="S4.SS2.p10.2.m2.1a"><msubsup id="S4.SS2.p10.2.m2.1.1" xref="S4.SS2.p10.2.m2.1.1.cmml"><mi id="S4.SS2.p10.2.m2.1.1.2.2" xref="S4.SS2.p10.2.m2.1.1.2.2.cmml">b</mi><mn id="S4.SS2.p10.2.m2.1.1.2.3" xref="S4.SS2.p10.2.m2.1.1.2.3.cmml">2</mn><mi id="S4.SS2.p10.2.m2.1.1.3" xref="S4.SS2.p10.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p10.2.m2.1b"><apply id="S4.SS2.p10.2.m2.1.1.cmml" xref="S4.SS2.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p10.2.m2.1.1.1.cmml" xref="S4.SS2.p10.2.m2.1.1">superscript</csymbol><apply id="S4.SS2.p10.2.m2.1.1.2.cmml" 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Subsequently, these mapping functions are optimized jointly through an OVA training <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib27" title="">27</a>]</cite>. With this joint training of a shared EDL model, there is no need to deploy separate models on MCUs, thereby significantly reducing memory costs.</p> </div> <div class="ltx_para" id="S4.SS2.p11"> <p class="ltx_p" id="S4.SS2.p11.1">In contrast to traditional softmax-based deep learning approaches, which force the Neural Networks (NNs) to predict a point estimation, we can replace the softmax layer of the neural network with a ReLU layer (or an exponential function but softplus is not available in the MCU library). This adjustment ensures that the outputs remain non-negative, aligning with the positive <math alttext="{\alpha}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p11.1.m1.1"><semantics id="S4.SS2.p11.1.m1.1a"><msup id="S4.SS2.p11.1.m1.1.1" xref="S4.SS2.p11.1.m1.1.1.cmml"><mi id="S4.SS2.p11.1.m1.1.1.2" xref="S4.SS2.p11.1.m1.1.1.2.cmml">α</mi><mi id="S4.SS2.p11.1.m1.1.1.3" xref="S4.SS2.p11.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p11.1.m1.1b"><apply id="S4.SS2.p11.1.m1.1.1.cmml" xref="S4.SS2.p11.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p11.1.m1.1.1.1.cmml" xref="S4.SS2.p11.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.p11.1.m1.1.1.2.cmml" xref="S4.SS2.p11.1.m1.1.1.2">𝛼</ci><ci id="S4.SS2.p11.1.m1.1.1.3.cmml" xref="S4.SS2.p11.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p11.1.m1.1c">{\alpha}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p11.1.m1.1d">italic_α start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and enabling the NNs to predict distributions for each event task.</p> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS3.5.1.1">IV-C</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS3.6.2">Uncertainty-aware training and optimization</span> </h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">Focusing on the training and optimization of the EDL framework for the proposed multi-event WED, we draw inspiration from Eq. <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.E3" title="3 ‣ IV-A Evidential Deep Learning ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">3</span></a> and propose using the binary cross entropy and Beta loss for each binary classifier of event <math alttext="c" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mi id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><ci id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_c</annotation></semantics></math> as: </p> </div> <div class="ltx_para" id="S4.SS3.p2"> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\min_{\theta}\mathcal{L}=\frac{1}{N}\sum_{i}^{N}BCE\left(\psi_{c}^{i}/S_{c}^{i% },y_{c}^{i}\right)-\lambda\cdot H\left(B\left(\psi_{c}^{i}\right)\right)" 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xref="S4.E8.m1.3.3.3.3.3.1">⋅</ci><ci id="S4.E8.m1.3.3.3.3.3.2.cmml" xref="S4.E8.m1.3.3.3.3.3.2">𝜆</ci><ci id="S4.E8.m1.3.3.3.3.3.3.cmml" xref="S4.E8.m1.3.3.3.3.3.3">𝐻</ci></apply><apply id="S4.E8.m1.3.3.3.3.1.1.1.cmml" xref="S4.E8.m1.3.3.3.3.1.1"><times id="S4.E8.m1.3.3.3.3.1.1.1.2.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.2"></times><ci id="S4.E8.m1.3.3.3.3.1.1.1.3.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.3">𝐵</ci><apply id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1">superscript</csymbol><apply id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.1.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1">subscript</csymbol><ci id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.2.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.2">𝜓</ci><ci id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.3.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.2.3">𝑐</ci></apply><ci id="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E8.m1.3.3.3.3.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.3c">\min_{\theta}\mathcal{L}=\frac{1}{N}\sum_{i}^{N}BCE\left(\psi_{c}^{i}/S_{c}^{i% },y_{c}^{i}\right)-\lambda\cdot H\left(B\left(\psi_{c}^{i}\right)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.3d">roman_min start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT caligraphic_L = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_B italic_C italic_E ( italic_ψ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_y start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) - italic_λ ⋅ italic_H ( italic_B ( italic_ψ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.8">where <math alttext="\psi_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><msubsup id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml"><mi id="S4.SS3.p3.1.m1.1.1.2.2" xref="S4.SS3.p3.1.m1.1.1.2.2.cmml">ψ</mi><mi id="S4.SS3.p3.1.m1.1.1.2.3" xref="S4.SS3.p3.1.m1.1.1.2.3.cmml">c</mi><mi id="S4.SS3.p3.1.m1.1.1.3" xref="S4.SS3.p3.1.m1.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><apply id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.1.m1.1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">superscript</csymbol><apply id="S4.SS3.p3.1.m1.1.1.2.cmml" xref="S4.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.1.m1.1.1.2.1.cmml" xref="S4.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p3.1.m1.1.1.2.2.cmml" xref="S4.SS3.p3.1.m1.1.1.2.2">𝜓</ci><ci id="S4.SS3.p3.1.m1.1.1.2.3.cmml" xref="S4.SS3.p3.1.m1.1.1.2.3">𝑐</ci></apply><ci id="S4.SS3.p3.1.m1.1.1.3.cmml" xref="S4.SS3.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.1c">\psi_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.1d">italic_ψ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> symbolizes the Beta distribution parameters <math alttext="(\alpha_{c}^{i},\beta_{c}^{i})" class="ltx_Math" display="inline" id="S4.SS3.p3.2.m2.2"><semantics id="S4.SS3.p3.2.m2.2a"><mrow id="S4.SS3.p3.2.m2.2.2.2" xref="S4.SS3.p3.2.m2.2.2.3.cmml"><mo id="S4.SS3.p3.2.m2.2.2.2.3" stretchy="false" xref="S4.SS3.p3.2.m2.2.2.3.cmml">(</mo><msubsup id="S4.SS3.p3.2.m2.1.1.1.1" xref="S4.SS3.p3.2.m2.1.1.1.1.cmml"><mi id="S4.SS3.p3.2.m2.1.1.1.1.2.2" xref="S4.SS3.p3.2.m2.1.1.1.1.2.2.cmml">α</mi><mi id="S4.SS3.p3.2.m2.1.1.1.1.2.3" xref="S4.SS3.p3.2.m2.1.1.1.1.2.3.cmml">c</mi><mi id="S4.SS3.p3.2.m2.1.1.1.1.3" xref="S4.SS3.p3.2.m2.1.1.1.1.3.cmml">i</mi></msubsup><mo id="S4.SS3.p3.2.m2.2.2.2.4" xref="S4.SS3.p3.2.m2.2.2.3.cmml">,</mo><msubsup id="S4.SS3.p3.2.m2.2.2.2.2" xref="S4.SS3.p3.2.m2.2.2.2.2.cmml"><mi id="S4.SS3.p3.2.m2.2.2.2.2.2.2" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2.cmml">β</mi><mi id="S4.SS3.p3.2.m2.2.2.2.2.2.3" xref="S4.SS3.p3.2.m2.2.2.2.2.2.3.cmml">c</mi><mi id="S4.SS3.p3.2.m2.2.2.2.2.3" xref="S4.SS3.p3.2.m2.2.2.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p3.2.m2.2.2.2.5" stretchy="false" xref="S4.SS3.p3.2.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.2.m2.2b"><interval closure="open" id="S4.SS3.p3.2.m2.2.2.3.cmml" xref="S4.SS3.p3.2.m2.2.2.2"><apply id="S4.SS3.p3.2.m2.1.1.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.2.m2.1.1.1.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1">superscript</csymbol><apply id="S4.SS3.p3.2.m2.1.1.1.1.2.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.2.m2.1.1.1.1.2.1.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.SS3.p3.2.m2.1.1.1.1.2.2.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1.2.2">𝛼</ci><ci id="S4.SS3.p3.2.m2.1.1.1.1.2.3.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1.2.3">𝑐</ci></apply><ci id="S4.SS3.p3.2.m2.1.1.1.1.3.cmml" xref="S4.SS3.p3.2.m2.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS3.p3.2.m2.2.2.2.2.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p3.2.m2.2.2.2.2.1.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2">superscript</csymbol><apply id="S4.SS3.p3.2.m2.2.2.2.2.2.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p3.2.m2.2.2.2.2.2.1.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2">subscript</csymbol><ci id="S4.SS3.p3.2.m2.2.2.2.2.2.2.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.2">𝛽</ci><ci id="S4.SS3.p3.2.m2.2.2.2.2.2.3.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.2.3">𝑐</ci></apply><ci id="S4.SS3.p3.2.m2.2.2.2.2.3.cmml" xref="S4.SS3.p3.2.m2.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.2.m2.2c">(\alpha_{c}^{i},\beta_{c}^{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.2.m2.2d">( italic_α start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_β start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math>, <span class="ltx_text ltx_markedasmath" id="S4.SS3.p3.8.1">BCE</span> is the binary cross-entropy loss, <math alttext="H" class="ltx_Math" display="inline" id="S4.SS3.p3.4.m4.1"><semantics id="S4.SS3.p3.4.m4.1a"><mi id="S4.SS3.p3.4.m4.1.1" xref="S4.SS3.p3.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.4.m4.1b"><ci id="S4.SS3.p3.4.m4.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.4.m4.1d">italic_H</annotation></semantics></math> represents the entropy of a Beta distribution <math alttext="B" class="ltx_Math" display="inline" id="S4.SS3.p3.5.m5.1"><semantics id="S4.SS3.p3.5.m5.1a"><mi id="S4.SS3.p3.5.m5.1.1" xref="S4.SS3.p3.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.5.m5.1b"><ci id="S4.SS3.p3.5.m5.1.1.cmml" xref="S4.SS3.p3.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.5.m5.1d">italic_B</annotation></semantics></math> parameterized by <math alttext="\psi_{c}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p3.6.m6.1"><semantics id="S4.SS3.p3.6.m6.1a"><msubsup id="S4.SS3.p3.6.m6.1.1" xref="S4.SS3.p3.6.m6.1.1.cmml"><mi id="S4.SS3.p3.6.m6.1.1.2.2" xref="S4.SS3.p3.6.m6.1.1.2.2.cmml">ψ</mi><mi id="S4.SS3.p3.6.m6.1.1.2.3" xref="S4.SS3.p3.6.m6.1.1.2.3.cmml">c</mi><mi id="S4.SS3.p3.6.m6.1.1.3" xref="S4.SS3.p3.6.m6.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.6.m6.1b"><apply id="S4.SS3.p3.6.m6.1.1.cmml" xref="S4.SS3.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.6.m6.1.1.1.cmml" xref="S4.SS3.p3.6.m6.1.1">superscript</csymbol><apply id="S4.SS3.p3.6.m6.1.1.2.cmml" xref="S4.SS3.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p3.6.m6.1.1.2.1.cmml" xref="S4.SS3.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS3.p3.6.m6.1.1.2.2.cmml" xref="S4.SS3.p3.6.m6.1.1.2.2">𝜓</ci><ci id="S4.SS3.p3.6.m6.1.1.2.3.cmml" xref="S4.SS3.p3.6.m6.1.1.2.3">𝑐</ci></apply><ci id="S4.SS3.p3.6.m6.1.1.3.cmml" xref="S4.SS3.p3.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.6.m6.1c">\psi_{c}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.6.m6.1d">italic_ψ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.SS3.p3.7.m7.1"><semantics id="S4.SS3.p3.7.m7.1a"><mi id="S4.SS3.p3.7.m7.1.1" xref="S4.SS3.p3.7.m7.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.7.m7.1b"><ci id="S4.SS3.p3.7.m7.1.1.cmml" xref="S4.SS3.p3.7.m7.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.7.m7.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.7.m7.1d">italic_λ</annotation></semantics></math> serves as a balancing weight between the cross-entropy loss and entropy of the Beta loss. For all <math alttext="C" class="ltx_Math" display="inline" id="S4.SS3.p3.8.m8.1"><semantics id="S4.SS3.p3.8.m8.1a"><mi id="S4.SS3.p3.8.m8.1.1" xref="S4.SS3.p3.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.8.m8.1b"><ci id="S4.SS3.p3.8.m8.1.1.cmml" xref="S4.SS3.p3.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.8.m8.1d">italic_C</annotation></semantics></math> events, we collectively optimize all binary classifiers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib28" title="">28</a>]</cite>, enabling the model to perform inference with just a single forward pass.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Cascade learning</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">This section discusses designing efficient neural networks for UR2M. We explore the benefits of the early-exit strategy and architecture search method for single-event sharing on MCUs, reducing computational and memory costs. We also examine multiple-event sharing and detail the training pipeline using cascade learning, with all search and training <span class="ltx_text ltx_font_italic" id="S5.p1.1.1">on the server</span>.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS1.5.1.1">V-A</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS1.6.2">Single-event Sharing</span> </h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.1">For many DL tasks, some input samples, referred to as “easy” samples, can be effectively classified using shallower layers of the representation. This indicates that these shallower representations can identify “easy” samples, thus avoiding extra computation, whereas more “difficult” samples require processing through deeper layers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib29" title="">29</a>]</cite>. However, unlike edge GPUs, designing model sharing on MCUs is challenging given the limited computing power, memory, and library support.</p> </div> <div class="ltx_para" id="S5.SS1.p2"> <p class="ltx_p" id="S5.SS1.p2.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.SS1.p2.1.1">Using Early-exits to Share Shallower Layers.</span> We propose a nested architecture featuring three early exits (sub-networks), which include shallow, medium, and deep models designed for single-event (intra-event) sharing, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2.F2" title="Figure 2 ‣ II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a> for MCUs. Each sub-network is designed using identical blocks of neural network layers, inspired by efficient neural networks for edge devices <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib30" title="">30</a>]</cite>. Existing early-exit methods usually rely on accuracy as a criterion to prune model branches. However, uncertainty can act as a crucial indicator for reliable prediction: we propose using uncertainty as a metric to determine whether to exit at each sub-network. As demonstrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2.F2" title="Figure 2 ‣ II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>, uncertainty thresholds are applied at the output of both shallow and medium models to facilitate early exits for data with low uncertainty (i.e., reliable predictions), thereby saving on MCU overheads.</p> </div> <div class="ltx_para" id="S5.SS1.p3"> <p class="ltx_p" id="S5.SS1.p3.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.SS1.p3.1.1">Uncertainty-aware Architecture Search<span class="ltx_text ltx_font_upright" id="S5.SS1.p3.1.1.1">.</span></span> To find efficient neural networks that minimize MCU overhead, recent studies have shown that the number of operations (OPS) and channel sizes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib14" title="">14</a>]</cite> are two crucial factors. Considering this, we propose an effective yet straightforward architecture search method to identify optimal neural networks for the early-exit models (i.e., shallow, medium, and deep models) in single-event sharing.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm1"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm1.23"> <div class="ltx_listingline" id="algorithm1.4.4"> <span class="ltx_text" id="algorithm1.4.4.1" style="color:#000000;"><span class="ltx_text ltx_font_bold" id="algorithm1.4.4.1.1">Input:</span> </span>Channel <math alttext="L" class="ltx_Math" display="inline" id="algorithm1.1.1.m1.1"><semantics id="algorithm1.1.1.m1.1a"><mi id="algorithm1.1.1.m1.1.1" xref="algorithm1.1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="algorithm1.1.1.m1.1b"><ci id="algorithm1.1.1.m1.1.1.cmml" xref="algorithm1.1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="algorithm1.1.1.m1.1d">italic_L</annotation></semantics></math>, OPS size <math alttext="O" class="ltx_Math" display="inline" id="algorithm1.2.2.m2.1"><semantics id="algorithm1.2.2.m2.1a"><mi id="algorithm1.2.2.m2.1.1" xref="algorithm1.2.2.m2.1.1.cmml">O</mi><annotation-xml encoding="MathML-Content" id="algorithm1.2.2.m2.1b"><ci id="algorithm1.2.2.m2.1.1.cmml" xref="algorithm1.2.2.m2.1.1">𝑂</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.2.2.m2.1c">O</annotation><annotation encoding="application/x-llamapun" id="algorithm1.2.2.m2.1d">italic_O</annotation></semantics></math>, <math alttext="\mathcal{D}^{TRAIN}" class="ltx_Math" display="inline" id="algorithm1.3.3.m3.1"><semantics id="algorithm1.3.3.m3.1a"><msup id="algorithm1.3.3.m3.1.1" xref="algorithm1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.3.3.m3.1.1.2" xref="algorithm1.3.3.m3.1.1.2.cmml">𝒟</mi><mrow id="algorithm1.3.3.m3.1.1.3" xref="algorithm1.3.3.m3.1.1.3.cmml"><mi id="algorithm1.3.3.m3.1.1.3.2" xref="algorithm1.3.3.m3.1.1.3.2.cmml">T</mi><mo id="algorithm1.3.3.m3.1.1.3.1" xref="algorithm1.3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.3.3.m3.1.1.3.3" xref="algorithm1.3.3.m3.1.1.3.3.cmml">R</mi><mo id="algorithm1.3.3.m3.1.1.3.1a" xref="algorithm1.3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.3.3.m3.1.1.3.4" xref="algorithm1.3.3.m3.1.1.3.4.cmml">A</mi><mo id="algorithm1.3.3.m3.1.1.3.1b" xref="algorithm1.3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.3.3.m3.1.1.3.5" xref="algorithm1.3.3.m3.1.1.3.5.cmml">I</mi><mo id="algorithm1.3.3.m3.1.1.3.1c" xref="algorithm1.3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.3.3.m3.1.1.3.6" xref="algorithm1.3.3.m3.1.1.3.6.cmml">N</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="algorithm1.3.3.m3.1b"><apply id="algorithm1.3.3.m3.1.1.cmml" xref="algorithm1.3.3.m3.1.1"><csymbol cd="ambiguous" id="algorithm1.3.3.m3.1.1.1.cmml" xref="algorithm1.3.3.m3.1.1">superscript</csymbol><ci id="algorithm1.3.3.m3.1.1.2.cmml" xref="algorithm1.3.3.m3.1.1.2">𝒟</ci><apply id="algorithm1.3.3.m3.1.1.3.cmml" xref="algorithm1.3.3.m3.1.1.3"><times id="algorithm1.3.3.m3.1.1.3.1.cmml" xref="algorithm1.3.3.m3.1.1.3.1"></times><ci id="algorithm1.3.3.m3.1.1.3.2.cmml" xref="algorithm1.3.3.m3.1.1.3.2">𝑇</ci><ci id="algorithm1.3.3.m3.1.1.3.3.cmml" xref="algorithm1.3.3.m3.1.1.3.3">𝑅</ci><ci id="algorithm1.3.3.m3.1.1.3.4.cmml" xref="algorithm1.3.3.m3.1.1.3.4">𝐴</ci><ci id="algorithm1.3.3.m3.1.1.3.5.cmml" xref="algorithm1.3.3.m3.1.1.3.5">𝐼</ci><ci id="algorithm1.3.3.m3.1.1.3.6.cmml" xref="algorithm1.3.3.m3.1.1.3.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.3.3.m3.1c">\mathcal{D}^{TRAIN}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.3.3.m3.1d">caligraphic_D start_POSTSUPERSCRIPT italic_T italic_R italic_A italic_I italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\mathcal{D}^{TEST}" class="ltx_Math" display="inline" id="algorithm1.4.4.m4.1"><semantics id="algorithm1.4.4.m4.1a"><msup id="algorithm1.4.4.m4.1.1" xref="algorithm1.4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.4.4.m4.1.1.2" xref="algorithm1.4.4.m4.1.1.2.cmml">𝒟</mi><mrow id="algorithm1.4.4.m4.1.1.3" xref="algorithm1.4.4.m4.1.1.3.cmml"><mi id="algorithm1.4.4.m4.1.1.3.2" xref="algorithm1.4.4.m4.1.1.3.2.cmml">T</mi><mo id="algorithm1.4.4.m4.1.1.3.1" xref="algorithm1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.4.4.m4.1.1.3.3" xref="algorithm1.4.4.m4.1.1.3.3.cmml">E</mi><mo id="algorithm1.4.4.m4.1.1.3.1a" xref="algorithm1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.4.4.m4.1.1.3.4" xref="algorithm1.4.4.m4.1.1.3.4.cmml">S</mi><mo id="algorithm1.4.4.m4.1.1.3.1b" xref="algorithm1.4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.4.4.m4.1.1.3.5" xref="algorithm1.4.4.m4.1.1.3.5.cmml">T</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="algorithm1.4.4.m4.1b"><apply id="algorithm1.4.4.m4.1.1.cmml" xref="algorithm1.4.4.m4.1.1"><csymbol cd="ambiguous" id="algorithm1.4.4.m4.1.1.1.cmml" xref="algorithm1.4.4.m4.1.1">superscript</csymbol><ci id="algorithm1.4.4.m4.1.1.2.cmml" xref="algorithm1.4.4.m4.1.1.2">𝒟</ci><apply id="algorithm1.4.4.m4.1.1.3.cmml" xref="algorithm1.4.4.m4.1.1.3"><times id="algorithm1.4.4.m4.1.1.3.1.cmml" xref="algorithm1.4.4.m4.1.1.3.1"></times><ci id="algorithm1.4.4.m4.1.1.3.2.cmml" xref="algorithm1.4.4.m4.1.1.3.2">𝑇</ci><ci id="algorithm1.4.4.m4.1.1.3.3.cmml" xref="algorithm1.4.4.m4.1.1.3.3">𝐸</ci><ci id="algorithm1.4.4.m4.1.1.3.4.cmml" xref="algorithm1.4.4.m4.1.1.3.4">𝑆</ci><ci id="algorithm1.4.4.m4.1.1.3.5.cmml" xref="algorithm1.4.4.m4.1.1.3.5">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.4.4.m4.1c">\mathcal{D}^{TEST}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.4.4.m4.1d">caligraphic_D start_POSTSUPERSCRIPT italic_T italic_E italic_S italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.6.6"> <span class="ltx_text" id="algorithm1.6.6.1" style="color:#000000;"><span class="ltx_text ltx_font_bold" id="algorithm1.6.6.1.1">Output:</span> </span>Event prediction <math alttext="y" class="ltx_Math" display="inline" id="algorithm1.5.5.m1.1"><semantics id="algorithm1.5.5.m1.1a"><mi id="algorithm1.5.5.m1.1.1" xref="algorithm1.5.5.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="algorithm1.5.5.m1.1b"><ci id="algorithm1.5.5.m1.1.1.cmml" xref="algorithm1.5.5.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.5.5.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="algorithm1.5.5.m1.1d">italic_y</annotation></semantics></math> and uncertainty <math alttext="u" class="ltx_Math" display="inline" id="algorithm1.6.6.m2.1"><semantics id="algorithm1.6.6.m2.1a"><mi id="algorithm1.6.6.m2.1.1" xref="algorithm1.6.6.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="algorithm1.6.6.m2.1b"><ci id="algorithm1.6.6.m2.1.1.cmml" xref="algorithm1.6.6.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.6.6.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="algorithm1.6.6.m2.1d">italic_u</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.7.7"> <span class="ltx_text" id="algorithm1.7.7.1" style="color:#000000;"><span class="ltx_text ltx_font_bold" id="algorithm1.7.7.1.1">Data:</span> </span>Training data <math alttext="\mathcal{D}^{TRAIN}" class="ltx_Math" display="inline" id="algorithm1.7.7.m1.1"><semantics id="algorithm1.7.7.m1.1a"><msup id="algorithm1.7.7.m1.1.1" xref="algorithm1.7.7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.7.7.m1.1.1.2" xref="algorithm1.7.7.m1.1.1.2.cmml">𝒟</mi><mrow id="algorithm1.7.7.m1.1.1.3" xref="algorithm1.7.7.m1.1.1.3.cmml"><mi id="algorithm1.7.7.m1.1.1.3.2" xref="algorithm1.7.7.m1.1.1.3.2.cmml">T</mi><mo id="algorithm1.7.7.m1.1.1.3.1" xref="algorithm1.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.7.7.m1.1.1.3.3" xref="algorithm1.7.7.m1.1.1.3.3.cmml">R</mi><mo id="algorithm1.7.7.m1.1.1.3.1a" xref="algorithm1.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.7.7.m1.1.1.3.4" xref="algorithm1.7.7.m1.1.1.3.4.cmml">A</mi><mo id="algorithm1.7.7.m1.1.1.3.1b" xref="algorithm1.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.7.7.m1.1.1.3.5" xref="algorithm1.7.7.m1.1.1.3.5.cmml">I</mi><mo id="algorithm1.7.7.m1.1.1.3.1c" xref="algorithm1.7.7.m1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.7.7.m1.1.1.3.6" xref="algorithm1.7.7.m1.1.1.3.6.cmml">N</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="algorithm1.7.7.m1.1b"><apply id="algorithm1.7.7.m1.1.1.cmml" xref="algorithm1.7.7.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.7.7.m1.1.1.1.cmml" xref="algorithm1.7.7.m1.1.1">superscript</csymbol><ci id="algorithm1.7.7.m1.1.1.2.cmml" xref="algorithm1.7.7.m1.1.1.2">𝒟</ci><apply id="algorithm1.7.7.m1.1.1.3.cmml" xref="algorithm1.7.7.m1.1.1.3"><times id="algorithm1.7.7.m1.1.1.3.1.cmml" xref="algorithm1.7.7.m1.1.1.3.1"></times><ci id="algorithm1.7.7.m1.1.1.3.2.cmml" xref="algorithm1.7.7.m1.1.1.3.2">𝑇</ci><ci id="algorithm1.7.7.m1.1.1.3.3.cmml" xref="algorithm1.7.7.m1.1.1.3.3">𝑅</ci><ci id="algorithm1.7.7.m1.1.1.3.4.cmml" xref="algorithm1.7.7.m1.1.1.3.4">𝐴</ci><ci id="algorithm1.7.7.m1.1.1.3.5.cmml" xref="algorithm1.7.7.m1.1.1.3.5">𝐼</ci><ci id="algorithm1.7.7.m1.1.1.3.6.cmml" xref="algorithm1.7.7.m1.1.1.3.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.7.7.m1.1c">\mathcal{D}^{TRAIN}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.7.7.m1.1d">caligraphic_D start_POSTSUPERSCRIPT italic_T italic_R italic_A italic_I italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.23.24"> <span class="ltx_text ltx_font_typewriter" id="algorithm1.23.24.1" style="font-size:80%;color:#0000FF;">/* </span><span class="ltx_text ltx_font_typewriter" id="algorithm1.23.24.2" style="font-size:80%;color:#0000FF;">search single-event model */</span> </div> <div class="ltx_listingline" id="algorithm1.8.8"> <span class="ltx_tag ltx_tag_listingline">1</span> best_backbone, best_score = False, 0 <span class="ltx_text ltx_font_bold" id="algorithm1.8.8.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.8.8.1"><math alttext="i" class="ltx_Math" display="inline" id="algorithm1.8.8.1.m1.1"><semantics id="algorithm1.8.8.1.m1.1a"><mi id="algorithm1.8.8.1.m1.1.1" xref="algorithm1.8.8.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="algorithm1.8.8.1.m1.1b"><ci id="algorithm1.8.8.1.m1.1.1.cmml" xref="algorithm1.8.8.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.8.8.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="algorithm1.8.8.1.m1.1d">italic_i</annotation></semantics></math> in L</em> <span class="ltx_text ltx_font_bold" id="algorithm1.8.8.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.9.9"> <span class="ltx_tag ltx_tag_listingline">2</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.9.9.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.9.9.1"><math alttext="j" class="ltx_Math" display="inline" id="algorithm1.9.9.1.m1.1"><semantics id="algorithm1.9.9.1.m1.1a"><mi id="algorithm1.9.9.1.m1.1.1" xref="algorithm1.9.9.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="algorithm1.9.9.1.m1.1b"><ci id="algorithm1.9.9.1.m1.1.1.cmml" xref="algorithm1.9.9.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.9.9.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="algorithm1.9.9.1.m1.1d">italic_j</annotation></semantics></math> in O</em> <span class="ltx_text ltx_font_bold" id="algorithm1.9.9.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.10.10"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_typewriter" id="algorithm1.10.10.2" style="font-size:80%;color:#0000FF;">// </span><span class="ltx_text ltx_font_typewriter" id="algorithm1.10.10.1" style="font-size:80%;color:#0000FF;">Train candidate NNs backbone (<math alttext="\mathbf{b}_{ij}" class="ltx_Math" display="inline" id="algorithm1.10.10.1.m1.1"><semantics id="algorithm1.10.10.1.m1.1a"><msub id="algorithm1.10.10.1.m1.1.1" xref="algorithm1.10.10.1.m1.1.1.cmml"><mi id="algorithm1.10.10.1.m1.1.1.2" mathcolor="#0000FF" xref="algorithm1.10.10.1.m1.1.1.2.cmml">𝐛</mi><mrow id="algorithm1.10.10.1.m1.1.1.3" xref="algorithm1.10.10.1.m1.1.1.3.cmml"><mi id="algorithm1.10.10.1.m1.1.1.3.2" mathcolor="#0000FF" xref="algorithm1.10.10.1.m1.1.1.3.2.cmml">i</mi><mo id="algorithm1.10.10.1.m1.1.1.3.1" mathcolor="#0000FF" mathvariant="monospace" xref="algorithm1.10.10.1.m1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.10.10.1.m1.1.1.3.3" mathcolor="#0000FF" xref="algorithm1.10.10.1.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="algorithm1.10.10.1.m1.1b"><apply id="algorithm1.10.10.1.m1.1.1.cmml" xref="algorithm1.10.10.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.10.10.1.m1.1.1.1.cmml" xref="algorithm1.10.10.1.m1.1.1">subscript</csymbol><ci id="algorithm1.10.10.1.m1.1.1.2.cmml" xref="algorithm1.10.10.1.m1.1.1.2">𝐛</ci><apply id="algorithm1.10.10.1.m1.1.1.3.cmml" xref="algorithm1.10.10.1.m1.1.1.3"><times id="algorithm1.10.10.1.m1.1.1.3.1.cmml" xref="algorithm1.10.10.1.m1.1.1.3.1"></times><ci id="algorithm1.10.10.1.m1.1.1.3.2.cmml" xref="algorithm1.10.10.1.m1.1.1.3.2">𝑖</ci><ci id="algorithm1.10.10.1.m1.1.1.3.3.cmml" xref="algorithm1.10.10.1.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.10.10.1.m1.1c">\mathbf{b}_{ij}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.10.10.1.m1.1d">bold_b start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>) </span> </div> <div class="ltx_listingline" id="algorithm1.16.16"> <span class="ltx_tag ltx_tag_listingline">3</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> NN<math alttext="\leftarrow" class="ltx_Math" display="inline" id="algorithm1.11.11.m1.1"><semantics id="algorithm1.11.11.m1.1a"><mo id="algorithm1.11.11.m1.1.1" stretchy="false" xref="algorithm1.11.11.m1.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="algorithm1.11.11.m1.1b"><ci id="algorithm1.11.11.m1.1.1.cmml" xref="algorithm1.11.11.m1.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.11.11.m1.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm1.11.11.m1.1d">←</annotation></semantics></math><math alttext="\mathbf{b}_{ij}(\mathbf{W}_{ij},L_{i},O_{j})" class="ltx_Math" display="inline" id="algorithm1.12.12.m2.3"><semantics id="algorithm1.12.12.m2.3a"><mrow id="algorithm1.12.12.m2.3.3" xref="algorithm1.12.12.m2.3.3.cmml"><msub id="algorithm1.12.12.m2.3.3.5" xref="algorithm1.12.12.m2.3.3.5.cmml"><mi id="algorithm1.12.12.m2.3.3.5.2" xref="algorithm1.12.12.m2.3.3.5.2.cmml">𝐛</mi><mrow id="algorithm1.12.12.m2.3.3.5.3" xref="algorithm1.12.12.m2.3.3.5.3.cmml"><mi id="algorithm1.12.12.m2.3.3.5.3.2" xref="algorithm1.12.12.m2.3.3.5.3.2.cmml">i</mi><mo id="algorithm1.12.12.m2.3.3.5.3.1" xref="algorithm1.12.12.m2.3.3.5.3.1.cmml">⁢</mo><mi id="algorithm1.12.12.m2.3.3.5.3.3" xref="algorithm1.12.12.m2.3.3.5.3.3.cmml">j</mi></mrow></msub><mo id="algorithm1.12.12.m2.3.3.4" xref="algorithm1.12.12.m2.3.3.4.cmml">⁢</mo><mrow id="algorithm1.12.12.m2.3.3.3.3" xref="algorithm1.12.12.m2.3.3.3.4.cmml"><mo id="algorithm1.12.12.m2.3.3.3.3.4" stretchy="false" xref="algorithm1.12.12.m2.3.3.3.4.cmml">(</mo><msub id="algorithm1.12.12.m2.1.1.1.1.1" xref="algorithm1.12.12.m2.1.1.1.1.1.cmml"><mi id="algorithm1.12.12.m2.1.1.1.1.1.2" xref="algorithm1.12.12.m2.1.1.1.1.1.2.cmml">𝐖</mi><mrow id="algorithm1.12.12.m2.1.1.1.1.1.3" xref="algorithm1.12.12.m2.1.1.1.1.1.3.cmml"><mi id="algorithm1.12.12.m2.1.1.1.1.1.3.2" xref="algorithm1.12.12.m2.1.1.1.1.1.3.2.cmml">i</mi><mo id="algorithm1.12.12.m2.1.1.1.1.1.3.1" xref="algorithm1.12.12.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.12.12.m2.1.1.1.1.1.3.3" xref="algorithm1.12.12.m2.1.1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="algorithm1.12.12.m2.3.3.3.3.5" xref="algorithm1.12.12.m2.3.3.3.4.cmml">,</mo><msub id="algorithm1.12.12.m2.2.2.2.2.2" xref="algorithm1.12.12.m2.2.2.2.2.2.cmml"><mi id="algorithm1.12.12.m2.2.2.2.2.2.2" xref="algorithm1.12.12.m2.2.2.2.2.2.2.cmml">L</mi><mi id="algorithm1.12.12.m2.2.2.2.2.2.3" xref="algorithm1.12.12.m2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="algorithm1.12.12.m2.3.3.3.3.6" xref="algorithm1.12.12.m2.3.3.3.4.cmml">,</mo><msub id="algorithm1.12.12.m2.3.3.3.3.3" xref="algorithm1.12.12.m2.3.3.3.3.3.cmml"><mi id="algorithm1.12.12.m2.3.3.3.3.3.2" xref="algorithm1.12.12.m2.3.3.3.3.3.2.cmml">O</mi><mi id="algorithm1.12.12.m2.3.3.3.3.3.3" xref="algorithm1.12.12.m2.3.3.3.3.3.3.cmml">j</mi></msub><mo id="algorithm1.12.12.m2.3.3.3.3.7" stretchy="false" xref="algorithm1.12.12.m2.3.3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.12.12.m2.3b"><apply id="algorithm1.12.12.m2.3.3.cmml" xref="algorithm1.12.12.m2.3.3"><times id="algorithm1.12.12.m2.3.3.4.cmml" xref="algorithm1.12.12.m2.3.3.4"></times><apply id="algorithm1.12.12.m2.3.3.5.cmml" xref="algorithm1.12.12.m2.3.3.5"><csymbol cd="ambiguous" id="algorithm1.12.12.m2.3.3.5.1.cmml" xref="algorithm1.12.12.m2.3.3.5">subscript</csymbol><ci id="algorithm1.12.12.m2.3.3.5.2.cmml" xref="algorithm1.12.12.m2.3.3.5.2">𝐛</ci><apply id="algorithm1.12.12.m2.3.3.5.3.cmml" xref="algorithm1.12.12.m2.3.3.5.3"><times id="algorithm1.12.12.m2.3.3.5.3.1.cmml" xref="algorithm1.12.12.m2.3.3.5.3.1"></times><ci id="algorithm1.12.12.m2.3.3.5.3.2.cmml" xref="algorithm1.12.12.m2.3.3.5.3.2">𝑖</ci><ci id="algorithm1.12.12.m2.3.3.5.3.3.cmml" xref="algorithm1.12.12.m2.3.3.5.3.3">𝑗</ci></apply></apply><vector id="algorithm1.12.12.m2.3.3.3.4.cmml" xref="algorithm1.12.12.m2.3.3.3.3"><apply id="algorithm1.12.12.m2.1.1.1.1.1.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.12.12.m2.1.1.1.1.1.1.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1">subscript</csymbol><ci id="algorithm1.12.12.m2.1.1.1.1.1.2.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1.2">𝐖</ci><apply id="algorithm1.12.12.m2.1.1.1.1.1.3.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1.3"><times id="algorithm1.12.12.m2.1.1.1.1.1.3.1.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1.3.1"></times><ci id="algorithm1.12.12.m2.1.1.1.1.1.3.2.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1.3.2">𝑖</ci><ci id="algorithm1.12.12.m2.1.1.1.1.1.3.3.cmml" xref="algorithm1.12.12.m2.1.1.1.1.1.3.3">𝑗</ci></apply></apply><apply id="algorithm1.12.12.m2.2.2.2.2.2.cmml" xref="algorithm1.12.12.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm1.12.12.m2.2.2.2.2.2.1.cmml" xref="algorithm1.12.12.m2.2.2.2.2.2">subscript</csymbol><ci id="algorithm1.12.12.m2.2.2.2.2.2.2.cmml" xref="algorithm1.12.12.m2.2.2.2.2.2.2">𝐿</ci><ci id="algorithm1.12.12.m2.2.2.2.2.2.3.cmml" xref="algorithm1.12.12.m2.2.2.2.2.2.3">𝑖</ci></apply><apply id="algorithm1.12.12.m2.3.3.3.3.3.cmml" xref="algorithm1.12.12.m2.3.3.3.3.3"><csymbol cd="ambiguous" id="algorithm1.12.12.m2.3.3.3.3.3.1.cmml" xref="algorithm1.12.12.m2.3.3.3.3.3">subscript</csymbol><ci id="algorithm1.12.12.m2.3.3.3.3.3.2.cmml" xref="algorithm1.12.12.m2.3.3.3.3.3.2">𝑂</ci><ci id="algorithm1.12.12.m2.3.3.3.3.3.3.cmml" xref="algorithm1.12.12.m2.3.3.3.3.3.3">𝑗</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.12.12.m2.3c">\mathbf{b}_{ij}(\mathbf{W}_{ij},L_{i},O_{j})</annotation><annotation encoding="application/x-llamapun" id="algorithm1.12.12.m2.3d">bold_b start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( bold_W start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_O start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> accuracy <math alttext="\leftarrow" class="ltx_Math" display="inline" id="algorithm1.13.13.m3.1"><semantics id="algorithm1.13.13.m3.1a"><mo id="algorithm1.13.13.m3.1.1" stretchy="false" xref="algorithm1.13.13.m3.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="algorithm1.13.13.m3.1b"><ci id="algorithm1.13.13.m3.1.1.cmml" xref="algorithm1.13.13.m3.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.13.13.m3.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm1.13.13.m3.1d">←</annotation></semantics></math> NN(<math alttext="\mathcal{D}^{TRAIN}" class="ltx_Math" display="inline" id="algorithm1.14.14.m4.1"><semantics id="algorithm1.14.14.m4.1a"><msup id="algorithm1.14.14.m4.1.1" xref="algorithm1.14.14.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.14.14.m4.1.1.2" xref="algorithm1.14.14.m4.1.1.2.cmml">𝒟</mi><mrow id="algorithm1.14.14.m4.1.1.3" xref="algorithm1.14.14.m4.1.1.3.cmml"><mi id="algorithm1.14.14.m4.1.1.3.2" xref="algorithm1.14.14.m4.1.1.3.2.cmml">T</mi><mo id="algorithm1.14.14.m4.1.1.3.1" xref="algorithm1.14.14.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.14.14.m4.1.1.3.3" xref="algorithm1.14.14.m4.1.1.3.3.cmml">R</mi><mo id="algorithm1.14.14.m4.1.1.3.1a" xref="algorithm1.14.14.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.14.14.m4.1.1.3.4" xref="algorithm1.14.14.m4.1.1.3.4.cmml">A</mi><mo id="algorithm1.14.14.m4.1.1.3.1b" xref="algorithm1.14.14.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.14.14.m4.1.1.3.5" xref="algorithm1.14.14.m4.1.1.3.5.cmml">I</mi><mo id="algorithm1.14.14.m4.1.1.3.1c" xref="algorithm1.14.14.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.14.14.m4.1.1.3.6" xref="algorithm1.14.14.m4.1.1.3.6.cmml">N</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="algorithm1.14.14.m4.1b"><apply id="algorithm1.14.14.m4.1.1.cmml" xref="algorithm1.14.14.m4.1.1"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1">superscript</csymbol><ci id="algorithm1.14.14.m4.1.1.2.cmml" xref="algorithm1.14.14.m4.1.1.2">𝒟</ci><apply id="algorithm1.14.14.m4.1.1.3.cmml" xref="algorithm1.14.14.m4.1.1.3"><times id="algorithm1.14.14.m4.1.1.3.1.cmml" xref="algorithm1.14.14.m4.1.1.3.1"></times><ci id="algorithm1.14.14.m4.1.1.3.2.cmml" xref="algorithm1.14.14.m4.1.1.3.2">𝑇</ci><ci id="algorithm1.14.14.m4.1.1.3.3.cmml" xref="algorithm1.14.14.m4.1.1.3.3">𝑅</ci><ci id="algorithm1.14.14.m4.1.1.3.4.cmml" xref="algorithm1.14.14.m4.1.1.3.4">𝐴</ci><ci id="algorithm1.14.14.m4.1.1.3.5.cmml" xref="algorithm1.14.14.m4.1.1.3.5">𝐼</ci><ci id="algorithm1.14.14.m4.1.1.3.6.cmml" xref="algorithm1.14.14.m4.1.1.3.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.14.14.m4.1c">\mathcal{D}^{TRAIN}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.14.14.m4.1d">caligraphic_D start_POSTSUPERSCRIPT italic_T italic_R italic_A italic_I italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>) tradeoff <math alttext="\leftarrow" class="ltx_Math" display="inline" id="algorithm1.15.15.m5.1"><semantics id="algorithm1.15.15.m5.1a"><mo id="algorithm1.15.15.m5.1.1" stretchy="false" xref="algorithm1.15.15.m5.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="algorithm1.15.15.m5.1b"><ci id="algorithm1.15.15.m5.1.1.cmml" xref="algorithm1.15.15.m5.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.15.15.m5.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm1.15.15.m5.1d">←</annotation></semantics></math> accuracy/OPS <span class="ltx_text ltx_font_bold" id="algorithm1.16.16.2">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.16.16.1">tradeoff <math alttext="<" class="ltx_Math" display="inline" id="algorithm1.16.16.1.m1.1"><semantics id="algorithm1.16.16.1.m1.1a"><mo id="algorithm1.16.16.1.m1.1.1" mathvariant="normal" xref="algorithm1.16.16.1.m1.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="algorithm1.16.16.1.m1.1b"><lt id="algorithm1.16.16.1.m1.1.1.cmml" xref="algorithm1.16.16.1.m1.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.16.16.1.m1.1c"><</annotation><annotation encoding="application/x-llamapun" id="algorithm1.16.16.1.m1.1d"><</annotation></semantics></math> best_score</em> <span class="ltx_text ltx_font_bold" id="algorithm1.16.16.3">then</span> </div> <div class="ltx_listingline" id="algorithm1.23.25"> <span class="ltx_tag ltx_tag_listingline">4</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> best_NN, best_score = NN, tradeoff </div> <div class="ltx_listingline" id="algorithm1.23.26"> <span class="ltx_tag ltx_tag_listingline">5</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.23.26.1">return</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.23.26.2">best_NN</em> </div> <div class="ltx_listingline" id="algorithm1.23.27"> <span class="ltx_tag ltx_tag_listingline">6</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm1.23.28"> <span class="ltx_text ltx_font_typewriter" id="algorithm1.23.28.1" style="font-size:80%;color:#0000FF;">/* </span><span class="ltx_text ltx_font_typewriter" id="algorithm1.23.28.2" style="font-size:80%;color:#0000FF;">train with cascade learning */</span> </div> <div class="ltx_listingline" id="algorithm1.17.17"> <span class="ltx_tag ltx_tag_listingline">7</span> <span class="ltx_text ltx_font_bold" id="algorithm1.17.17.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.17.17.1"><math alttext="l=0,1,2" class="ltx_Math" display="inline" id="algorithm1.17.17.1.m1.3"><semantics id="algorithm1.17.17.1.m1.3a"><mrow id="algorithm1.17.17.1.m1.3.4" xref="algorithm1.17.17.1.m1.3.4.cmml"><mi id="algorithm1.17.17.1.m1.3.4.2" xref="algorithm1.17.17.1.m1.3.4.2.cmml">l</mi><mo id="algorithm1.17.17.1.m1.3.4.1" mathvariant="normal" xref="algorithm1.17.17.1.m1.3.4.1.cmml">=</mo><mrow id="algorithm1.17.17.1.m1.3.4.3.2" xref="algorithm1.17.17.1.m1.3.4.3.1.cmml"><mn id="algorithm1.17.17.1.m1.1.1" mathvariant="normal" xref="algorithm1.17.17.1.m1.1.1.cmml">0</mn><mo id="algorithm1.17.17.1.m1.3.4.3.2.1" mathvariant="normal" xref="algorithm1.17.17.1.m1.3.4.3.1.cmml">,</mo><mn id="algorithm1.17.17.1.m1.2.2" mathvariant="normal" xref="algorithm1.17.17.1.m1.2.2.cmml">1</mn><mo id="algorithm1.17.17.1.m1.3.4.3.2.2" mathvariant="normal" xref="algorithm1.17.17.1.m1.3.4.3.1.cmml">,</mo><mn id="algorithm1.17.17.1.m1.3.3" mathvariant="normal" xref="algorithm1.17.17.1.m1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.17.17.1.m1.3b"><apply id="algorithm1.17.17.1.m1.3.4.cmml" xref="algorithm1.17.17.1.m1.3.4"><eq id="algorithm1.17.17.1.m1.3.4.1.cmml" xref="algorithm1.17.17.1.m1.3.4.1"></eq><ci id="algorithm1.17.17.1.m1.3.4.2.cmml" xref="algorithm1.17.17.1.m1.3.4.2">𝑙</ci><list id="algorithm1.17.17.1.m1.3.4.3.1.cmml" xref="algorithm1.17.17.1.m1.3.4.3.2"><cn id="algorithm1.17.17.1.m1.1.1.cmml" type="integer" xref="algorithm1.17.17.1.m1.1.1">0</cn><cn id="algorithm1.17.17.1.m1.2.2.cmml" type="integer" xref="algorithm1.17.17.1.m1.2.2">1</cn><cn id="algorithm1.17.17.1.m1.3.3.cmml" type="integer" xref="algorithm1.17.17.1.m1.3.3">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.17.17.1.m1.3c">l=0,1,2</annotation><annotation encoding="application/x-llamapun" id="algorithm1.17.17.1.m1.3d">italic_l = 0 , 1 , 2</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm1.17.17.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.23.29"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_typewriter" id="algorithm1.23.29.1" style="font-size:80%;color:#0000FF;">// </span><span class="ltx_text ltx_font_typewriter" id="algorithm1.23.29.2" style="font-size:80%;color:#0000FF;">take each output as next exit’s input </span> </div> <div class="ltx_listingline" id="algorithm1.22.22"> <span class="ltx_tag ltx_tag_listingline">8</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="u" class="ltx_Math" display="inline" id="algorithm1.18.18.m1.1"><semantics id="algorithm1.18.18.m1.1a"><mi id="algorithm1.18.18.m1.1.1" xref="algorithm1.18.18.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="algorithm1.18.18.m1.1b"><ci id="algorithm1.18.18.m1.1.1.cmml" xref="algorithm1.18.18.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.18.18.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="algorithm1.18.18.m1.1d">italic_u</annotation></semantics></math>, output <math alttext="\leftarrow" class="ltx_Math" display="inline" id="algorithm1.19.19.m2.1"><semantics id="algorithm1.19.19.m2.1a"><mo id="algorithm1.19.19.m2.1.1" stretchy="false" xref="algorithm1.19.19.m2.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="algorithm1.19.19.m2.1b"><ci id="algorithm1.19.19.m2.1.1.cmml" xref="algorithm1.19.19.m2.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.19.19.m2.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm1.19.19.m2.1d">←</annotation></semantics></math><math alttext="(\mathbf{b}_{l}\;(\mathbf{W}_{l}),\mathcal{D}^{TRAIN}" class="ltx_math_unparsed" display="inline" id="algorithm1.20.20.m3.1"><semantics id="algorithm1.20.20.m3.1a"><mrow id="algorithm1.20.20.m3.1b"><mo id="algorithm1.20.20.m3.1.1" stretchy="false">(</mo><msub id="algorithm1.20.20.m3.1.2"><mi id="algorithm1.20.20.m3.1.2.2">𝐛</mi><mi id="algorithm1.20.20.m3.1.2.3">l</mi></msub><mrow id="algorithm1.20.20.m3.1.3"><mo id="algorithm1.20.20.m3.1.3.1" stretchy="false">(</mo><msub id="algorithm1.20.20.m3.1.3.2"><mi id="algorithm1.20.20.m3.1.3.2.2">𝐖</mi><mi id="algorithm1.20.20.m3.1.3.2.3">l</mi></msub><mo id="algorithm1.20.20.m3.1.3.3" stretchy="false">)</mo></mrow><mo id="algorithm1.20.20.m3.1.4">,</mo><msup id="algorithm1.20.20.m3.1.5"><mi class="ltx_font_mathcaligraphic" id="algorithm1.20.20.m3.1.5.2">𝒟</mi><mrow id="algorithm1.20.20.m3.1.5.3"><mi id="algorithm1.20.20.m3.1.5.3.2">T</mi><mo id="algorithm1.20.20.m3.1.5.3.1">⁢</mo><mi id="algorithm1.20.20.m3.1.5.3.3">R</mi><mo id="algorithm1.20.20.m3.1.5.3.1a">⁢</mo><mi id="algorithm1.20.20.m3.1.5.3.4">A</mi><mo id="algorithm1.20.20.m3.1.5.3.1b">⁢</mo><mi id="algorithm1.20.20.m3.1.5.3.5">I</mi><mo id="algorithm1.20.20.m3.1.5.3.1c">⁢</mo><mi id="algorithm1.20.20.m3.1.5.3.6">N</mi></mrow></msup></mrow><annotation encoding="application/x-tex" id="algorithm1.20.20.m3.1c">(\mathbf{b}_{l}\;(\mathbf{W}_{l}),\mathcal{D}^{TRAIN}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.20.20.m3.1d">( bold_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( bold_W start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) , caligraphic_D start_POSTSUPERSCRIPT italic_T italic_R italic_A italic_I italic_N end_POSTSUPERSCRIPT</annotation></semantics></math>) <math alttext="\mathcal{D}^{TRAIN}" class="ltx_Math" display="inline" id="algorithm1.21.21.m4.1"><semantics id="algorithm1.21.21.m4.1a"><msup id="algorithm1.21.21.m4.1.1" xref="algorithm1.21.21.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.21.21.m4.1.1.2" xref="algorithm1.21.21.m4.1.1.2.cmml">𝒟</mi><mrow id="algorithm1.21.21.m4.1.1.3" xref="algorithm1.21.21.m4.1.1.3.cmml"><mi id="algorithm1.21.21.m4.1.1.3.2" xref="algorithm1.21.21.m4.1.1.3.2.cmml">T</mi><mo id="algorithm1.21.21.m4.1.1.3.1" xref="algorithm1.21.21.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.21.21.m4.1.1.3.3" xref="algorithm1.21.21.m4.1.1.3.3.cmml">R</mi><mo id="algorithm1.21.21.m4.1.1.3.1a" xref="algorithm1.21.21.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.21.21.m4.1.1.3.4" xref="algorithm1.21.21.m4.1.1.3.4.cmml">A</mi><mo id="algorithm1.21.21.m4.1.1.3.1b" xref="algorithm1.21.21.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.21.21.m4.1.1.3.5" xref="algorithm1.21.21.m4.1.1.3.5.cmml">I</mi><mo id="algorithm1.21.21.m4.1.1.3.1c" xref="algorithm1.21.21.m4.1.1.3.1.cmml">⁢</mo><mi id="algorithm1.21.21.m4.1.1.3.6" xref="algorithm1.21.21.m4.1.1.3.6.cmml">N</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="algorithm1.21.21.m4.1b"><apply id="algorithm1.21.21.m4.1.1.cmml" xref="algorithm1.21.21.m4.1.1"><csymbol cd="ambiguous" id="algorithm1.21.21.m4.1.1.1.cmml" xref="algorithm1.21.21.m4.1.1">superscript</csymbol><ci id="algorithm1.21.21.m4.1.1.2.cmml" xref="algorithm1.21.21.m4.1.1.2">𝒟</ci><apply id="algorithm1.21.21.m4.1.1.3.cmml" xref="algorithm1.21.21.m4.1.1.3"><times id="algorithm1.21.21.m4.1.1.3.1.cmml" xref="algorithm1.21.21.m4.1.1.3.1"></times><ci id="algorithm1.21.21.m4.1.1.3.2.cmml" xref="algorithm1.21.21.m4.1.1.3.2">𝑇</ci><ci id="algorithm1.21.21.m4.1.1.3.3.cmml" xref="algorithm1.21.21.m4.1.1.3.3">𝑅</ci><ci id="algorithm1.21.21.m4.1.1.3.4.cmml" xref="algorithm1.21.21.m4.1.1.3.4">𝐴</ci><ci id="algorithm1.21.21.m4.1.1.3.5.cmml" xref="algorithm1.21.21.m4.1.1.3.5">𝐼</ci><ci id="algorithm1.21.21.m4.1.1.3.6.cmml" xref="algorithm1.21.21.m4.1.1.3.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.21.21.m4.1c">\mathcal{D}^{TRAIN}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.21.21.m4.1d">caligraphic_D start_POSTSUPERSCRIPT italic_T italic_R italic_A italic_I italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> <math alttext="\leftarrow" class="ltx_Math" display="inline" id="algorithm1.22.22.m5.1"><semantics id="algorithm1.22.22.m5.1a"><mo id="algorithm1.22.22.m5.1.1" stretchy="false" xref="algorithm1.22.22.m5.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="algorithm1.22.22.m5.1b"><ci id="algorithm1.22.22.m5.1.1.cmml" xref="algorithm1.22.22.m5.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.22.22.m5.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm1.22.22.m5.1d">←</annotation></semantics></math> output <span class="ltx_text ltx_font_bold" id="algorithm1.22.22.1">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.22.22.2">converge</em> <span class="ltx_text ltx_font_bold" id="algorithm1.22.22.3">then</span> </div> <div class="ltx_listingline" id="algorithm1.23.23"> <span class="ltx_tag ltx_tag_listingline">9</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.23.23.2">return</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.23.23.1"><math alttext="\mathbf{W}_{l}" class="ltx_Math" display="inline" id="algorithm1.23.23.1.m1.1"><semantics id="algorithm1.23.23.1.m1.1a"><msub id="algorithm1.23.23.1.m1.1.1" xref="algorithm1.23.23.1.m1.1.1.cmml"><mi id="algorithm1.23.23.1.m1.1.1.2" xref="algorithm1.23.23.1.m1.1.1.2.cmml">𝐖</mi><mi id="algorithm1.23.23.1.m1.1.1.3" xref="algorithm1.23.23.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.23.23.1.m1.1b"><apply id="algorithm1.23.23.1.m1.1.1.cmml" xref="algorithm1.23.23.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.23.23.1.m1.1.1.1.cmml" xref="algorithm1.23.23.1.m1.1.1">subscript</csymbol><ci id="algorithm1.23.23.1.m1.1.1.2.cmml" xref="algorithm1.23.23.1.m1.1.1.2">𝐖</ci><ci id="algorithm1.23.23.1.m1.1.1.3.cmml" xref="algorithm1.23.23.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.23.23.1.m1.1c">\mathbf{W}_{l}</annotation><annotation encoding="application/x-llamapun" id="algorithm1.23.23.1.m1.1d">bold_W start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></em> </div> <div class="ltx_listingline" id="algorithm1.23.30"> <span class="ltx_tag ltx_tag_listingline">10</span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm1.23.31"> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm1.25.1.1">Algorithm 1</span> </span>The Search and training of UR2M</figcaption> </figure> <div class="ltx_para" id="S5.SS1.p4"> <p class="ltx_p" id="S5.SS1.p4.3">Specifically, we employ the Depthwise block as the OPS to control model depth, as it serves as an ideal proxy for managing model latency on MCUs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib14" title="">14</a>]</cite>. The structure of each block consists of 1<math alttext="\times" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.1"><semantics id="S5.SS1.p4.1.m1.1a"><mo id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.1b"><times id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.1d">×</annotation></semantics></math>1 Convolutions, 3<math alttext="\times" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.1"><semantics id="S5.SS1.p4.2.m2.1a"><mo id="S5.SS1.p4.2.m2.1.1" xref="S5.SS1.p4.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.2.m2.1b"><times id="S5.SS1.p4.2.m2.1.1.cmml" xref="S5.SS1.p4.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.2.m2.1d">×</annotation></semantics></math>3 Depthwise Convolutions, and 1<math alttext="\times" class="ltx_Math" display="inline" id="S5.SS1.p4.3.m3.1"><semantics id="S5.SS1.p4.3.m3.1a"><mo id="S5.SS1.p4.3.m3.1.1" xref="S5.SS1.p4.3.m3.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.3.m3.1b"><times id="S5.SS1.p4.3.m3.1.1.cmml" xref="S5.SS1.p4.3.m3.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.3.m3.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.3.m3.1d">×</annotation></semantics></math>1 Convolutions. We design each block using a 2D convolutional layer to to effectively handle various input types and extract the initial features. Subsequently, we use a consistent padding strategy to control the depth of OPS, ensuring that the output of each block matches its input. Lastly, we incorporate a linear classifier in each block as the output layer for single-event detection.</p> </div> <div class="ltx_para" id="S5.SS1.p5"> <p class="ltx_p" id="S5.SS1.p5.4">To define the model search space for efficient architectures on edge devices, we configure channel sizes <math alttext="L" class="ltx_Math" display="inline" id="S5.SS1.p5.1.m1.1"><semantics id="S5.SS1.p5.1.m1.1a"><mi id="S5.SS1.p5.1.m1.1.1" xref="S5.SS1.p5.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.1.m1.1b"><ci id="S5.SS1.p5.1.m1.1.1.cmml" xref="S5.SS1.p5.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.1.m1.1d">italic_L</annotation></semantics></math> (ranging from 32 to 512) and OPS sizes <math alttext="O" class="ltx_Math" display="inline" id="S5.SS1.p5.2.m2.1"><semantics id="S5.SS1.p5.2.m2.1a"><mi id="S5.SS1.p5.2.m2.1.1" xref="S5.SS1.p5.2.m2.1.1.cmml">O</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.2.m2.1b"><ci id="S5.SS1.p5.2.m2.1.1.cmml" xref="S5.SS1.p5.2.m2.1.1">𝑂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.2.m2.1c">O</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.2.m2.1d">italic_O</annotation></semantics></math> (3 to 7), drawing from models like MobileNet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib30" title="">30</a>]</cite>, DSCNN <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib31" title="">31</a>]</cite> for mobile devices, and MicroNets <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib14" title="">14</a>]</cite> for MCUs. This leads to 60 potential configurations (<math alttext="N=L\times O" class="ltx_Math" display="inline" id="S5.SS1.p5.3.m3.1"><semantics id="S5.SS1.p5.3.m3.1a"><mrow id="S5.SS1.p5.3.m3.1.1" xref="S5.SS1.p5.3.m3.1.1.cmml"><mi id="S5.SS1.p5.3.m3.1.1.2" xref="S5.SS1.p5.3.m3.1.1.2.cmml">N</mi><mo id="S5.SS1.p5.3.m3.1.1.1" xref="S5.SS1.p5.3.m3.1.1.1.cmml">=</mo><mrow id="S5.SS1.p5.3.m3.1.1.3" xref="S5.SS1.p5.3.m3.1.1.3.cmml"><mi id="S5.SS1.p5.3.m3.1.1.3.2" xref="S5.SS1.p5.3.m3.1.1.3.2.cmml">L</mi><mo id="S5.SS1.p5.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.SS1.p5.3.m3.1.1.3.1.cmml">×</mo><mi id="S5.SS1.p5.3.m3.1.1.3.3" xref="S5.SS1.p5.3.m3.1.1.3.3.cmml">O</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.3.m3.1b"><apply id="S5.SS1.p5.3.m3.1.1.cmml" xref="S5.SS1.p5.3.m3.1.1"><eq id="S5.SS1.p5.3.m3.1.1.1.cmml" xref="S5.SS1.p5.3.m3.1.1.1"></eq><ci id="S5.SS1.p5.3.m3.1.1.2.cmml" xref="S5.SS1.p5.3.m3.1.1.2">𝑁</ci><apply id="S5.SS1.p5.3.m3.1.1.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3"><times id="S5.SS1.p5.3.m3.1.1.3.1.cmml" xref="S5.SS1.p5.3.m3.1.1.3.1"></times><ci id="S5.SS1.p5.3.m3.1.1.3.2.cmml" xref="S5.SS1.p5.3.m3.1.1.3.2">𝐿</ci><ci id="S5.SS1.p5.3.m3.1.1.3.3.cmml" xref="S5.SS1.p5.3.m3.1.1.3.3">𝑂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.3.m3.1c">N=L\times O</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.3.m3.1d">italic_N = italic_L × italic_O</annotation></semantics></math>), each comprising three sub-networks. Our objective is to identify the optimal configuration <math alttext="N^{*}" class="ltx_Math" display="inline" id="S5.SS1.p5.4.m4.1"><semantics id="S5.SS1.p5.4.m4.1a"><msup id="S5.SS1.p5.4.m4.1.1" xref="S5.SS1.p5.4.m4.1.1.cmml"><mi id="S5.SS1.p5.4.m4.1.1.2" xref="S5.SS1.p5.4.m4.1.1.2.cmml">N</mi><mo id="S5.SS1.p5.4.m4.1.1.3" xref="S5.SS1.p5.4.m4.1.1.3.cmml">*</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.p5.4.m4.1b"><apply id="S5.SS1.p5.4.m4.1.1.cmml" xref="S5.SS1.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S5.SS1.p5.4.m4.1.1.1.cmml" xref="S5.SS1.p5.4.m4.1.1">superscript</csymbol><ci id="S5.SS1.p5.4.m4.1.1.2.cmml" xref="S5.SS1.p5.4.m4.1.1.2">𝑁</ci><times id="S5.SS1.p5.4.m4.1.1.3.cmml" xref="S5.SS1.p5.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p5.4.m4.1c">N^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p5.4.m4.1d">italic_N start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT</annotation></semantics></math> that balances minimal OPS with maximal accuracy. As outlined in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#algorithm1" title="1 ‣ V-A Single-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a> (Lines 1-9), the search process involves initially setting a best backbone and score (Line 1), iterating through combinations of channel and OPS sizes (Lines 2-3), and assessing candidate NNs based on accuracy and operational space trade-offs (Lines 5-8), to ultimately select the most efficient and accurate NN backbone (Line 9).</p> </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S5.SS2.5.1.1">V-B</span> </span><span class="ltx_text ltx_font_italic" id="S5.SS2.6.2">Multiple-event Sharing</span> </h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.2">For a <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mi id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><ci id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_C</annotation></semantics></math> multi-event detection task, a common approach is to develop individual models to ensure reusability across different applications or use cases and to optimize efficiency for each model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib12" title="">12</a>]</cite>. These models can occupy <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.1"><semantics id="S5.SS2.p1.2.m2.1a"><mi id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.1b"><ci id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.1d">italic_C</annotation></semantics></math> times the MCU memory and computation cost compared to a single-event model. However, some singular events may share similar characteristics, which can be captured by an identical network for feature extraction. For example, EEG signals are often used to detect alpha waves (event 1) and beta waves (event 2) using two independent models, despite the fact that both waves describe brain activities and can share certain information.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.3"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.SS2.p2.3.1">Using Heads to Share Entire Backbone.</span> We propose our multi-event detection models, which share three sub-networks (i.e., shallow, medium, and deep) and consist of <math alttext="C*3" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><mrow id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml"><mi id="S5.SS2.p2.1.m1.1.1.2" xref="S5.SS2.p2.1.m1.1.1.2.cmml">C</mi><mo id="S5.SS2.p2.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.SS2.p2.1.m1.1.1.1.cmml">*</mo><mn id="S5.SS2.p2.1.m1.1.1.3" xref="S5.SS2.p2.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><apply id="S5.SS2.p2.1.m1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1"><times id="S5.SS2.p2.1.m1.1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1.1"></times><ci id="S5.SS2.p2.1.m1.1.1.2.cmml" xref="S5.SS2.p2.1.m1.1.1.2">𝐶</ci><cn id="S5.SS2.p2.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.p2.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">C*3</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">italic_C * 3</annotation></semantics></math> adaptive classifiers (cf. Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2.F2" title="Figure 2 ‣ II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>). Compared to multi-class classification, our multi-event sharing framework allows for more flexibility in single-event detection, which is especially preferred on low-power MCUs to ensure efficiency and reusability across multiple applications. Specifically, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S2.F2" title="Figure 2 ‣ II Related Works ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">2</span></a>, for each shared shallow, medium, and deep backbone NNs, we design <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.1"><semantics id="S5.SS2.p2.2.m2.1a"><mi id="S5.SS2.p2.2.m2.1.1" xref="S5.SS2.p2.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.1b"><ci id="S5.SS2.p2.2.m2.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.1d">italic_C</annotation></semantics></math> independent classifiers to distinguish the <math alttext="C" class="ltx_Math" display="inline" id="S5.SS2.p2.3.m3.1"><semantics id="S5.SS2.p2.3.m3.1a"><mi id="S5.SS2.p2.3.m3.1.1" xref="S5.SS2.p2.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.3.m3.1b"><ci id="S5.SS2.p2.3.m3.1.1.cmml" xref="S5.SS2.p2.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.3.m3.1d">italic_C</annotation></semantics></math> events. Each classifier is composed of an adaptive pooling layer and a linear layer. The adaptive pooling layer aims to adjust the different output sizes from the searched shallow, medium, and deep sub-networks to match the input size of the classifiers. We optimize all the classifiers in a multi-task learning paradigm.</p> </div> <div class="ltx_para" id="S5.SS2.p3"> <p class="ltx_p" id="S5.SS2.p3.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.SS2.p3.1.1">Uncertainty-aware Cascade Learning<span class="ltx_text ltx_font_upright" id="S5.SS2.p3.1.1.1">.</span></span> To train the aforementioned shallow, medium, and deep models for MCUs, we propose an uncertainty-aware cascade model inspired by deep cascade learning for training our early-exit models. As illustrated in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#algorithm1" title="1 ‣ V-A Single-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a> (Lines 10-14), we employ three optimizers for the three exits, with each exit representing one-third of the model layers. Initially, we train the first one-third of the layers in the searched backbone model and then utilize its output to train the second exit. Finally, we optimize the third exit.</p> </div> <figure class="ltx_figure" id="S5.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_flex_size_1 ltx_align_center" id="S5.F3.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="116" id="S5.F3.sf1.g1" src="x3.png" width="346"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(a) </span>Uncertainty operators using TFLM</figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_flex_size_1 ltx_align_center" id="S5.F3.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="99" id="S5.F3.sf2.g1" src="x4.png" width="622"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(b) </span>MCUs library optimization</figcaption> </figure> </div> <div class="ltx_flex_break"></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Deployment stage. (a) Uncertainty deployment on MCU based on multiple operators to calculate uncertainty and classification results. (b) MCU library space before optimization (top) and after optimization (bottom). </figcaption> </figure> <div class="ltx_para" id="S5.SS2.p4"> <p class="ltx_p" id="S5.SS2.p4.1">During each exit, we apply a single-layer linear layer (referred to as a head) for each event, which takes input maps of the output dimensions of the early-exits. Each early exit produces two outputs: the prediction and the uncertainty. We optimize all sub-networks concurrently on the server.</p> </div> <div class="ltx_para" id="S5.SS2.p5"> <p class="ltx_p" id="S5.SS2.p5.1">Our design is supported by the MCU libraries of Tensorflow Lite Micro (TFLM) in terms of multi-tenancy (e.g., enabling model deployment in a cascade manner) and memory planner (e.g., reusing the same operator’s memory). This coherence can significantly reduce the overheads compared to the conventional multi-event detection models. Overall, our approach aims to optimize the performance of the models while accounting for uncertainty and providing early exits for faster inference.</p> </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span><span class="ltx_text ltx_font_smallcaps" id="S6.1.1">Implementation</span> </h2> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S6.SS1.5.1.1">VI-A</span> </span><span class="ltx_text ltx_font_italic" id="S6.SS1.6.2">System Implementation</span> </h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S6.SS1.p1.1.1">Hardware<span class="ltx_text ltx_font_upright" id="S6.SS1.p1.1.1.1">.</span></span> The training stage of our system is implemented and tested on a Linux server equipped with an Intel Xeon Gold 5218 CPU and NVIDIA Quadro RTX 8000 GPU. The shared backbone and multiple heads are pre-trained during this stage. Afterwards, in the deployment stage, we deploy the shared backbone and heads on two MCUs. The first one is the STM32F446ZE (or F446ZE), which has an ARM Cortex M4 processor with 128 KB of SRAM and 512 KB of eFlash. The other one is the STM32H747XI (or 747XI), featuring a dual-core processor (ARM Cortex M4 and M7) with 1 MB of SRAM and 2 MB of eFlash. Our evaluation only utilizes one core (ARM Cortex M7) since MCUs are typically equipped with only one CPU core. This setup limits the usage space of SRAM and eFlash to 512 KB and 1 MB, respectively.</p> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.1">We developed and assessed our system’s training stage using PyTorch 1.8, and tested various baselines on a Linux server. The evidential uncertainty module is implemented with Python and NumPy. We adopted TensorFlow Lite Micro (TFLM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib26" title="">26</a>]</cite> for MCU deployment due to its portability, ease of use, and support for numerous neural network layers and optimized kernels. UR2M’s deployment stage and online optimization scheme are developed in C++ on two MCUs (ARM Cortex M4 and M7). To deploy a PyTorch model on MCUs, we convert it to TensorFlow Lite (TF Lite) using ONNX representation and the TF Lite converter. The model is run on MCUs using TFLM and Mbed OS. Additionally, the CMSIS-DSP software library processes raw signals to generate model inputs (e.g., MFCC features), and the CMSIS-NN kernels in TFLM facilitate efficient neural network operations on MCUs.</p> </div> <div class="ltx_para" id="S6.SS1.p3"> <p class="ltx_p" id="S6.SS1.p3.3"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S6.SS1.p3.3.1">Multi-tenancy Deployment<span class="ltx_text ltx_font_upright" id="S6.SS1.p3.3.1.1">.</span></span> To facilitate multi-event sharing on MCUs with limited memory, we develop a multi-tenancy deployment for early-exit models using TFLM. UR2M utilizes multiple model interpreters to allocate memory from a unified space, ensuring efficient model operation. During evaluation, this deployment strategy is applied to all baselines and the UR2M model. For example, Deep Ensembles have five models, potentially using 5<math alttext="\times" class="ltx_Math" display="inline" id="S6.SS1.p3.1.m1.1"><semantics id="S6.SS1.p3.1.m1.1a"><mo id="S6.SS1.p3.1.m1.1.1" xref="S6.SS1.p3.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.1.m1.1b"><times id="S6.SS1.p3.1.m1.1.1.cmml" xref="S6.SS1.p3.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.1.m1.1d">×</annotation></semantics></math> eFlash space. However, with optimization, it only consumes 2<math alttext="\times" class="ltx_Math" display="inline" id="S6.SS1.p3.2.m2.1"><semantics id="S6.SS1.p3.2.m2.1a"><mo id="S6.SS1.p3.2.m2.1.1" xref="S6.SS1.p3.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.2.m2.1b"><times id="S6.SS1.p3.2.m2.1.1.cmml" xref="S6.SS1.p3.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.2.m2.1d">×</annotation></semantics></math> more SRAM (cf. <math alttext="\lx@sectionsign" class="ltx_Math" display="inline" id="S6.SS1.p3.3.m3.1"><semantics id="S6.SS1.p3.3.m3.1a"><mi id="S6.SS1.p3.3.m3.1.1" mathvariant="normal" xref="S6.SS1.p3.3.m3.1.1.cmml">§</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.3.m3.1b"><ci id="S6.SS1.p3.3.m3.1.1.cmml" xref="S6.SS1.p3.3.m3.1.1">§</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.3.m3.1c">\lx@sectionsign</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.3.m3.1d">§</annotation></semantics></math><a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS3" title="VIII-C End-to-end System Efficiency ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">VIII-C</span></span></a>) due to multi-tenancy deployment.</p> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S6.SS2.5.1.1">VI-B</span> </span><span class="ltx_text ltx_font_italic" id="S6.SS2.6.2">Uncertainty Operator Implementation</span> </h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.1">To capture the uncertainty at inference time on MCUs, we only use TFLM-supported operations. First, we utilize a ReLU operator to regulate the distribution of the output as non-negatives. Then, based on these outputs, we follow Eq.<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S4.E6" title="6 ‣ IV-B Efficient Evidential Modeling for Event Detection on MCUs ‣ IV Efficient Uncertainty Quantification ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">6</span></a> to generate uncertainties. Specifically, calculating uncertainty first requires the sum of reduced dimensions. Although the <span class="ltx_text ltx_font_italic" id="S6.SS2.p1.1.1">reduced_sum</span> operator is supported, it is not available for TFLM. To solve this, we use a <span class="ltx_text ltx_font_italic" id="S6.SS2.p1.1.2">squeeze</span> operator to reduce the output dimensions, followed by a <span class="ltx_text ltx_font_italic" id="S6.SS2.p1.1.3">sum</span> operator. Finally, we apply a <span class="ltx_text ltx_font_italic" id="S6.SS2.p1.1.4">divide</span> operator to generate the uncertainty. We wrap the above-mentioned operators within the model and implement them in the TFLM library to save the overhead of uncertainty prediction. The overall uncertainty implementation is shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5.F3.sf1" title="3a ‣ Figure 3 ‣ V-B Multiple-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">3a</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S6.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S6.SS3.5.1.1">VI-C</span> </span><span class="ltx_text ltx_font_italic" id="S6.SS3.6.2">MCU Library Optimization</span> </h3> <div class="ltx_para" id="S6.SS3.p1"> <p class="ltx_p" id="S6.SS3.p1.1">Unlike mobile devices’ memory architecture that employs large off-chip main memory (e.g., DRAM), MCUs consist of only small-sized on-chip memory (e.g., SRAM and eFlash) (cf. Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S1.F1" title="Figure 1 ‣ I Introduction ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a>). To understand the memory requirements of our model to fit in MCUs, we first compute the memory usage of UR2M. For a searched shallow model with 8-bit int quantization, we observe that TFLM requires 79 KB of SRAM and 203 KB of eFlash, which falls within the tight memory budgets of many MCUs, for example, 64 KB of SRAM and 128 KB of eFlash of STM32F205VB as described in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S1.F1" title="Figure 1 ‣ I Introduction ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">1</span></a>. In particular, on SRAM, the memory usage includes intermediate tensors (30 KB), persistent buffers (3 KB), runtime overhead of the TFLM interpreter (6 KB), and MBed OS and other libraries (10 KB). Additionally, Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5.F3.sf2" title="3b ‣ Figure 3 ‣ V-B Multiple-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">3b</span></a> top shows the on-chip eFlash architecture of an F446ZE MCU and how TFLM allocates memory space to run a shallow model on an MCU.</p> </div> <div class="ltx_para" id="S6.SS3.p2"> <p class="ltx_p" id="S6.SS3.p2.1"><span class="ltx_text ltx_font_italic" id="S6.SS3.p2.1.1">Note that since we only conduct 8-bit post-quantization, we only observe a maximum of 1% performance drop between the pre-and post-quantization stages among all methods.</span> </p> </div> <div class="ltx_para" id="S6.SS3.p3"> <p class="ltx_p" id="S6.SS3.p3.1">Given the limited memory space for searching the optimal model parameters, we propose optimizing the TFLM library. First, we removed all operation-related files that did not impact our backbone. Then, we reordered the operations files based on our backbone structure. As shown at the bottom of Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S5.F3.sf2" title="3b ‣ Figure 3 ‣ V-B Multiple-event Sharing ‣ V Cascade learning ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">3b</span></a>, our MCU library significantly optimized the TFLM interpreter’s runtime overhead, reducing it from 122 KB to 32 KB (3.8<math alttext="\times" class="ltx_Math" display="inline" id="S6.SS3.p3.1.m1.1"><semantics id="S6.SS3.p3.1.m1.1a"><mo id="S6.SS3.p3.1.m1.1.1" xref="S6.SS3.p3.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.SS3.p3.1.m1.1b"><times id="S6.SS3.p3.1.m1.1.1.cmml" xref="S6.SS3.p3.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p3.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p3.1.m1.1d">×</annotation></semantics></math> smaller). Moreover, the graph definition was reduced by 2 KB, from 8 KB to 6 KB, in the eFlash memory. After the optimization, a total of 104 KB of memory is used, which can now fit into the STM32F205VB and many other MCUs. Overall, UR2M optimizes 49% of eFlash memory compared to the baseline TFLM library.</p> </div> <div class="ltx_para" id="S6.SS3.p4"> <p class="ltx_p" id="S6.SS3.p4.1"><span class="ltx_text ltx_font_italic" id="S6.SS3.p4.1.1">Note that during the evaluation, we applied the same MCU library optimization strategy to all baselines as well as the UR2M model.</span></p> </div> </section> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VII </span><span class="ltx_text ltx_font_smallcaps" id="S7.1.1">Evaluation settings</span> </h2> <figure class="ltx_figure" id="S7.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_flex_size_3 ltx_align_center" id="S7.F4.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="120" id="S7.F4.sf1.g1" src="x5.png" width="134"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(a) </span>Model sizes <span class="ltx_text ltx_font_italic" id="S7.F4.sf1.2.1">vs.</span> Acc.</figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_flex_size_3 ltx_align_center" id="S7.F4.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="122" id="S7.F4.sf2.g1" src="x6.png" width="134"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(b) </span>Acc. on Oesense</figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <figure class="ltx_figure ltx_flex_size_3 ltx_align_center" id="S7.F4.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="124" id="S7.F4.sf3.g1" src="x7.png" width="254"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(c) </span>Acc. on KWS</figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_flex_size_1 ltx_align_center" id="S7.F4.sf4"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="117" id="S7.F4.sf4.g1" src="x8.png" width="144"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">(d) </span>Acc. on ECG5000</figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Model sizes <span class="ltx_text ltx_font_italic" id="S7.F4.2.1">vs.</span> accuracy and early exit result for single events. Note that the ECG5000 UB event has only one test sample.</figcaption> </figure> <section class="ltx_subsection" id="S7.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S7.SS1.5.1.1">VII-A</span> </span><span class="ltx_text ltx_font_italic" id="S7.SS1.6.2">Evaluated Datasets</span> </h3> <div class="ltx_para" id="S7.SS1.p1"> <p class="ltx_p" id="S7.SS1.p1.1">Our target application scenarios are focused on WED applications. Specifically, we evaluate three wearable datasets, including in-ear activity recognition <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib32" title="">32</a>]</cite>, audio event keyword spotting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib33" title="">33</a>]</cite>, and heart disorder event detection <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib34" title="">34</a>]</cite>. We experiment with these three datasets, each featuring different data modalities that suit UR2M settings. For imbalanced datasets, we use SMOTE <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib35" title="">35</a>]</cite> to upsample the training data.</p> </div> <div class="ltx_para" id="S7.SS1.p2"> <p class="ltx_p" id="S7.SS1.p2.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS1.p2.2.1">In-ear Dataset<span class="ltx_text ltx_font_upright" id="S7.SS1.p2.2.1.1">.</span></span> Oesense <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib32" title="">32</a>]</cite> contains an in-ear audio dataset for activity recognition (including five events: “walk”, “run”, “still”, “drink”, and “chew”) among 31 subjects. For preprocessing, we first segment the original audio into one-second segments and set the sampling rate at 4 kHz. Then, we extract the 2-D MFCC features for each segment. 10 MFCC features are then obtained from an audio frame with a length of 80 ms and a stride of 40 ms, yielding an input dimension of 1<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p2.1.m1.1"><semantics id="S7.SS1.p2.1.m1.1a"><mo id="S7.SS1.p2.1.m1.1.1" xref="S7.SS1.p2.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.1.m1.1b"><times id="S7.SS1.p2.1.m1.1.1.cmml" xref="S7.SS1.p2.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.1.m1.1d">×</annotation></semantics></math>10<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p2.2.m2.1"><semantics id="S7.SS1.p2.2.m2.1a"><mo id="S7.SS1.p2.2.m2.1.1" xref="S7.SS1.p2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.2.m2.1b"><times id="S7.SS1.p2.2.m2.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.2.m2.1d">×</annotation></semantics></math>21. After preprocessing each event, we obtained 40,064 training samples (90%) and 4,452 test samples (10%) for all five activities.</p> </div> <div class="ltx_para" id="S7.SS1.p3"> <p class="ltx_p" id="S7.SS1.p3.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS1.p3.2.1">KWS Dataset<span class="ltx_text ltx_font_upright" id="S7.SS1.p3.2.1.1">.</span></span> The Keywords Spotting (KWS) V2 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib33" title="">33</a>]</cite> dataset contains 105,829 utterances from 2,618 speakers. There are 35 words split into 12 classes, including ten keyword spotting classes and an ’unknown’ class (remaining 24 words). For preprocessing, we first constrained all event samples to one second by segmentation or zero-padding and set the sampling rate at 16 kHz. Then we extracted MFCC features using 640 FFT points and 320 points of hop length. We obtained 10 MFCC features from an audio frame with a length of 40ms and a sliding window of 20ms, yielding the input dimension of 1<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p3.1.m1.1"><semantics id="S7.SS1.p3.1.m1.1a"><mo id="S7.SS1.p3.1.m1.1.1" xref="S7.SS1.p3.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.1.m1.1b"><times id="S7.SS1.p3.1.m1.1.1.cmml" xref="S7.SS1.p3.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.1.m1.1d">×</annotation></semantics></math>10<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p3.2.m2.1"><semantics id="S7.SS1.p3.2.m2.1a"><mo id="S7.SS1.p3.2.m2.1.1" xref="S7.SS1.p3.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.2.m2.1b"><times id="S7.SS1.p3.2.m2.1.1.cmml" xref="S7.SS1.p3.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.2.m2.1d">×</annotation></semantics></math>51. After preprocessing, we obtained 92,502 total event training samples (90%) and 10,278 test samples (10%).</p> </div> <div class="ltx_para" id="S7.SS1.p4"> <p class="ltx_p" id="S7.SS1.p4.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS1.p4.2.1">ECG5000 Dataset<span class="ltx_text ltx_font_upright" id="S7.SS1.p4.2.1.1">.</span></span> The ECG5000 dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib34" title="">34</a>]</cite> is a 20-hour long one-channel ECG dataset that contains 92,584 heartbeats, including five different types of heart events: Normal (NM) (58.4%), R-on-T Premature Ventricular Contraction (RTPVC) (35.3%), Premature Ventricular Contraction (PVC) (3.9%), Supra-ventricular Premature or Ectopic Beat (SPEB) (2%), and Unclassified Beat (UB) (0.5%). For preprocessing, we resample the input duration of 0.56s with 140 samples into 560 samples. Then we reshape the input into 10 channels, yielding the input dimension of 1<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p4.1.m1.1"><semantics id="S7.SS1.p4.1.m1.1a"><mo id="S7.SS1.p4.1.m1.1.1" xref="S7.SS1.p4.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.1.m1.1b"><times id="S7.SS1.p4.1.m1.1.1.cmml" xref="S7.SS1.p4.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.1.m1.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.1.m1.1d">×</annotation></semantics></math>10<math alttext="\times" class="ltx_Math" display="inline" id="S7.SS1.p4.2.m2.1"><semantics id="S7.SS1.p4.2.m2.1a"><mo id="S7.SS1.p4.2.m2.1.1" xref="S7.SS1.p4.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.2.m2.1b"><times id="S7.SS1.p4.2.m2.1.1.cmml" xref="S7.SS1.p4.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.2.m2.1d">×</annotation></semantics></math>56. After the preprocessing, we obtained 4,500 total event training samples (90%) and 500 (10%) test samples. Note that UB has only one test sample.</p> </div> </section> <section class="ltx_subsection" id="S7.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S7.SS2.5.1.1">VII-B</span> </span><span class="ltx_text ltx_font_italic" id="S7.SS2.6.2">Uncertainty Metrics</span> </h3> <div class="ltx_para" id="S7.SS2.p1"> <p class="ltx_p" id="S7.SS2.p1.1">We compare UR2M using three important uncertainty metrics: Brier score, Negative Log-Likelihood (NLL), and Expected Calibration Error (ECE), to examine the uncertainty estimation performance.</p> </div> </section> <section class="ltx_subsection" id="S7.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S7.SS3.5.1.1">VII-C</span> </span><span class="ltx_text ltx_font_italic" id="S7.SS3.6.2">Uncertainty Quantification Baselines</span> </h3> <div class="ltx_para" id="S7.SS3.p1"> <p class="ltx_p" id="S7.SS3.p1.1">We evaluate the proposed method by comparing it to three baseline uncertainty solutions: the traditional softmax-based models, deep ensembles and data augmentation. It is important to note that MCDP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib10" title="">10</a>]</cite> is not available for the MCUs library TFLM because it stores models as binary files that cannot be modified. Moreover, its computational costs are similar to or greater than deep ensembles, while its uncertainty performance is lower than that of deep ensembles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib9" title="">9</a>]</cite>.</p> </div> <div class="ltx_para" id="S7.SS3.p2"> <p class="ltx_p" id="S7.SS3.p2.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS3.p2.1.1">Vanilla EDL</span>. Vanilla EDL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib22" title="">22</a>]</cite> is the state-of-the-art (SOTA) model to <span class="ltx_text ltx_font_italic" id="S7.SS3.p2.1.2">efficiently</span> quantify uncertainty and can be implemented on MCUs.</p> </div> <div class="ltx_para" id="S7.SS3.p3"> <p class="ltx_p" id="S7.SS3.p3.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS3.p3.2.1">Deep Ensembles</span>. Deep ensembles approach (denoted as D(Softmax)+Ense) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib7" title="">7</a>]</cite> is the SOTA model to <span class="ltx_text ltx_font_italic" id="S7.SS3.p3.2.2">accurately</span> quantify uncertainty estimation, which typically ensembles <math alttext="N" class="ltx_Math" display="inline" id="S7.SS3.p3.1.m1.1"><semantics id="S7.SS3.p3.1.m1.1a"><mi id="S7.SS3.p3.1.m1.1.1" xref="S7.SS3.p3.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p3.1.m1.1b"><ci id="S7.SS3.p3.1.m1.1.1.cmml" xref="S7.SS3.p3.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p3.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p3.1.m1.1d">italic_N</annotation></semantics></math> deterministic Softmax models with random weight initializations. We use <math alttext="N=5" class="ltx_Math" display="inline" id="S7.SS3.p3.2.m2.1"><semantics id="S7.SS3.p3.2.m2.1a"><mrow id="S7.SS3.p3.2.m2.1.1" xref="S7.SS3.p3.2.m2.1.1.cmml"><mi id="S7.SS3.p3.2.m2.1.1.2" xref="S7.SS3.p3.2.m2.1.1.2.cmml">N</mi><mo id="S7.SS3.p3.2.m2.1.1.1" xref="S7.SS3.p3.2.m2.1.1.1.cmml">=</mo><mn id="S7.SS3.p3.2.m2.1.1.3" xref="S7.SS3.p3.2.m2.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p3.2.m2.1b"><apply id="S7.SS3.p3.2.m2.1.1.cmml" xref="S7.SS3.p3.2.m2.1.1"><eq id="S7.SS3.p3.2.m2.1.1.1.cmml" xref="S7.SS3.p3.2.m2.1.1.1"></eq><ci id="S7.SS3.p3.2.m2.1.1.2.cmml" xref="S7.SS3.p3.2.m2.1.1.2">𝑁</ci><cn id="S7.SS3.p3.2.m2.1.1.3.cmml" type="integer" xref="S7.SS3.p3.2.m2.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p3.2.m2.1c">N=5</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p3.2.m2.1d">italic_N = 5</annotation></semantics></math> which is widely adopted in recent efficient studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib36" title="">36</a>]</cite>.</p> </div> <div class="ltx_para" id="S7.SS3.p4"> <p class="ltx_p" id="S7.SS3.p4.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S7.SS3.p4.2.1">Data Augmentation</span>. Test time data augmentation (denoted as D(Softmax)+InAug) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib37" title="">37</a>]</cite> is a <span class="ltx_text ltx_font_italic" id="S7.SS3.p4.2.2">memory-efficient</span> uncertainty quantification method generating multiple test samples by applying data augmentation techniques through a single model. We utilize five augmented samples, incorporating Jittering, with a mean <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S7.SS3.p4.1.m1.1"><semantics id="S7.SS3.p4.1.m1.1a"><mi id="S7.SS3.p4.1.m1.1.1" xref="S7.SS3.p4.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.1.m1.1b"><ci id="S7.SS3.p4.1.m1.1.1.cmml" xref="S7.SS3.p4.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.1.m1.1d">italic_ε</annotation></semantics></math> of 0 and a standard deviation <math alttext="\sigma" class="ltx_Math" display="inline" id="S7.SS3.p4.2.m2.1"><semantics id="S7.SS3.p4.2.m2.1a"><mi id="S7.SS3.p4.2.m2.1.1" xref="S7.SS3.p4.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.2.m2.1b"><ci id="S7.SS3.p4.2.m2.1.1.cmml" xref="S7.SS3.p4.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.2.m2.1d">italic_σ</annotation></semantics></math> of 0.03, which are added to the test data. </p> </div> </section> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VIII </span><span class="ltx_text ltx_font_smallcaps" id="S8.1.1">Results</span> </h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.1">This section will discuss the results and answer the following questions: (1) How efficient is UR2M for typical MCUs? (2) How robust is UR2M compared with traditional point prediction models?</p> </div> <section class="ltx_subsection" id="S8.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S8.SS1.5.1.1">VIII-A</span> </span><span class="ltx_text ltx_font_italic" id="S8.SS1.6.2">Performance of Event Detection</span> </h3> <div class="ltx_para" id="S8.SS1.p1"> <p class="ltx_p" id="S8.SS1.p1.2">Utilizing the Adam optimizer with a learning rate of <math alttext="1\mathrm{e}{-3}" class="ltx_Math" display="inline" id="S8.SS1.p1.1.m1.1"><semantics id="S8.SS1.p1.1.m1.1a"><mrow id="S8.SS1.p1.1.m1.1.1" xref="S8.SS1.p1.1.m1.1.1.cmml"><mrow id="S8.SS1.p1.1.m1.1.1.2" xref="S8.SS1.p1.1.m1.1.1.2.cmml"><mn id="S8.SS1.p1.1.m1.1.1.2.2" xref="S8.SS1.p1.1.m1.1.1.2.2.cmml">1</mn><mo id="S8.SS1.p1.1.m1.1.1.2.1" xref="S8.SS1.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S8.SS1.p1.1.m1.1.1.2.3" mathvariant="normal" xref="S8.SS1.p1.1.m1.1.1.2.3.cmml">e</mi></mrow><mo id="S8.SS1.p1.1.m1.1.1.1" xref="S8.SS1.p1.1.m1.1.1.1.cmml">−</mo><mn id="S8.SS1.p1.1.m1.1.1.3" xref="S8.SS1.p1.1.m1.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S8.SS1.p1.1.m1.1b"><apply id="S8.SS1.p1.1.m1.1.1.cmml" xref="S8.SS1.p1.1.m1.1.1"><minus id="S8.SS1.p1.1.m1.1.1.1.cmml" xref="S8.SS1.p1.1.m1.1.1.1"></minus><apply id="S8.SS1.p1.1.m1.1.1.2.cmml" xref="S8.SS1.p1.1.m1.1.1.2"><times id="S8.SS1.p1.1.m1.1.1.2.1.cmml" xref="S8.SS1.p1.1.m1.1.1.2.1"></times><cn id="S8.SS1.p1.1.m1.1.1.2.2.cmml" type="integer" xref="S8.SS1.p1.1.m1.1.1.2.2">1</cn><ci id="S8.SS1.p1.1.m1.1.1.2.3.cmml" xref="S8.SS1.p1.1.m1.1.1.2.3">e</ci></apply><cn id="S8.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S8.SS1.p1.1.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.SS1.p1.1.m1.1c">1\mathrm{e}{-3}</annotation><annotation encoding="application/x-llamapun" id="S8.SS1.p1.1.m1.1d">1 roman_e - 3</annotation></semantics></math>, a 32 batch size, and an early stopping of 5 epochs, we train our networks, showcased in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.F4.sf1" title="4a ‣ Figure 4 ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">4a</span></a> and Figures <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.F4.sf2" title="4b ‣ Figure 4 ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">4b</span></a>- <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.F4.sf4" title="4d ‣ Figure 4 ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">4d</span></a>. While system accuracy generally increases with OPS across all datasets, significant increases in overhead do not invariably equate to notable accuracy improvements, as observed in the KWS and ECG5000 datasets. For instance, a shallow Oesense model (accuracy: 0.83, parameters: 0.38 MB) contrasts with the medium and deep models, which respectively present 0.87/0.58 MB and 0.91/0.76 MB in accuracy/parameters. The 2% accuracy enhancement when transitioning from medium to deep models incurs a 31% overhead spike. Similarly, for ECG5000, a 1% accuracy improvement requires doubling the model sizes. Shallow models across all datasets exhibit proficient performance (e.g., <math alttext=">" class="ltx_Math" display="inline" id="S8.SS1.p1.2.m2.1"><semantics id="S8.SS1.p1.2.m2.1a"><mo id="S8.SS1.p1.2.m2.1.1" xref="S8.SS1.p1.2.m2.1.1.cmml">></mo><annotation-xml encoding="MathML-Content" id="S8.SS1.p1.2.m2.1b"><gt id="S8.SS1.p1.2.m2.1.1.cmml" xref="S8.SS1.p1.2.m2.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S8.SS1.p1.2.m2.1c">></annotation><annotation encoding="application/x-llamapun" id="S8.SS1.p1.2.m2.1d">></annotation></semantics></math>80%) with minimized model size, hinting that UR2M could deliver effective performance with modest overheads.</p> </div> <div class="ltx_para" id="S8.SS1.p2"> <p class="ltx_p" id="S8.SS1.p2.1">Regarding the channel sizes, our searched model yields the output shape for each OPS as [5,11] for Oesense, [5, 26] for KWS, and [5, 29] for ECG5000, respectively. Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S7.F4" title="Figure 4 ‣ VII Evaluation settings ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">4</span></a> further illustrates the UR2M performance for single event detection using shallow, medium and deep network structures.</p> </div> <figure class="ltx_figure" id="S8.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="358" id="S8.F5.g1" src="x9.png" width="871"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Comparing Vanilla EDL, data augmentation, deep ensembles (SOTA), and UR2M using uncertainty, error rate, and memory usage metrics across three datasets. eF1 and SR1 refer to the eFlash and SRAM usage of H747XI, while eF2 and SR2 refer to those of F464ZE, respectively. For all metrics, lower values are preferred.</figcaption> </figure> <div class="ltx_para" id="S8.SS1.p3"> <p class="ltx_p" id="S8.SS1.p3.1">Based on Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F5" title="Figure 5 ‣ VIII-A Performance of Event Detection ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">5</span></a>, we can observe that UR2M’s uncertainty metrics are better than Data Augmentation (D(Softmax)+InAug) baseline across all three datasets, with up to 22% lower NLL scores (0.65 to 0.53). This improvement indicates that the proposed method produces better-calibrated models that are less prone to overconfidence errors. Compared to the D(Softmax)+Ense model, UR2M achieves similar performance in terms of both uncertainty estimation and prediction accuracy. For instance, UR2M outperforms D(Softmax)+Ense by 8.0% in terms of Brier score for KWS, and achieves 1.7% and 2.4% relative improvements in NLL for Oesense and ECG5000, respectively.</p> </div> <div class="ltx_para" id="S8.SS1.p4"> <p class="ltx_p" id="S8.SS1.p4.1"><span class="ltx_text ltx_font_italic" id="S8.SS1.p4.1.1">Notably, UR2M achieves these results while using up to only half of the memory, much less energy, and latency required by SOTA method deep ensembles (cf. Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F5" title="Figure 5 ‣ VIII-A Performance of Event Detection ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">5</span></a> and §<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.SS3" title="VIII-C End-to-end System Efficiency ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">VIII-C</span></span></a>), demonstrating the computational efficiency of UR2M without compromising uncertainty estimation.</span></p> </div> </section> <section class="ltx_subsection" id="S8.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S8.SS2.5.1.1">VIII-B</span> </span><span class="ltx_text ltx_font_italic" id="S8.SS2.6.2">Impact of different Uncertainty Thresholds</span> </h3> <div class="ltx_para" id="S8.SS2.p1"> <p class="ltx_p" id="S8.SS2.p1.1">Users can decide on the uncertainty threshold according to their specific applications. For example, in healthcare applications (e.g., heart attack detection), we prefer a low uncertainty (e.g., <math alttext="u" class="ltx_Math" display="inline" id="S8.SS2.p1.1.m1.1"><semantics id="S8.SS2.p1.1.m1.1a"><mi id="S8.SS2.p1.1.m1.1.1" xref="S8.SS2.p1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S8.SS2.p1.1.m1.1b"><ci id="S8.SS2.p1.1.m1.1.1.cmml" xref="S8.SS2.p1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS2.p1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S8.SS2.p1.1.m1.1d">italic_u</annotation></semantics></math>=0.05) for detected heart attacks to avoid disastrous consequences. This tradeoff is depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F7" title="Figure 7 ‣ VIII-B Impact of different Uncertainty Thresholds ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">7</span></a>. In other scenarios (e.g., running detection), a higher uncertainty threshold can be tolerated to save battery life by exiting through shallow layers. Similarly, it is observed that increasing the threshold gradually reduces latency across all three datasets when evaluated on the F446ZE and H747XI MCUs. With a higher uncertainty threshold, more samples are filtered out by the shallow and medium sub-networks, and fewer samples pass through deep models, leading to reduced latency. This indicates that selecting different uncertainty thresholds allows users to obtain a personalized model, increasing the usability of UR2M.</p> </div> <div class="ltx_para" id="S8.SS2.p2"> <p class="ltx_p" id="S8.SS2.p2.1"><span class="ltx_text ltx_font_italic" id="S8.SS2.p2.1.1">In sum, our model design, which allows users to define the threshold, can help determine the optimized threshold to balance the tradeoff, thereby achieving personalized models.</span></p> </div> <figure class="ltx_figure" id="S8.F7"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Uncertainty impact</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><span class="ltx_inline-para ltx_minipage ltx_flex_size_2 ltx_align_center ltx_align_middle" id="S8.F7.1" style="width:216.8pt;"> <span class="ltx_para ltx_align_center" id="S8.F7.1.p1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="102" id="S8.F7.1.p1.g1" src="x10.png" width="146"/> </span></span></div> <div class="ltx_flex_cell ltx_flex_size_2"><span class="ltx_inline-para ltx_minipage ltx_flex_size_2 ltx_align_center ltx_align_middle" id="S8.F7.2" style="width:216.8pt;"> <span class="ltx_para ltx_align_center" id="S8.F7.2.p1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="89" id="S8.F7.2.p1.g1" src="x11.png" width="157"/> </span></span></div> <div class="ltx_flex_break"></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Uncertainty impact</figcaption> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>End-to-end deployment</figcaption> </figure> </section> <section class="ltx_subsection" id="S8.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S8.SS3.5.1.1">VIII-C</span> </span><span class="ltx_text ltx_font_italic" id="S8.SS3.6.2">End-to-end System Efficiency</span> </h3> <div class="ltx_para" id="S8.SS3.p1"> <p class="ltx_p" id="S8.SS3.p1.1">Following the optimization of all baselines and UR2M using techniques including multi-tenancy deployment, model quantization, and MCU library optimization, we evaluate their runtime efficiency during deployment on MCUs (Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F7" title="Figure 7 ‣ VIII-B Impact of different Uncertainty Thresholds ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">7</span></a>). Our evaluation encompasses the entire system, including signal acquisition, feature extraction, and memory usage in terms of SRAM and eFlash required for model execution. We conducted experiments with various datasets and two typical resource-constrained MCUs, the F446ZE and H747XI. Although the focus is primarily on the ECG5000 dataset due to page limits, note that consistent outcomes were observed across all three datasets.</p> </div> <div class="ltx_para" id="S8.SS3.p2"> <p class="ltx_p" id="S8.SS3.p2.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S8.SS3.p2.1.1">Model Inference Memory Footprint<span class="ltx_text ltx_font_upright" id="S8.SS3.p2.1.1.1">.</span></span> Based on our implementation, UR2M consumes only 49 KB and 51 KB of SRAM (38.5% and 9.9% of the total SRAM of F446ZE and H747XI, respectively) as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F5" title="Figure 5 ‣ VIII-A Performance of Event Detection ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">5</span></a>. Additionally, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F5" title="Figure 5 ‣ VIII-A Performance of Event Detection ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">5</span></a>, UR2M requires 142 KB and 145 KB of eFlash (27.7% and 14.1% of the total eFlash of F446ZE and H747XI, respectively). These results demonstrate that UR2M consumes only a small portion of the limited resources of MCUs, leaving enough resources for other applications to be supported simultaneously. Furthermore, UR2M requires only 66-67% of SRAM (49 KB vs. 75 KB for F446ZE and 51 KB vs. 75 KB for H747XI) and 51% of eFlash (142 KB vs. 280 KB for F446ZE and 145 KB vs. 283 KB for H747XI) compared to the deep ensembles baseline.</p> </div> <div class="ltx_para" id="S8.SS3.p3"> <p class="ltx_p" id="S8.SS3.p3.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S8.SS3.p3.1.1">Signal Acquisition Overheads.</span> To evaluate signal acquisition overheads for the F446ZE MCU, we employ an INMP441 MEMS microphone. For the H747XI, we use the MP34DT05-A built-in microphone on the H747I-DISCO evaluation board (Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F7" title="Figure 7 ‣ VIII-B Impact of different Uncertainty Thresholds ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">7</span></a>). We assess energy consumption and memory usage as key factors. Energy consumption (J) is computed as the product of time/latency (t) and power (W). Power is determined from input voltage (V) and current measurements (A), conducted with a Fluke 87V digital multimeter. For the F446ZE, we record a power consumption of 24.6 mA at 3.3V, resulting in 81.18 mW for one second of audio signal acquisition. Memory-wise, it uses 4KB of SRAM and 32KB of eflash. In contrast, the H747XI consumes 31.6 mA at 3.3V, totaling 104.28 mW in power. It utilizes 29KB of SRAM and 66KB of eflash. Overall, signal acquisition overheads for these two MCUs are minimal.</p> </div> <div class="ltx_para" id="S8.SS3.p4"> <p class="ltx_p" id="S8.SS3.p4.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S8.SS3.p4.1.1">Feature Extraction Overheads.</span> The feature extraction step for both UR2M and the baselines is the same, using MFCC features as inputs. The extraction process is fast, taking only 4.505 ms and 10.913 ms per extraction for the H747XI across two datasets, indicating minimal overhead.</p> </div> <div class="ltx_para" id="S8.SS3.p5"> <p class="ltx_p" id="S8.SS3.p5.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S8.SS3.p5.1.1">Model Inference Latency<span class="ltx_text ltx_font_upright" id="S8.SS3.p5.1.1.1">.</span></span> Using the MBed Timer API to measure latency on MCUs, Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F8" title="Figure 8 ‣ VIII-C End-to-end System Efficiency ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">8</span></a> illustrates UR2M’s and baseline inference results across three datasets and two MCUs. With uncertainty thresholds (<math alttext="u" class="ltx_Math" display="inline" id="S8.SS3.p5.1.m1.1"><semantics id="S8.SS3.p5.1.m1.1a"><mi id="S8.SS3.p5.1.m1.1.1" xref="S8.SS3.p5.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S8.SS3.p5.1.m1.1b"><ci id="S8.SS3.p5.1.m1.1.1.cmml" xref="S8.SS3.p5.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS3.p5.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S8.SS3.p5.1.m1.1d">italic_u</annotation></semantics></math>) ranging from 1 to 0, UR2Mpresents latencies from lowest to highest, respectively. While baseline approaches, like deep ensemble, yield reliable uncertainty estimations, they exhibit high inference latencies of 717.2-717.4 ms on F446ZE and 171.1-179.3 ms on H747XI per sample. Conversely, UR2Mensures both reliable uncertainty and minimized latency, cutting inference latencies up to 864% (83.0 ms vs. 717.2 ms) on F446ZE and 835% (20.2 ms vs. 171.1 ms) on H747XI. Moreover, UR2Menhances latency by approximately 456% against other baselines, even without uncertainty filtering.</p> </div> <div class="ltx_para" id="S8.SS3.p6"> <p class="ltx_p" id="S8.SS3.p6.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S8.SS3.p6.1.1">Model Inference Energy Consumption<span class="ltx_text ltx_font_upright" id="S8.SS3.p6.1.1.1">.</span></span> Similar to the latency results, UR2M significantly reduces energy consumption compared to the baselines, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F8" title="Figure 8 ‣ VIII-C End-to-end System Efficiency ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">8</span></a>. For example, UR2M decreases energy consumption by up to 834% (116.0 mJ vs. 13.9 mJ) on F446ZE and 857% (39.4 mJ vs. 4.6 mJ) on H747XI when compared to the best-performing benchmark uncertainty-aware baselines. Also, we observe that UR2M achieves around 450% energy improvement compared to the baselines without uncertainty filtering.</p> </div> <figure class="ltx_figure" id="S8.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="305" id="S8.F8.g1" src="x12.png" width="847"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Comparison of latency and energy consumption of uncertainty-aware methods on two MCUs.</figcaption> </figure> </section> <section class="ltx_subsection" id="S8.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S8.SS4.5.1.1">VIII-D</span> </span><span class="ltx_text ltx_font_italic" id="S8.SS4.6.2">Robustness Against Signal Uncertainties</span> </h3> <div class="ltx_para" id="S8.SS4.p1"> <p class="ltx_p" id="S8.SS4.p1.1">We evaluate UR2M in the context of two types of signal uncertainties: signal missing (replace as zero) and noise (gaussian noise). Due to page limitations, we compare our method with traditional softmax-based NNs having the same model structure. As demonstrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#S8.F9" title="Figure 9 ‣ VIII-D Robustness Against Signal Uncertainties ‣ VIII Results ‣ UR2M: Uncertainty and Resource-Aware Event Detection on Microcontrollers"><span class="ltx_text ltx_ref_tag">9</span></a>, for a correct event signal “Chew”, the absence of signal and the presence of random noise can lead softmax-based NNs to predict incorrectly. In contrast, UR2M can accurately predict most corrupted signals. When predictions are incorrect, UR2M also exhibits high uncertainty (e.g., <math alttext="u" class="ltx_Math" display="inline" id="S8.SS4.p1.1.m1.1"><semantics id="S8.SS4.p1.1.m1.1a"><mi id="S8.SS4.p1.1.m1.1.1" xref="S8.SS4.p1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S8.SS4.p1.1.m1.1b"><ci id="S8.SS4.p1.1.m1.1.1.cmml" xref="S8.SS4.p1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.SS4.p1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S8.SS4.p1.1.m1.1d">italic_u</annotation></semantics></math>=1.0), which could be used for alerting the system to potential misclassifications or triggering additional validation steps.</p> </div> <figure class="ltx_figure" id="S8.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="269" id="S8.F9.g1" src="x13.png" width="611"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Uncertainty estimation towards signal missing and noise. Labels in red indicate wrong predictions.</figcaption> </figure> </section> </section> <section class="ltx_section" id="S9"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IX </span><span class="ltx_text ltx_font_smallcaps" id="S9.1.1">Discussion</span> </h2> <div class="ltx_para" id="S9.p1"> <p class="ltx_p" id="S9.p1.1">In this section, we discuss several possible future directions for our work.</p> </div> <div class="ltx_para" id="S9.p2"> <p class="ltx_p" id="S9.p2.1"><span class="ltx_text ltx_font_bold" id="S9.p2.1.1">Generalizing UR2M to other sensors and higher-end MCUs.</span> Ideally, UR2M could be generalized to any wearable sensors driven by MCUs. However, sensor signal complexity and limited MCU memory size pose limitations. More complex signals usually require larger model sizes, challenging the deployment on the constrained memory of MCUs. Fortunately, recent work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2402.09264v3#bib.bib12" title="">12</a>]</cite> shows that by investigating compressive sensing, key patterns of primitives in signals can be compressed and extracted, which indicates it can reduce the model size to save system overhead. Therefore, we will study how compressive sensing combined with UR2M could further reduce system overhead to generalize to ultra low-end MCUs. We envision our method could also benefit higher-end MCUs, e.g. STM32F4, which has 1MB flash and 192KB SRAM. Since less memory is required, higher-end MCUs could experience improvements in latency and energy efficiency.</p> </div> <div class="ltx_para" id="S9.p3"> <p class="ltx_p" id="S9.p3.1"><span class="ltx_text ltx_font_bold" id="S9.p3.1.1">Impact of UR2M on future WED systems.</span> Our work has illustrated that uncertainty is a key criterion to ensure reliable prediction in WED systems. Therefore, an important and urgent question is how to define uncertainty tolerance thresholds for specific applications. Fortunately, for healthcare applications, we can design this criterion through a doctor-in-the-loop strategy to select the optimal threshold.</p> </div> </section> <section class="ltx_section" id="S10"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">X </span><span class="ltx_text ltx_font_smallcaps" id="S10.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S10.p1"> <p class="ltx_p" id="S10.p1.1">In this paper, we have proposed UR2M, a resource and uncertainty-aware framework which can efficiently and reliably enable wearable event detection and related uncertainty on MCUs. By exploiting evidential uncertainty theory, cascade learning, and system optimization, UR2M significantly improves energy and memory efficiency for MCUs without sacrificing accuracy, enabling real-time and reliable event detection.</p> </div> </section> <section class="ltx_section" id="S11"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">XI </span><span class="ltx_text ltx_font_smallcaps" id="S11.1.1">Acknowledgment</span> </h2> <div class="ltx_para" id="S11.p1"> <p class="ltx_p" id="S11.p1.1">This work is supported by ERC through Project 833296 (EAR), and Nokia Bell Labs through a donation. </p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Arash Alavi, Gireesh K Bogu, Meng Wang, Ekanath Srihari Rangan, Andrew W Brooks, Qiwen Wang, Emily Higgs, Alessandra Celli, Tejaswini Mishra, Ahmed A Metwally, et al. </span> <span class="ltx_bibblock">Real-time alerting system for covid-19 and other stress events using wearable data. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Nature medicine</span>, 28(1):175–184, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> Christian Holz and Edward J. 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