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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches</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.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.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_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2409.06327v1/"/></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/2409.06327v1#S1" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">1 Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2 Vulnerability of contemporary ASV model to simultaneous threats</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.SS1" title="In 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2.1 Testing dataset</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.SS2" title="In 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2.2 Preliminary experimental result of contemporary ASV model</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3 Proposed integrated approach</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS1" title="In 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.1 Overview</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS2" title="In 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.2 Model architecture and objectives</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS2.SSS1" title="In 3.2 Model architecture and objectives ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.2.1 Asymmetric dual-path feature extractor</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS2.SSS2" title="In 3.2 Model architecture and objectives ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.2.2 Speaker classifier and spoof classifier</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS2.SSS3" title="In 3.2 Model architecture and objectives ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.2.3 SASV binary classifier</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS3" title="In 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.3 Learning paradigm</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS3.SSS1" title="In 3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.3.1 Outer loop</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS3.SSS2" title="In 3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.3.2 Inner loop</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4 Experimental results</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS1" title="In 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.1 Dataset</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS2" title="In 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.2 Experimental settings</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS3" title="In 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.3 Results and analysis</a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS3.SSS1" title="In 4.3 Results and analysis ‣ 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.3.1 Result for channel mismatch scenario</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS3.SSS2" title="In 4.3 Results and analysis ‣ 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.3.2 Result for spoofing attack scenario</a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4.SS3.SSS3" title="In 4.3 Results and analysis ‣ 4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4.3.3 Result for domain mismatch scenario</a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S5" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">5 Conclusion</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S6" title="In Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">6 ACKNOWLEDGMENTS</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches</h1> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> In real-world applications, it is challenging to build a speaker verification system that is simultaneously robust against common threats, including spoofing attacks, channel mismatch, and domain mismatch. Traditional automatic speaker verification (ASV) systems often tackle these issues separately, leading to suboptimal performance when faced with simultaneous challenges. In this paper, we propose an integrated framework that incorporates pair-wise learning and spoofing attack simulation into the meta-learning paradigm to enhance robustness against these multifaceted threats. This novel approach employs an asymmetric dual-path model and a multi-task learning strategy to handle ASV, anti-spoofing, and spoofing-aware ASV tasks concurrently. A new testing dataset, CNComplex, is introduced to evaluate system performance under these combined threats. Experimental results demonstrate that our integrated model significantly improves performance over traditional ASV systems across various scenarios, showcasing its potential for real-world deployment. Additionally, the proposed framework’s ability to generalize across different conditions highlights its robustness and reliability, making it a promising solution for practical ASV applications. </div> <div class="ltx_para" id="p1"> Index Terms— Speaker verification, robustness, multi-task learning, meta-learning </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 1 Introduction</h2> <div class="ltx_para" id="S1.p1"> Automatic speaker verification (ASV) aims to accurately identify the speaker’s identity by analyzing voice characteristics within speech signals, providing robust support for personalized services. Mainstream ASV solutions have evolved from traditional statistics-based methods such as GMM-UBM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib1" title="">1</a>]</cite> and I-Vector <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib2" title="">2</a>]</cite> to deep neural network-based approaches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib4" title="">4</a>]</cite>. For example, time-delay neural network (TDNN) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib7" title="">7</a>]</cite> and ResNet <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib9" title="">9</a>]</cite> have dominated the realm of extracting speaker embeddings since they can efficiently leverage large amounts of speech data to capture the properties of speakers accurately. Additionally, the back-end is another important component of an ASV system. Neural network-based methods <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib12" title="">12</a>]</cite> have gradually replaced the widely used PLDA model in recent years. </div> <div class="ltx_para" id="S1.p2"> However, as speaker recognition has rapidly evolved, its focus has shifted from improving performance on simple datasets such as TIMIT and LibriSpeech <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib14" title="">14</a>]</cite> to addressing challenges in what is often referred to as “recognition in the wild” <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib17" title="">17</a>]</cite>. In this context, numerous issues have surfaced, including channel mismatch <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib21" title="">21</a>]</cite>, vulnerability to spoofing attacks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib24" title="">24</a>]</cite>, and domain mismatches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib25" title="">25</a>]</cite>, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S1.F1" style="font-size:90%;" title="Figure 1 ‣ 1 Introduction ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">1</a>. While substantial progress has been made in addressing each of these challenges individually <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib28" title="">28</a>]</cite>, the practical deployment of ASV systems demands solutions that can effectively handle these multifaceted threats. Given the potential for the ASV system to encounter simultaneous attacks, there is a critical need to amalgamate all these functionalities within the ASV system itself. </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="116" id="S1.F1.g1" src="x1.png" width="436"/> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;">Fig. 1: The primary threats to ASV systems. These threats can be categorized into channel mismatch, spoofing attacks, and domain mismatch.</figcaption> </figure> <div class="ltx_para" id="S1.p3"> However, the integration is hindered by two factors. <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> 1. <div class="ltx_para" id="S1.I1.i1.p1"> Presently, no dataset encompasses the presence of three concurrent threats. For instance, widely used datasets for speaker verification such as VoxCeleb 1&2 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib29" title="">29</a>]</cite> and CNCeleb 1&2 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib30" title="">30</a>]</cite> exclusively feature scenarios involving channel mismatch. Meanwhile, datasets designed for the ASVspoof challenge series <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib23" title="">23</a>]</cite> solely tackle spoofing attacks. Hence, it is imperative to develop a new dataset and evaluation protocol encompassing all three considerations. </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> 2. <div class="ltx_para" id="S1.I1.i2.p1"> One apparent method to incorporate all functionalities involves cascading individual components, each dedicated to addressing a specific challenge. For instance, positioning the countermeasures module ahead of the ASV module serves to detect synthesized input utterances. Additionally, regarding the ASV module as comprising a neural speaker encoder and a back-end, assigning domain mismatch handling to the former and channel mismatch to the latter seems intuitive. However, as discussed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>]</cite>, this cascading approach overlooks the interactions among these modules, leading to suboptimal system performance. Consequently, a novel framework must be devised to facilitate and leverage these interactions effectively. </div> </li> </ol> </div> <div class="ltx_para" id="S1.p4"> In response to this practical need, in this paper we introduce an integrated framework that harmonizes the techniques elucidated in previous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib28" title="">28</a>]</cite>. This includes the pair-wise learning paradigm, which simulates channel mismatch, the fusion module, which replicates spoofing attacks, and the meta-learning paradigm, which mimics domain mismatch. The aim is to establish a robust defense against channel variability, spoofing attacks, and domain mismatch, all in one comprehensive approach. This represents an end-to-end methodology, optimizing resilience across these challenges by utilizing a shared model. To the best of our knowledge, this is the first attempt to integrate robustness into a single model by explicitly exposing the model to all threats. The primary contribution lies in amalgamating the pair-wise learning, spoofing simulation, and meta-learning paradigms into a unified optimization strategy. </div> <div class="ltx_para" id="S1.p5"> The subsequent sections of this paper are structured as follows. We commence with a preliminary experiment with the ECAPA-TDNN model on a new spoofing-aware ASV evaluation dataset, which contains testing trials for both channel and domain mismatches, offering evidence of the vulnerability of contemporary SOTA ASV models when facing three threats simultaneously. Next, an overview of the proposed integrated approach is given in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3" style="font-size:90%;" title="3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3</a>, including the overview of it to offer insights into the integration methodology in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS1" style="font-size:90%;" title="3.1 Overview ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.1</a>, the specifics of the model architecture and its objectives in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS2" style="font-size:90%;" title="3.2 Model architecture and objectives ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.2</a>, and the learning paradigm for E2E optimization in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.SS3" style="font-size:90%;" title="3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3.3</a>. The experimental setup, along with the results and their analysis, is detailed in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S4" style="font-size:90%;" title="4 Experimental results ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4</a>. Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S5" style="font-size:90%;" title="5 Conclusion ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">5</a> concludes the paper with a comprehensive summary. </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 2 Vulnerability of contemporary ASV model to simultaneous threats</h2> <figure class="ltx_table" id="S2.T1"> <figcaption class="ltx_caption" style="font-size:90%;">Table 1: Genre group division. Ten genres are randomly divided into four groups.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S2.T1.5"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T1.5.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S2.T1.5.1.1.1">Group</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S2.T1.5.1.1.2">Genre Types</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T1.5.2.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T1.5.2.1.1">Group I</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T1.5.2.1.2">drama (dr), vlog (vl), speech (sp)</td> </tr> <tr class="ltx_tr" id="S2.T1.5.3.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T1.5.3.2.1">Group II</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T1.5.3.2.2">entertainment (en), interview (in), play (pl)</td> </tr> <tr class="ltx_tr" id="S2.T1.5.4.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T1.5.4.3.1">Group III</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T1.5.4.3.2">live broadcast (lb), movie (mo)</td> </tr> <tr class="ltx_tr" id="S2.T1.5.5.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S2.T1.5.5.4.1">Group IV</th> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S2.T1.5.5.4.2">singing (si), recitation (re)</td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="S2.T2"> <figcaption class="ltx_caption" style="font-size:90%;">Table 2: Cross-genre protocols (CGPs) established via K-fold validation. For each protocol, the training dataset only contains seen genre groups, while the testing dataset contains both seen and unseen genre groups. The unseen genre in the testing dataset results in a cross-genre scenario. Note that speakers in the training and testing datasets overlap.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S2.T2.5"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T2.5.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S2.T2.5.1.1.1" style="padding-left:4.0pt;padding-right:4.0pt;">CGP</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S2.T2.5.1.1.2" style="padding-left:4.0pt;padding-right:4.0pt;">Seen Genres</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S2.T2.5.1.1.3" style="padding-left:4.0pt;padding-right:4.0pt;">Unseen Genres</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T2.5.2.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T2.5.2.1.1" style="padding-left:4.0pt;padding-right:4.0pt;">CGP I</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.2.1.2" style="padding-left:4.0pt;padding-right:4.0pt;">Group I, Group II, Group III</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.2.1.3" style="padding-left:4.0pt;padding-right:4.0pt;">Group IV</td> </tr> <tr class="ltx_tr" id="S2.T2.5.3.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T2.5.3.2.1" style="padding-left:4.0pt;padding-right:4.0pt;">CGP II</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.3.2.2" style="padding-left:4.0pt;padding-right:4.0pt;">Group I, Group II, Group IV</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.3.2.3" style="padding-left:4.0pt;padding-right:4.0pt;">Group III</td> </tr> <tr class="ltx_tr" id="S2.T2.5.4.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T2.5.4.3.1" style="padding-left:4.0pt;padding-right:4.0pt;">CGP III</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.4.3.2" style="padding-left:4.0pt;padding-right:4.0pt;">Group I, Group III, Group IV</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T2.5.4.3.3" style="padding-left:4.0pt;padding-right:4.0pt;">Group II</td> </tr> <tr class="ltx_tr" id="S2.T2.5.5.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S2.T2.5.5.4.1" style="padding-left:4.0pt;padding-right:4.0pt;">CGP IV</th> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S2.T2.5.5.4.2" style="padding-left:4.0pt;padding-right:4.0pt;">Group II, Group III, Group IV</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S2.T2.5.5.4.3" style="padding-left:4.0pt;padding-right:4.0pt;">Group I</td> </tr> </tbody> </table> </figure> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 2.1 Testing dataset</h3> <div class="ltx_para" id="S2.SS1.p1"> In light of the absence of an existing dataset that aligns with requirements for evaluating system performance in the challenging scenario involving all three threats simultaneously, it is imperative to create a new testing dataset and devise a corresponding evaluation protocol tailored to this scenario. This newly developed test dataset was constructed based on the original CNCeleb test dataset, as CNCeleb has multiple genres, making it well-suited as a basis for building complex test sets. Within this dataset, the enrollment utterances remain unchanged. However, the testing utterances are subject to random substitution with re-vocoded data sourced from the CNSpoof dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib31" title="">31</a>]</cite>. The new testing dataset is referred to as CNComplex. </div> <div class="ltx_para" id="S2.SS1.p2"> In terms of establishing the ground truth, we adhere to the definition outlined in SASVC 2022 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib32" title="">32</a>]</cite>. Specifically, if the testing utterance is genuine and its corresponding speaker label matches the enrolled identity, it is labeled as true; otherwise, it is marked as false. Consequently, the new testing dataset now encompasses both channel mismatch and spoofing attacks. </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 2.2 Preliminary experimental result of contemporary ASV model</h3> <div class="ltx_para" id="S2.SS2.p1"> To demonstrate the vulnerability of the contemporary ASV model when faced with the CNComplex testing dataset, an ECAPA-TDNN speaker encoder was trained, incorporating the AM-Softmax loss function <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib34" title="">34</a>]</cite>, on two separate datasets: CNCeleb 1&2 and the combination of CNCeleb 1&2 and CNSpoof datasets according to the cross-genre protocols (CGPs) described in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib31" title="">31</a>]</cite>. Each protocol excludes data with certain genres from the training dataset. In detail, the ten genres present in the training data were categorized into four groups, as outlined in Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.T1" style="font-size:90%;" title="Table 1 ‣ 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">1</a>. It is worth noting that the ”advertisement” genre was excluded from the dataset due to its limited amount of data. For each CGP, <math alttext="660,000" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.2"><semantics id="S2.SS2.p1.1.m1.2a"><mrow id="S2.SS2.p1.1.m1.2.3.2" xref="S2.SS2.p1.1.m1.2.3.1.cmml"><mn id="S2.SS2.p1.1.m1.1.1" mathsize="90%" xref="S2.SS2.p1.1.m1.1.1.cmml">660</mn><mo id="S2.SS2.p1.1.m1.2.3.2.1" mathsize="90%" xref="S2.SS2.p1.1.m1.2.3.1.cmml">,</mo><mn id="S2.SS2.p1.1.m1.2.2" mathsize="90%" xref="S2.SS2.p1.1.m1.2.2.cmml">000</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.2b"><list id="S2.SS2.p1.1.m1.2.3.1.cmml" xref="S2.SS2.p1.1.m1.2.3.2"><cn id="S2.SS2.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1">660</cn><cn id="S2.SS2.p1.1.m1.2.2.cmml" type="integer" xref="S2.SS2.p1.1.m1.2.2">000</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.2c">660,000</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.2d">660 , 000</annotation></semantics></math> utterances were randomly sampled to compile a training dataset, which encompassed three of the genre groups. Importantly, in this way, each training dataset has an unseen genre group in the testing dataset. This strategy is detailed in Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.T2" style="font-size:90%;" title="Table 2 ‣ 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2</a>. </div> <div class="ltx_para" id="S2.SS2.p2"> The results of this experiment evaluated by cosine similarity are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.T3" style="font-size:90%;" title="Table 3 ‣ 2.2 Preliminary experimental result of contemporary ASV model ‣ 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">3</a>. We subjected these well-trained models to evaluation on both the original CNCeleb testing dataset (CNCeleb.Eval) and the newly introduced CNComplex dataset. </div> <figure class="ltx_table" id="S2.T3"> <figcaption class="ltx_caption" style="font-size:80%;">Table 3: Performance of the ECAPA-TDNN model trained on two distinct sets of data: CNCeleb 1&2 and the combination of (CNCeleb 1&2, CNSpoof) according to CGP. Subsequently, evaluations on both the original CNCeleb testing dataset and the newly developed testing dataset were conducted. The evaluation metrics employed for this assessment are the SV-EER and SASV-EER defined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>]</cite>.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S2.T3.8"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S2.T3.8.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S2.T3.8.1.1.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">Protocol</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S2.T3.8.1.1.2" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">Training dataset</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S2.T3.8.1.1.3" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb.Eval</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S2.T3.8.1.1.4" style="padding-left:2.3pt;padding-right:2.3pt;">CNComplex</th> </tr> <tr class="ltx_tr" id="S2.T3.8.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S2.T3.8.2.2.1" style="padding-left:2.3pt;padding-right:2.3pt;">SV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S2.T3.8.2.2.2" style="padding-left:2.3pt;padding-right:2.3pt;">SASV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S2.T3.8.2.2.3" style="padding-left:2.3pt;padding-right:2.3pt;">SV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S2.T3.8.2.2.4" style="padding-left:2.3pt;padding-right:2.3pt;">SASV-EER</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.T3.8.3.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S2.T3.8.3.1.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP I</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T3.8.3.1.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T3.8.3.1.3" style="padding-left:2.3pt;padding-right:2.3pt;">9.38</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T3.8.3.1.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T3.8.3.1.5" style="padding-left:2.3pt;padding-right:2.3pt;">10.05</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.T3.8.3.1.6" style="padding-left:2.3pt;padding-right:2.3pt;">37.64</td> </tr> <tr class="ltx_tr" id="S2.T3.8.4.2"> <td class="ltx_td ltx_align_center" id="S2.T3.8.4.2.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.4.2.2" style="padding-left:2.3pt;padding-right:2.3pt;">9.02</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.4.2.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.4.2.4" style="padding-left:2.3pt;padding-right:2.3pt;">9.56</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.4.2.5" style="padding-left:2.3pt;padding-right:2.3pt;">36.97</td> </tr> <tr class="ltx_tr" id="S2.T3.8.5.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S2.T3.8.5.3.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP II</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.5.3.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.5.3.3" style="padding-left:2.3pt;padding-right:2.3pt;">10.04</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.5.3.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.5.3.5" style="padding-left:2.3pt;padding-right:2.3pt;">10.85</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.5.3.6" style="padding-left:2.3pt;padding-right:2.3pt;">40.76</td> </tr> <tr class="ltx_tr" id="S2.T3.8.6.4"> <td class="ltx_td ltx_align_center" id="S2.T3.8.6.4.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.6.4.2" style="padding-left:2.3pt;padding-right:2.3pt;">9.75</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.6.4.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.6.4.4" style="padding-left:2.3pt;padding-right:2.3pt;">10.79</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.6.4.5" style="padding-left:2.3pt;padding-right:2.3pt;">40.17</td> </tr> <tr class="ltx_tr" id="S2.T3.8.7.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S2.T3.8.7.5.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP III</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.7.5.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.7.5.3" style="padding-left:2.3pt;padding-right:2.3pt;">10.12</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.7.5.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.7.5.5" style="padding-left:2.3pt;padding-right:2.3pt;">10.67</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.7.5.6" style="padding-left:2.3pt;padding-right:2.3pt;">39.62</td> </tr> <tr class="ltx_tr" id="S2.T3.8.8.6"> <td class="ltx_td ltx_align_center" id="S2.T3.8.8.6.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.8.6.2" style="padding-left:2.3pt;padding-right:2.3pt;">9.33</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.8.6.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.8.6.4" style="padding-left:2.3pt;padding-right:2.3pt;">10.22</td> <td class="ltx_td ltx_align_center" id="S2.T3.8.8.6.5" style="padding-left:2.3pt;padding-right:2.3pt;">38.49</td> </tr> <tr class="ltx_tr" id="S2.T3.8.9.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_tt" id="S2.T3.8.9.7.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP IV</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.9.7.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.9.7.3" style="padding-left:2.3pt;padding-right:2.3pt;">9.59</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.9.7.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.9.7.5" style="padding-left:2.3pt;padding-right:2.3pt;">10.24</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S2.T3.8.9.7.6" style="padding-left:2.3pt;padding-right:2.3pt;">37.97</td> </tr> <tr class="ltx_tr" id="S2.T3.8.10.8"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S2.T3.8.10.8.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S2.T3.8.10.8.2" style="padding-left:2.3pt;padding-right:2.3pt;">9.21</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S2.T3.8.10.8.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S2.T3.8.10.8.4" style="padding-left:2.3pt;padding-right:2.3pt;">9.86</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S2.T3.8.10.8.5" style="padding-left:2.3pt;padding-right:2.3pt;">37.42</td> </tr> </tbody> </table> </figure> <div class="ltx_para" id="S2.SS2.p3"> First and foremost, it is noteworthy that the ECAPA-TDNN model trained on the combination of CNCeleb 1&2 and CNSpoof datasets exhibited a slightly consistent performance improvement on the CNCeleb.Eval dataset compared to that of the model trained solely on the CNCeleb 1&2 dataset under all CGPs. This observation suggests that the copy-synthesis method <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib36" title="">36</a>]</cite> employed to generate the CNSpoof dataset can be considered as an effective data augmentation strategy in training an ASV model. </div> <div class="ltx_para" id="S2.SS2.p4"> However, the critical focus shifts to the results obtained from the CNComplex dataset. As anticipated, the performance substantially deteriorated on this new dataset under all CGPs, regardless of which training dataset was used. This unequivocally indicates that the contemporary state-of-the-art ASV model is ill-equipped to address the complexities inherent in the new testing dataset, which contains simultaneous threats. </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 3 Proposed integrated approach</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 3.1 Overview</h3> <div class="ltx_para" id="S3.SS1.p1"> As described in the preceding section, ASV models were observed to exhibit vulnerabilities when confronted with complex scenarios. However, in real-world settings, ASV systems often encounter these threats simultaneously. Therefore, it is imperative to develop a robust system that can effectively counter these challenges in a concurrent manner. </div> <div class="ltx_para" id="S3.SS1.p2"> In this section, building upon the foundation laid by the previous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib28" title="">28</a>]</cite>, we present an integrated solution designed to address these challenges comprehensively. The proposed solution utilizes an asymmetric dual-path model equipped with a multi-task learning strategy, as depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.F2" style="font-size:90%;" title="Figure 2 ‣ 3.1 Overview ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2</a>. This model is tasked with simultaneously handling the ASV task, the anti-spoofing task, and the spoofing-aware ASV task. Furthermore, we incorporate pair-wise learning and spoofing attack simulation into the meta-learning paradigms to train the model so that the model is equipped with the capability to address channel mismatch, spoofing attacks, and domain mismatch issues. </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="332" id="S3.F2.g1" src="x2.png" width="436"/> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;">Fig. 2: Architecture of the dual-path model consisting of four submodules: a generic feature extractor, a speaker classifier, a spoof classifier, and a SASV binary classifier. Notably, the feature extractor adopts an asymmetric dual-path structure.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p3"> In comparison to traditional robust systems targeting a single threat, the proposed approach presents two distinct advantages. Firstly, it capitalizes on the interactions between the spoofing-aware attention back-end and the meta-learning-based generalization, resulting in the enhancement of both subsystems. Secondly, the end-to-end training methodology enables more efficient signal propagation to train a system robust against multiple risks. The initial gains observed in individual tasks in the previous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib28" title="">28</a>]</cite> clearly underscore the promising potential of this integrated approach. </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 3.2 Model architecture and objectives</h3> <div class="ltx_para" id="S3.SS2.p1"> Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.F2" style="font-size:90%;" title="Figure 2 ‣ 3.1 Overview ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2</a> provides a visual representation of the architecture of the proposed spoofing-aware ASV model. This model consists of four main components: <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S3.I1.i1.p1"> Asymmetric dual-path feature extractor </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S3.I1.i2.p1"> Speaker classifier </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="S3.I1.i3.p1"> Spoof classifier </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> • <div class="ltx_para" id="S3.I1.i4.p1"> SASV binary classifier </div> </li> </ul> Each of these components will be elaborated upon in detail in the subsequent section. </div> <section class="ltx_subsubsection" id="S3.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 3.2.1 Asymmetric dual-path feature extractor</h4> <div class="ltx_para" id="S3.SS2.SSS1.p1"> As depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.F2" style="font-size:90%;" title="Figure 2 ‣ 3.1 Overview ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2</a>, the asymmetric dual-path feature extractor comprises N blocks, each with distinct parameters but an identical architecture. Within each block, two different branches share the same input, denoted as <math alttext="\boldsymbol{X}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.1.m1.1"><semantics id="S3.SS2.SSS1.p1.1.m1.1a"><mi id="S3.SS2.SSS1.p1.1.m1.1.1" mathsize="90%" xref="S3.SS2.SSS1.p1.1.m1.1.1.cmml">𝑿</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.1.m1.1b"><ci id="S3.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p1.1.m1.1.1">𝑿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.1.m1.1c">\boldsymbol{X}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.1.m1.1d">bold_italic_X</annotation></semantics></math>. The left multi-head attention branch, inspired by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib40" title="">40</a>]</cite>, primarily focuses on extracting features for the anti-spoofing task. Conversely, the right SE-Res2Block, proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib7" title="">7</a>]</cite>, is primarily responsible for feature extraction for the ASV task. The outputs of these two branches are combined through addition and layer normalization operations. The computational process can be expressed as follows: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx1"> <tbody id="S3.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{left}^{n}" class="ltx_Math" display="inline" id="S3.E1.m1.1"><semantics id="S3.E1.m1.1a"><msubsup id="S3.E1.m1.1.1" xref="S3.E1.m1.1.1.cmml"><mi id="S3.E1.m1.1.1.2.2" mathsize="90%" xref="S3.E1.m1.1.1.2.2.cmml">𝑯</mi><mrow id="S3.E1.m1.1.1.2.3" xref="S3.E1.m1.1.1.2.3.cmml"><mi id="S3.E1.m1.1.1.2.3.2" mathsize="90%" xref="S3.E1.m1.1.1.2.3.2.cmml">l</mi><mo id="S3.E1.m1.1.1.2.3.1" xref="S3.E1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.E1.m1.1.1.2.3.3" mathsize="90%" xref="S3.E1.m1.1.1.2.3.3.cmml">e</mi><mo id="S3.E1.m1.1.1.2.3.1a" xref="S3.E1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.E1.m1.1.1.2.3.4" mathsize="90%" xref="S3.E1.m1.1.1.2.3.4.cmml">f</mi><mo 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xref="S3.E1.m1.1.1.2.3.5">𝑡</ci></apply></apply><ci id="S3.E1.m1.1.1.3.cmml" xref="S3.E1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.1c">\displaystyle\boldsymbol{H}_{left}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_l italic_e italic_f italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=f_{left}(\boldsymbol{X})," class="ltx_Math" display="inline" id="S3.E1.m2.2"><semantics id="S3.E1.m2.2a"><mrow id="S3.E1.m2.2.2.1" xref="S3.E1.m2.2.2.1.1.cmml"><mrow id="S3.E1.m2.2.2.1.1" xref="S3.E1.m2.2.2.1.1.cmml"><mi id="S3.E1.m2.2.2.1.1.2" xref="S3.E1.m2.2.2.1.1.2.cmml"></mi><mo id="S3.E1.m2.2.2.1.1.1" mathsize="90%" xref="S3.E1.m2.2.2.1.1.1.cmml">=</mo><mrow id="S3.E1.m2.2.2.1.1.3" xref="S3.E1.m2.2.2.1.1.3.cmml"><msub 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rowspan="1">(1)</td> </tr></tbody> <tbody id="S3.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{right}^{n}" class="ltx_Math" display="inline" id="S3.E2.m1.1"><semantics id="S3.E2.m1.1a"><msubsup id="S3.E2.m1.1.1" xref="S3.E2.m1.1.1.cmml"><mi id="S3.E2.m1.1.1.2.2" mathsize="90%" xref="S3.E2.m1.1.1.2.2.cmml">𝑯</mi><mrow id="S3.E2.m1.1.1.2.3" xref="S3.E2.m1.1.1.2.3.cmml"><mi id="S3.E2.m1.1.1.2.3.2" mathsize="90%" xref="S3.E2.m1.1.1.2.3.2.cmml">r</mi><mo id="S3.E2.m1.1.1.2.3.1" xref="S3.E2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.E2.m1.1.1.2.3.3" mathsize="90%" xref="S3.E2.m1.1.1.2.3.3.cmml">i</mi><mo id="S3.E2.m1.1.1.2.3.1a" xref="S3.E2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.E2.m1.1.1.2.3.4" mathsize="90%" xref="S3.E2.m1.1.1.2.3.4.cmml">g</mi><mo id="S3.E2.m1.1.1.2.3.1b" xref="S3.E2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.E2.m1.1.1.2.3.5" 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xref="S3.E2.m1.1.1.2.3.4">𝑔</ci><ci id="S3.E2.m1.1.1.2.3.5.cmml" xref="S3.E2.m1.1.1.2.3.5">ℎ</ci><ci id="S3.E2.m1.1.1.2.3.6.cmml" xref="S3.E2.m1.1.1.2.3.6">𝑡</ci></apply></apply><ci id="S3.E2.m1.1.1.3.cmml" xref="S3.E2.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.1c">\displaystyle\boldsymbol{H}_{right}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_r italic_i italic_g italic_h italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=f_{right}(\boldsymbol{X})," class="ltx_Math" display="inline" id="S3.E2.m2.2"><semantics id="S3.E2.m2.2a"><mrow id="S3.E2.m2.2.2.1" xref="S3.E2.m2.2.2.1.1.cmml"><mrow id="S3.E2.m2.2.2.1.1" xref="S3.E2.m2.2.2.1.1.cmml"><mi id="S3.E2.m2.2.2.1.1.2" xref="S3.E2.m2.2.2.1.1.2.cmml"></mi><mo id="S3.E2.m2.2.2.1.1.1" mathsize="90%" xref="S3.E2.m2.2.2.1.1.1.cmml">=</mo><mrow id="S3.E2.m2.2.2.1.1.3" xref="S3.E2.m2.2.2.1.1.3.cmml"><msub id="S3.E2.m2.2.2.1.1.3.2" xref="S3.E2.m2.2.2.1.1.3.2.cmml"><mi id="S3.E2.m2.2.2.1.1.3.2.2" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.2.cmml">f</mi><mrow id="S3.E2.m2.2.2.1.1.3.2.3" xref="S3.E2.m2.2.2.1.1.3.2.3.cmml"><mi id="S3.E2.m2.2.2.1.1.3.2.3.2" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.3.2.cmml">r</mi><mo id="S3.E2.m2.2.2.1.1.3.2.3.1" xref="S3.E2.m2.2.2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E2.m2.2.2.1.1.3.2.3.3" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.3.3.cmml">i</mi><mo id="S3.E2.m2.2.2.1.1.3.2.3.1a" xref="S3.E2.m2.2.2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E2.m2.2.2.1.1.3.2.3.4" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.3.4.cmml">g</mi><mo id="S3.E2.m2.2.2.1.1.3.2.3.1b" xref="S3.E2.m2.2.2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E2.m2.2.2.1.1.3.2.3.5" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.3.5.cmml">h</mi><mo id="S3.E2.m2.2.2.1.1.3.2.3.1c" xref="S3.E2.m2.2.2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E2.m2.2.2.1.1.3.2.3.6" mathsize="90%" xref="S3.E2.m2.2.2.1.1.3.2.3.6.cmml">t</mi></mrow></msub><mo id="S3.E2.m2.2.2.1.1.3.1" xref="S3.E2.m2.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S3.E2.m2.2.2.1.1.3.3.2" xref="S3.E2.m2.2.2.1.1.3.cmml"><mo id="S3.E2.m2.2.2.1.1.3.3.2.1" maxsize="90%" minsize="90%" xref="S3.E2.m2.2.2.1.1.3.cmml">(</mo><mi id="S3.E2.m2.1.1" mathsize="90%" xref="S3.E2.m2.1.1.cmml">𝑿</mi><mo id="S3.E2.m2.2.2.1.1.3.3.2.2" maxsize="90%" minsize="90%" xref="S3.E2.m2.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E2.m2.2.2.1.2" mathsize="90%" xref="S3.E2.m2.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E2.m2.2b"><apply id="S3.E2.m2.2.2.1.1.cmml" xref="S3.E2.m2.2.2.1"><eq id="S3.E2.m2.2.2.1.1.1.cmml" xref="S3.E2.m2.2.2.1.1.1"></eq><csymbol cd="latexml" id="S3.E2.m2.2.2.1.1.2.cmml" xref="S3.E2.m2.2.2.1.1.2">absent</csymbol><apply id="S3.E2.m2.2.2.1.1.3.cmml" xref="S3.E2.m2.2.2.1.1.3"><times id="S3.E2.m2.2.2.1.1.3.1.cmml" xref="S3.E2.m2.2.2.1.1.3.1"></times><apply id="S3.E2.m2.2.2.1.1.3.2.cmml" xref="S3.E2.m2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S3.E2.m2.2.2.1.1.3.2.1.cmml" xref="S3.E2.m2.2.2.1.1.3.2">subscript</csymbol><ci id="S3.E2.m2.2.2.1.1.3.2.2.cmml" xref="S3.E2.m2.2.2.1.1.3.2.2">𝑓</ci><apply id="S3.E2.m2.2.2.1.1.3.2.3.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3"><times id="S3.E2.m2.2.2.1.1.3.2.3.1.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.1"></times><ci id="S3.E2.m2.2.2.1.1.3.2.3.2.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.2">𝑟</ci><ci id="S3.E2.m2.2.2.1.1.3.2.3.3.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.3">𝑖</ci><ci id="S3.E2.m2.2.2.1.1.3.2.3.4.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.4">𝑔</ci><ci id="S3.E2.m2.2.2.1.1.3.2.3.5.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.5">ℎ</ci><ci id="S3.E2.m2.2.2.1.1.3.2.3.6.cmml" xref="S3.E2.m2.2.2.1.1.3.2.3.6">𝑡</ci></apply></apply><ci id="S3.E2.m2.1.1.cmml" xref="S3.E2.m2.1.1">𝑿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m2.2c">\displaystyle=f_{right}(\boldsymbol{X}),</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m2.2d">= italic_f start_POSTSUBSCRIPT italic_r italic_i italic_g italic_h italic_t end_POSTSUBSCRIPT ( bold_italic_X ) ,</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">(2)</td> </tr></tbody> <tbody id="S3.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{fuse}^{n}" class="ltx_Math" display="inline" id="S3.E3.m1.1"><semantics id="S3.E3.m1.1a"><msubsup id="S3.E3.m1.1.1" xref="S3.E3.m1.1.1.cmml"><mi id="S3.E3.m1.1.1.2.2" mathsize="90%" xref="S3.E3.m1.1.1.2.2.cmml">𝑯</mi><mrow id="S3.E3.m1.1.1.2.3" xref="S3.E3.m1.1.1.2.3.cmml"><mi id="S3.E3.m1.1.1.2.3.2" mathsize="90%" 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xref="S3.E3.m1.1.1.2.3"><times id="S3.E3.m1.1.1.2.3.1.cmml" xref="S3.E3.m1.1.1.2.3.1"></times><ci id="S3.E3.m1.1.1.2.3.2.cmml" xref="S3.E3.m1.1.1.2.3.2">𝑓</ci><ci id="S3.E3.m1.1.1.2.3.3.cmml" xref="S3.E3.m1.1.1.2.3.3">𝑢</ci><ci id="S3.E3.m1.1.1.2.3.4.cmml" xref="S3.E3.m1.1.1.2.3.4">𝑠</ci><ci id="S3.E3.m1.1.1.2.3.5.cmml" xref="S3.E3.m1.1.1.2.3.5">𝑒</ci></apply></apply><ci id="S3.E3.m1.1.1.3.cmml" xref="S3.E3.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.1c">\displaystyle\boldsymbol{H}_{fuse}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\text{LN}(\boldsymbol{H}_{spoof}^{n}+\boldsymbol{H}_{asv}^{n})," class="ltx_Math" display="inline" 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xref="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.1"></times><ci id="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.2.cmml" xref="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.2">𝑎</ci><ci id="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.3.cmml" xref="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.3">𝑠</ci><ci id="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.4.cmml" xref="S3.E3.m2.1.1.1.1.1.1.1.1.3.2.3.4">𝑣</ci></apply></apply><ci id="S3.E3.m2.1.1.1.1.1.1.1.1.3.3.cmml" xref="S3.E3.m2.1.1.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m2.1c">\displaystyle=\text{LN}(\boldsymbol{H}_{spoof}^{n}+\boldsymbol{H}_{asv}^{n}),</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m2.1d">= LN ( bold_italic_H start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + bold_italic_H start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n 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">(3)</td> </tr></tbody> </table> where <math alttext="\boldsymbol{H}_{left}^{n}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.2.m1.1"><semantics id="S3.SS2.SSS1.p1.2.m1.1a"><msubsup id="S3.SS2.SSS1.p1.2.m1.1.1" xref="S3.SS2.SSS1.p1.2.m1.1.1.cmml"><mi id="S3.SS2.SSS1.p1.2.m1.1.1.2.2" mathsize="90%" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS1.p1.2.m1.1.1.2.3" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.3.cmml"><mi id="S3.SS2.SSS1.p1.2.m1.1.1.2.3.2" mathsize="90%" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.3.2.cmml">l</mi><mo id="S3.SS2.SSS1.p1.2.m1.1.1.2.3.1" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.2.m1.1.1.2.3.3" mathsize="90%" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.3.3.cmml">e</mi><mo id="S3.SS2.SSS1.p1.2.m1.1.1.2.3.1a" xref="S3.SS2.SSS1.p1.2.m1.1.1.2.3.1.cmml">⁢</mo><mi 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alttext="\boldsymbol{H}_{fuse}^{n}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.4.m3.1"><semantics id="S3.SS2.SSS1.p1.4.m3.1a"><msubsup id="S3.SS2.SSS1.p1.4.m3.1.1" xref="S3.SS2.SSS1.p1.4.m3.1.1.cmml"><mi id="S3.SS2.SSS1.p1.4.m3.1.1.2.2" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS1.p1.4.m3.1.1.2.3" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.cmml"><mi id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.2" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.2.cmml">f</mi><mo id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.3" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.3.cmml">u</mi><mo id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1a" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.4" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.4.cmml">s</mi><mo id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1b" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.5" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.5.cmml">e</mi></mrow><mi id="S3.SS2.SSS1.p1.4.m3.1.1.3" mathsize="90%" xref="S3.SS2.SSS1.p1.4.m3.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.4.m3.1b"><apply id="S3.SS2.SSS1.p1.4.m3.1.1.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p1.4.m3.1.1.1.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1">superscript</csymbol><apply id="S3.SS2.SSS1.p1.4.m3.1.1.2.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p1.4.m3.1.1.2.1.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p1.4.m3.1.1.2.2.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.2">𝑯</ci><apply id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3"><times id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.1"></times><ci id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.2.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.2">𝑓</ci><ci id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.3.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.3">𝑢</ci><ci id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.4.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.4">𝑠</ci><ci id="S3.SS2.SSS1.p1.4.m3.1.1.2.3.5.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.2.3.5">𝑒</ci></apply></apply><ci id="S3.SS2.SSS1.p1.4.m3.1.1.3.cmml" xref="S3.SS2.SSS1.p1.4.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.4.m3.1c">\boldsymbol{H}_{fuse}^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.4.m3.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> represent the outputs of the left branch, right branch, and the fusion operation, respectively, within the <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.5.m4.1"><semantics id="S3.SS2.SSS1.p1.5.m4.1a"><mi id="S3.SS2.SSS1.p1.5.m4.1.1" mathsize="90%" xref="S3.SS2.SSS1.p1.5.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.5.m4.1b"><ci id="S3.SS2.SSS1.p1.5.m4.1.1.cmml" xref="S3.SS2.SSS1.p1.5.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.5.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.5.m4.1d">italic_n</annotation></semantics></math>-th block. The functions <math alttext="f_{left}(\cdot)" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.6.m5.1"><semantics id="S3.SS2.SSS1.p1.6.m5.1a"><mrow id="S3.SS2.SSS1.p1.6.m5.1.2" xref="S3.SS2.SSS1.p1.6.m5.1.2.cmml"><msub id="S3.SS2.SSS1.p1.6.m5.1.2.2" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.cmml"><mi id="S3.SS2.SSS1.p1.6.m5.1.2.2.2" mathsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.2.cmml">f</mi><mrow id="S3.SS2.SSS1.p1.6.m5.1.2.2.3" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.cmml"><mi id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.2" mathsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.2.cmml">l</mi><mo id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.3" mathsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.3.cmml">e</mi><mo id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1a" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.4" mathsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.4.cmml">f</mi><mo id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1b" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.5" mathsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.5.cmml">t</mi></mrow></msub><mo id="S3.SS2.SSS1.p1.6.m5.1.2.1" xref="S3.SS2.SSS1.p1.6.m5.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS1.p1.6.m5.1.2.3.2" xref="S3.SS2.SSS1.p1.6.m5.1.2.cmml"><mo id="S3.SS2.SSS1.p1.6.m5.1.2.3.2.1" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.cmml">(</mo><mo id="S3.SS2.SSS1.p1.6.m5.1.1" lspace="0em" mathsize="90%" rspace="0em" xref="S3.SS2.SSS1.p1.6.m5.1.1.cmml">⋅</mo><mo id="S3.SS2.SSS1.p1.6.m5.1.2.3.2.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.6.m5.1b"><apply id="S3.SS2.SSS1.p1.6.m5.1.2.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2"><times id="S3.SS2.SSS1.p1.6.m5.1.2.1.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.1"></times><apply id="S3.SS2.SSS1.p1.6.m5.1.2.2.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p1.6.m5.1.2.2.1.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2">subscript</csymbol><ci id="S3.SS2.SSS1.p1.6.m5.1.2.2.2.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.2">𝑓</ci><apply id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3"><times id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.1"></times><ci id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.2.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.2">𝑙</ci><ci id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.3.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.3">𝑒</ci><ci id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.4.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.4">𝑓</ci><ci id="S3.SS2.SSS1.p1.6.m5.1.2.2.3.5.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.2.2.3.5">𝑡</ci></apply></apply><ci id="S3.SS2.SSS1.p1.6.m5.1.1.cmml" xref="S3.SS2.SSS1.p1.6.m5.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.6.m5.1c">f_{left}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.6.m5.1d">italic_f start_POSTSUBSCRIPT italic_l italic_e italic_f italic_t end_POSTSUBSCRIPT ( ⋅ )</annotation></semantics></math> and <math alttext="f_{right}(\cdot)" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.7.m6.1"><semantics id="S3.SS2.SSS1.p1.7.m6.1a"><mrow id="S3.SS2.SSS1.p1.7.m6.1.2" xref="S3.SS2.SSS1.p1.7.m6.1.2.cmml"><msub id="S3.SS2.SSS1.p1.7.m6.1.2.2" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.cmml"><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.2" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.2.cmml">f</mi><mrow id="S3.SS2.SSS1.p1.7.m6.1.2.2.3" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.cmml"><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.2" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.2.cmml">r</mi><mo id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.3" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.3.cmml">i</mi><mo id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1a" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.4" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.4.cmml">g</mi><mo id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1b" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.5" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.5.cmml">h</mi><mo id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1c" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.6" mathsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.6.cmml">t</mi></mrow></msub><mo id="S3.SS2.SSS1.p1.7.m6.1.2.1" xref="S3.SS2.SSS1.p1.7.m6.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS1.p1.7.m6.1.2.3.2" xref="S3.SS2.SSS1.p1.7.m6.1.2.cmml"><mo id="S3.SS2.SSS1.p1.7.m6.1.2.3.2.1" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.cmml">(</mo><mo id="S3.SS2.SSS1.p1.7.m6.1.1" lspace="0em" mathsize="90%" rspace="0em" xref="S3.SS2.SSS1.p1.7.m6.1.1.cmml">⋅</mo><mo id="S3.SS2.SSS1.p1.7.m6.1.2.3.2.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.7.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.7.m6.1b"><apply id="S3.SS2.SSS1.p1.7.m6.1.2.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2"><times id="S3.SS2.SSS1.p1.7.m6.1.2.1.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.1"></times><apply id="S3.SS2.SSS1.p1.7.m6.1.2.2.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p1.7.m6.1.2.2.1.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2">subscript</csymbol><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.2.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.2">𝑓</ci><apply id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3"><times id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.1"></times><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.2.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.2">𝑟</ci><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.3.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.3">𝑖</ci><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.4.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.4">𝑔</ci><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.5.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.5">ℎ</ci><ci id="S3.SS2.SSS1.p1.7.m6.1.2.2.3.6.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.2.2.3.6">𝑡</ci></apply></apply><ci id="S3.SS2.SSS1.p1.7.m6.1.1.cmml" xref="S3.SS2.SSS1.p1.7.m6.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.7.m6.1c">f_{right}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.7.m6.1d">italic_f start_POSTSUBSCRIPT italic_r italic_i italic_g italic_h italic_t end_POSTSUBSCRIPT ( ⋅ )</annotation></semantics></math> correspond to the transformations in the two branches, while <math alttext="\text{LN}(\cdot)" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p1.8.m7.1"><semantics id="S3.SS2.SSS1.p1.8.m7.1a"><mrow id="S3.SS2.SSS1.p1.8.m7.1.2" xref="S3.SS2.SSS1.p1.8.m7.1.2.cmml"><mtext id="S3.SS2.SSS1.p1.8.m7.1.2.2" mathsize="90%" xref="S3.SS2.SSS1.p1.8.m7.1.2.2a.cmml">LN</mtext><mo id="S3.SS2.SSS1.p1.8.m7.1.2.1" xref="S3.SS2.SSS1.p1.8.m7.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS1.p1.8.m7.1.2.3.2" xref="S3.SS2.SSS1.p1.8.m7.1.2.cmml"><mo id="S3.SS2.SSS1.p1.8.m7.1.2.3.2.1" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.8.m7.1.2.cmml">(</mo><mo id="S3.SS2.SSS1.p1.8.m7.1.1" lspace="0em" mathsize="90%" rspace="0em" xref="S3.SS2.SSS1.p1.8.m7.1.1.cmml">⋅</mo><mo id="S3.SS2.SSS1.p1.8.m7.1.2.3.2.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS1.p1.8.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p1.8.m7.1b"><apply id="S3.SS2.SSS1.p1.8.m7.1.2.cmml" xref="S3.SS2.SSS1.p1.8.m7.1.2"><times id="S3.SS2.SSS1.p1.8.m7.1.2.1.cmml" xref="S3.SS2.SSS1.p1.8.m7.1.2.1"></times><ci id="S3.SS2.SSS1.p1.8.m7.1.2.2a.cmml" xref="S3.SS2.SSS1.p1.8.m7.1.2.2"><mtext id="S3.SS2.SSS1.p1.8.m7.1.2.2.cmml" mathsize="90%" xref="S3.SS2.SSS1.p1.8.m7.1.2.2">LN</mtext></ci><ci id="S3.SS2.SSS1.p1.8.m7.1.1.cmml" xref="S3.SS2.SSS1.p1.8.m7.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p1.8.m7.1c">\text{LN}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p1.8.m7.1d">LN ( ⋅ )</annotation></semantics></math> signifies the layer normalization, identical to the one used in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib41" title="">41</a>]</cite>. </div> <div class="ltx_para" id="S3.SS2.SSS1.p2"> Furthermore, a one-dimensional convolutional layer was employed to operate on the output of the fusion operation, enhancing the non-linearity of the entire block. In contrast to the use of fully connected layers in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib41" title="">41</a>]</cite>, we contend that the local correlations in speech features need to be emphasized. Thus, drawing inspiration from the operations described in FastSpeech <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib43" title="">43</a>]</cite>, convolutional layers were incorporated to capture these local correlations. Additionally, residual connections and layer normalization were utilized to facilitate more stable training of the model. The entire computational process is illustrated as follows: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S3.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{final}^{n}=\text{LN}((\text{ReLU}(\text{Conv}(% \boldsymbol{H}_{fuse}^{n}))+\boldsymbol{H}_{fuse}^{n})," class="ltx_math_unparsed" display="inline" id="S3.E4.m1.1"><semantics id="S3.E4.m1.1a"><mrow id="S3.E4.m1.1b"><msubsup id="S3.E4.m1.1.1"><mi id="S3.E4.m1.1.1.2.2" mathsize="90%">𝑯</mi><mrow id="S3.E4.m1.1.1.2.3"><mi id="S3.E4.m1.1.1.2.3.2" mathsize="90%">f</mi><mo id="S3.E4.m1.1.1.2.3.1">⁢</mo><mi id="S3.E4.m1.1.1.2.3.3" mathsize="90%">i</mi><mo id="S3.E4.m1.1.1.2.3.1a">⁢</mo><mi id="S3.E4.m1.1.1.2.3.4" mathsize="90%">n</mi><mo id="S3.E4.m1.1.1.2.3.1b">⁢</mo><mi id="S3.E4.m1.1.1.2.3.5" mathsize="90%">a</mi><mo id="S3.E4.m1.1.1.2.3.1c">⁢</mo><mi id="S3.E4.m1.1.1.2.3.6" mathsize="90%">l</mi></mrow><mi id="S3.E4.m1.1.1.3" mathsize="90%">n</mi></msubsup><mo id="S3.E4.m1.1.2" mathsize="90%">=</mo><mtext id="S3.E4.m1.1.3" mathsize="90%">LN</mtext><mrow id="S3.E4.m1.1.4"><mo id="S3.E4.m1.1.4.1" maxsize="90%" minsize="90%">(</mo><mrow id="S3.E4.m1.1.4.2"><mo id="S3.E4.m1.1.4.2.1" maxsize="90%" minsize="90%">(</mo><mtext id="S3.E4.m1.1.4.2.2" mathsize="90%">ReLU</mtext><mrow id="S3.E4.m1.1.4.2.3"><mo id="S3.E4.m1.1.4.2.3.1" maxsize="90%" minsize="90%">(</mo><mtext id="S3.E4.m1.1.4.2.3.2" mathsize="90%">Conv</mtext><mrow id="S3.E4.m1.1.4.2.3.3"><mo id="S3.E4.m1.1.4.2.3.3.1" maxsize="90%" minsize="90%">(</mo><msubsup id="S3.E4.m1.1.4.2.3.3.2"><mi id="S3.E4.m1.1.4.2.3.3.2.2.2" mathsize="90%">𝑯</mi><mrow id="S3.E4.m1.1.4.2.3.3.2.2.3"><mi id="S3.E4.m1.1.4.2.3.3.2.2.3.2" mathsize="90%">f</mi><mo id="S3.E4.m1.1.4.2.3.3.2.2.3.1">⁢</mo><mi id="S3.E4.m1.1.4.2.3.3.2.2.3.3" mathsize="90%">u</mi><mo id="S3.E4.m1.1.4.2.3.3.2.2.3.1a">⁢</mo><mi id="S3.E4.m1.1.4.2.3.3.2.2.3.4" mathsize="90%">s</mi><mo id="S3.E4.m1.1.4.2.3.3.2.2.3.1b">⁢</mo><mi id="S3.E4.m1.1.4.2.3.3.2.2.3.5" mathsize="90%">e</mi></mrow><mi id="S3.E4.m1.1.4.2.3.3.2.3" mathsize="90%">n</mi></msubsup><mo id="S3.E4.m1.1.4.2.3.3.3" maxsize="90%" minsize="90%">)</mo></mrow><mo id="S3.E4.m1.1.4.2.3.4" maxsize="90%" minsize="90%">)</mo></mrow><mo id="S3.E4.m1.1.4.2.4" mathsize="90%">+</mo><msubsup id="S3.E4.m1.1.4.2.5"><mi id="S3.E4.m1.1.4.2.5.2.2" mathsize="90%">𝑯</mi><mrow id="S3.E4.m1.1.4.2.5.2.3"><mi id="S3.E4.m1.1.4.2.5.2.3.2" mathsize="90%">f</mi><mo id="S3.E4.m1.1.4.2.5.2.3.1">⁢</mo><mi id="S3.E4.m1.1.4.2.5.2.3.3" mathsize="90%">u</mi><mo id="S3.E4.m1.1.4.2.5.2.3.1a">⁢</mo><mi id="S3.E4.m1.1.4.2.5.2.3.4" mathsize="90%">s</mi><mo id="S3.E4.m1.1.4.2.5.2.3.1b">⁢</mo><mi id="S3.E4.m1.1.4.2.5.2.3.5" mathsize="90%">e</mi></mrow><mi id="S3.E4.m1.1.4.2.5.3" mathsize="90%">n</mi></msubsup><mo id="S3.E4.m1.1.4.2.6" maxsize="90%" minsize="90%">)</mo></mrow><mo id="S3.E4.m1.1.4.3" mathsize="90%">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.E4.m1.1c">\displaystyle\boldsymbol{H}_{final}^{n}=\text{LN}((\text{ReLU}(\text{Conv}(% \boldsymbol{H}_{fuse}^{n}))+\boldsymbol{H}_{fuse}^{n}),</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_i italic_n italic_a italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = LN ( ( ReLU ( Conv ( bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ) + bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n 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">(4)</td> </tr></tbody> </table> where <math alttext="\boldsymbol{H}_{final}^{n}" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p2.1.m1.1"><semantics id="S3.SS2.SSS1.p2.1.m1.1a"><msubsup id="S3.SS2.SSS1.p2.1.m1.1.1" xref="S3.SS2.SSS1.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.2" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS1.p2.1.m1.1.1.2.3" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.cmml"><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.2" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.2.cmml">f</mi><mo id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.3" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.3.cmml">i</mi><mo id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1a" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.4" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.4.cmml">n</mi><mo id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1b" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.5" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.5.cmml">a</mi><mo id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1c" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.6" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.3.6.cmml">l</mi></mrow><mi id="S3.SS2.SSS1.p2.1.m1.1.1.3" mathsize="90%" xref="S3.SS2.SSS1.p2.1.m1.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p2.1.m1.1b"><apply id="S3.SS2.SSS1.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1">superscript</csymbol><apply id="S3.SS2.SSS1.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS1.p2.1.m1.1.1.2.1.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS1.p2.1.m1.1.1.2.2.cmml" xref="S3.SS2.SSS1.p2.1.m1.1.1.2.2">𝑯</ci><apply id="S3.SS2.SSS1.p2.1.m1.1.1.2.3.cmml" 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end_POSTSUPERSCRIPT</annotation></semantics></math> represents the final output of the <math alttext="n" class="ltx_Math" display="inline" id="S3.SS2.SSS1.p2.2.m2.1"><semantics id="S3.SS2.SSS1.p2.2.m2.1a"><mi id="S3.SS2.SSS1.p2.2.m2.1.1" mathsize="90%" xref="S3.SS2.SSS1.p2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS1.p2.2.m2.1b"><ci id="S3.SS2.SSS1.p2.2.m2.1.1.cmml" xref="S3.SS2.SSS1.p2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS1.p2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS1.p2.2.m2.1d">italic_n</annotation></semantics></math>-th block. </div> </section> <section class="ltx_subsubsection" id="S3.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 3.2.2 Speaker classifier and spoof classifier</h4> <div class="ltx_para" id="S3.SS2.SSS2.p1"> The architectures of both the speaker classifier and the spoof classifier in the model are nearly identical, differing only in the output nodes of the final fully connected layers, which are associated with the number of classes. Drawing inspiration from the successful utilization of multi-layer feature aggregation as seen in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib7" title="">7</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib44" title="">44</a>]</cite>, we have also incorporated this strategy to aggregate the output of each left branch and each right branch for the anti-spoofing and ASV sub-objectives, respectively. This aggregation can be expressed as follows: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S3.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{spoof}" class="ltx_Math" display="inline" id="S3.E5.m1.1"><semantics id="S3.E5.m1.1a"><msub id="S3.E5.m1.1.1" xref="S3.E5.m1.1.1.cmml"><mi id="S3.E5.m1.1.1.2" mathsize="90%" xref="S3.E5.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.E5.m1.1.1.3" xref="S3.E5.m1.1.1.3.cmml"><mi id="S3.E5.m1.1.1.3.2" mathsize="90%" xref="S3.E5.m1.1.1.3.2.cmml">s</mi><mo id="S3.E5.m1.1.1.3.1" xref="S3.E5.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E5.m1.1.1.3.3" mathsize="90%" xref="S3.E5.m1.1.1.3.3.cmml">p</mi><mo id="S3.E5.m1.1.1.3.1a" xref="S3.E5.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E5.m1.1.1.3.4" mathsize="90%" xref="S3.E5.m1.1.1.3.4.cmml">o</mi><mo id="S3.E5.m1.1.1.3.1b" 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encoding="application/x-tex" id="S3.E5.m1.1c">\displaystyle\boldsymbol{H}_{spoof}</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{H}_{left}^{n}," class="ltx_Math" display="inline" id="S3.E5.m2.1"><semantics id="S3.E5.m2.1a"><mrow id="S3.E5.m2.1.1.1" xref="S3.E5.m2.1.1.1.1.cmml"><mrow id="S3.E5.m2.1.1.1.1" xref="S3.E5.m2.1.1.1.1.cmml"><mi id="S3.E5.m2.1.1.1.1.2" xref="S3.E5.m2.1.1.1.1.2.cmml"></mi><mo id="S3.E5.m2.1.1.1.1.1" mathsize="90%" xref="S3.E5.m2.1.1.1.1.1.cmml">=</mo><mrow id="S3.E5.m2.1.1.1.1.3" xref="S3.E5.m2.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S3.E5.m2.1.1.1.1.3.2" xref="S3.E5.m2.1.1.1.1.3.2.cmml"><mfrac id="S3.E5.m2.1.1.1.1.3.2a" xref="S3.E5.m2.1.1.1.1.3.2.cmml"><mn id="S3.E5.m2.1.1.1.1.3.2.2" 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xref="S3.E5.m2.1.1.1.1.3.3.2.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m2.1c">\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{H}_{left}^{n},</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m2.1d">= divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT bold_italic_H start_POSTSUBSCRIPT italic_l italic_e italic_f italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n 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">(5)</td> </tr></tbody> <tbody id="S3.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{asv}" class="ltx_Math" display="inline" id="S3.E6.m1.1"><semantics id="S3.E6.m1.1a"><msub id="S3.E6.m1.1.1" xref="S3.E6.m1.1.1.cmml"><mi id="S3.E6.m1.1.1.2" mathsize="90%" xref="S3.E6.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.E6.m1.1.1.3" xref="S3.E6.m1.1.1.3.cmml"><mi id="S3.E6.m1.1.1.3.2" mathsize="90%" xref="S3.E6.m1.1.1.3.2.cmml">a</mi><mo id="S3.E6.m1.1.1.3.1" xref="S3.E6.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E6.m1.1.1.3.3" mathsize="90%" xref="S3.E6.m1.1.1.3.3.cmml">s</mi><mo id="S3.E6.m1.1.1.3.1a" xref="S3.E6.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E6.m1.1.1.3.4" mathsize="90%" xref="S3.E6.m1.1.1.3.4.cmml">v</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.E6.m1.1b"><apply id="S3.E6.m1.1.1.cmml" xref="S3.E6.m1.1.1"><csymbol cd="ambiguous" id="S3.E6.m1.1.1.1.cmml" xref="S3.E6.m1.1.1">subscript</csymbol><ci id="S3.E6.m1.1.1.2.cmml" xref="S3.E6.m1.1.1.2">𝑯</ci><apply id="S3.E6.m1.1.1.3.cmml" xref="S3.E6.m1.1.1.3"><times id="S3.E6.m1.1.1.3.1.cmml" xref="S3.E6.m1.1.1.3.1"></times><ci id="S3.E6.m1.1.1.3.2.cmml" xref="S3.E6.m1.1.1.3.2">𝑎</ci><ci id="S3.E6.m1.1.1.3.3.cmml" xref="S3.E6.m1.1.1.3.3">𝑠</ci><ci id="S3.E6.m1.1.1.3.4.cmml" xref="S3.E6.m1.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.1c">\displaystyle\boldsymbol{H}_{asv}</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{H}_{right}^{n}," class="ltx_Math" display="inline" id="S3.E6.m2.1"><semantics id="S3.E6.m2.1a"><mrow id="S3.E6.m2.1.1.1" xref="S3.E6.m2.1.1.1.1.cmml"><mrow id="S3.E6.m2.1.1.1.1" xref="S3.E6.m2.1.1.1.1.cmml"><mi id="S3.E6.m2.1.1.1.1.2" xref="S3.E6.m2.1.1.1.1.2.cmml"></mi><mo id="S3.E6.m2.1.1.1.1.1" mathsize="90%" 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start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT bold_italic_H start_POSTSUBSCRIPT italic_r italic_i italic_g italic_h italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n 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">(6)</td> </tr></tbody> </table> Here, <math alttext="\boldsymbol{H}_{asv}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p1.1.m1.1"><semantics id="S3.SS2.SSS2.p1.1.m1.1a"><msub id="S3.SS2.SSS2.p1.1.m1.1.1" xref="S3.SS2.SSS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS2.p1.1.m1.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p1.1.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS2.p1.1.m1.1.1.3" xref="S3.SS2.SSS2.p1.1.m1.1.1.3.cmml"><mi id="S3.SS2.SSS2.p1.1.m1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p1.1.m1.1.1.3.2.cmml">a</mi><mo id="S3.SS2.SSS2.p1.1.m1.1.1.3.1" xref="S3.SS2.SSS2.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi 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xref="S3.SS2.SSS2.p1.1.m1.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p1.1.m1.1c">\boldsymbol{H}_{asv}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p1.1.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{H}_{spoof}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p1.2.m2.1"><semantics id="S3.SS2.SSS2.p1.2.m2.1a"><msub id="S3.SS2.SSS2.p1.2.m2.1.1" xref="S3.SS2.SSS2.p1.2.m2.1.1.cmml"><mi id="S3.SS2.SSS2.p1.2.m2.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS2.p1.2.m2.1.1.3" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.cmml"><mi id="S3.SS2.SSS2.p1.2.m2.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.2.cmml">s</mi><mo id="S3.SS2.SSS2.p1.2.m2.1.1.3.1" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p1.2.m2.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.3.cmml">p</mi><mo id="S3.SS2.SSS2.p1.2.m2.1.1.3.1a" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p1.2.m2.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.4.cmml">o</mi><mo id="S3.SS2.SSS2.p1.2.m2.1.1.3.1b" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p1.2.m2.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.5.cmml">o</mi><mo id="S3.SS2.SSS2.p1.2.m2.1.1.3.1c" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p1.2.m2.1.1.3.6" mathsize="90%" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.6.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p1.2.m2.1b"><apply id="S3.SS2.SSS2.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p1.2.m2.1.1.2.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.2">𝑯</ci><apply id="S3.SS2.SSS2.p1.2.m2.1.1.3.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3"><times id="S3.SS2.SSS2.p1.2.m2.1.1.3.1.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.1"></times><ci id="S3.SS2.SSS2.p1.2.m2.1.1.3.2.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.2">𝑠</ci><ci id="S3.SS2.SSS2.p1.2.m2.1.1.3.3.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.3">𝑝</ci><ci id="S3.SS2.SSS2.p1.2.m2.1.1.3.4.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.4">𝑜</ci><ci id="S3.SS2.SSS2.p1.2.m2.1.1.3.5.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.5">𝑜</ci><ci id="S3.SS2.SSS2.p1.2.m2.1.1.3.6.cmml" xref="S3.SS2.SSS2.p1.2.m2.1.1.3.6">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p1.2.m2.1c">\boldsymbol{H}_{spoof}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p1.2.m2.1d">bold_italic_H start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT</annotation></semantics></math> represent the aggregated output for the ASV and anti-spoofing tasks, respectively. </div> <div class="ltx_para" id="S3.SS2.SSS2.p2"> Both of the aggregated features undergo transformation through a one-dimensional convolutional layer followed by an average pooling layer. Subsequently, the classification probabilities are computed based on the outputs <math alttext="\boldsymbol{e}_{asv}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.1.m1.1"><semantics id="S3.SS2.SSS2.p2.1.m1.1a"><msub id="S3.SS2.SSS2.p2.1.m1.1.1" xref="S3.SS2.SSS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS2.p2.1.m1.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.1.m1.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS2.p2.1.m1.1.1.3" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.1.m1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.2.cmml">a</mi><mo id="S3.SS2.SSS2.p2.1.m1.1.1.3.1" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.1.m1.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.3.cmml">s</mi><mo id="S3.SS2.SSS2.p2.1.m1.1.1.3.1a" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.1.m1.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.4.cmml">v</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.1.m1.1b"><apply id="S3.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.2">𝒆</ci><apply id="S3.SS2.SSS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.3"><times id="S3.SS2.SSS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.1"></times><ci id="S3.SS2.SSS2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.2">𝑎</ci><ci id="S3.SS2.SSS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.3">𝑠</ci><ci id="S3.SS2.SSS2.p2.1.m1.1.1.3.4.cmml" xref="S3.SS2.SSS2.p2.1.m1.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.1.m1.1c">\boldsymbol{e}_{asv}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.1.m1.1d">bold_italic_e start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{e}_{spoof}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.2.m2.1"><semantics id="S3.SS2.SSS2.p2.2.m2.1a"><msub id="S3.SS2.SSS2.p2.2.m2.1.1" xref="S3.SS2.SSS2.p2.2.m2.1.1.cmml"><mi id="S3.SS2.SSS2.p2.2.m2.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS2.p2.2.m2.1.1.3" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.2.m2.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.2.cmml">s</mi><mo id="S3.SS2.SSS2.p2.2.m2.1.1.3.1" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.2.m2.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.3.cmml">p</mi><mo id="S3.SS2.SSS2.p2.2.m2.1.1.3.1a" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.2.m2.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.4.cmml">o</mi><mo id="S3.SS2.SSS2.p2.2.m2.1.1.3.1b" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.2.m2.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.5.cmml">o</mi><mo id="S3.SS2.SSS2.p2.2.m2.1.1.3.1c" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.2.m2.1.1.3.6" mathsize="90%" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.6.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.2.m2.1b"><apply id="S3.SS2.SSS2.p2.2.m2.1.1.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.2">𝒆</ci><apply id="S3.SS2.SSS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3"><times id="S3.SS2.SSS2.p2.2.m2.1.1.3.1.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.1"></times><ci id="S3.SS2.SSS2.p2.2.m2.1.1.3.2.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.2">𝑠</ci><ci id="S3.SS2.SSS2.p2.2.m2.1.1.3.3.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.3">𝑝</ci><ci id="S3.SS2.SSS2.p2.2.m2.1.1.3.4.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.4">𝑜</ci><ci id="S3.SS2.SSS2.p2.2.m2.1.1.3.5.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.5">𝑜</ci><ci id="S3.SS2.SSS2.p2.2.m2.1.1.3.6.cmml" xref="S3.SS2.SSS2.p2.2.m2.1.1.3.6">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.2.m2.1c">\boldsymbol{e}_{spoof}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.2.m2.1d">bold_italic_e start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT</annotation></semantics></math> from the pooling layers. This computation can be expressed as follows: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx4"> <tbody id="S3.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P(\boldsymbol{e}_{spoof})" class="ltx_Math" display="inline" id="S3.E7.m1.1"><semantics id="S3.E7.m1.1a"><mrow id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml"><mi id="S3.E7.m1.1.1.3" mathsize="90%" xref="S3.E7.m1.1.1.3.cmml">P</mi><mo id="S3.E7.m1.1.1.2" xref="S3.E7.m1.1.1.2.cmml">⁢</mo><mrow id="S3.E7.m1.1.1.1.1" xref="S3.E7.m1.1.1.1.1.1.cmml"><mo id="S3.E7.m1.1.1.1.1.2" maxsize="90%" minsize="90%" xref="S3.E7.m1.1.1.1.1.1.cmml">(</mo><msub id="S3.E7.m1.1.1.1.1.1" xref="S3.E7.m1.1.1.1.1.1.cmml"><mi id="S3.E7.m1.1.1.1.1.1.2" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.2.cmml">𝒆</mi><mrow id="S3.E7.m1.1.1.1.1.1.3" xref="S3.E7.m1.1.1.1.1.1.3.cmml"><mi id="S3.E7.m1.1.1.1.1.1.3.2" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S3.E7.m1.1.1.1.1.1.3.1" xref="S3.E7.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.1.1.1.3.3" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.3.3.cmml">p</mi><mo id="S3.E7.m1.1.1.1.1.1.3.1a" xref="S3.E7.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.1.1.1.3.4" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.3.4.cmml">o</mi><mo id="S3.E7.m1.1.1.1.1.1.3.1b" xref="S3.E7.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.1.1.1.3.5" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.3.5.cmml">o</mi><mo id="S3.E7.m1.1.1.1.1.1.3.1c" xref="S3.E7.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E7.m1.1.1.1.1.1.3.6" mathsize="90%" xref="S3.E7.m1.1.1.1.1.1.3.6.cmml">f</mi></mrow></msub><mo id="S3.E7.m1.1.1.1.1.3" maxsize="90%" minsize="90%" xref="S3.E7.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m1.1b"><apply id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.1.1"><times id="S3.E7.m1.1.1.2.cmml" xref="S3.E7.m1.1.1.2"></times><ci id="S3.E7.m1.1.1.3.cmml" xref="S3.E7.m1.1.1.3">𝑃</ci><apply id="S3.E7.m1.1.1.1.1.1.cmml" xref="S3.E7.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E7.m1.1.1.1.1.1.1.cmml" xref="S3.E7.m1.1.1.1.1">subscript</csymbol><ci id="S3.E7.m1.1.1.1.1.1.2.cmml" xref="S3.E7.m1.1.1.1.1.1.2">𝒆</ci><apply id="S3.E7.m1.1.1.1.1.1.3.cmml" xref="S3.E7.m1.1.1.1.1.1.3"><times id="S3.E7.m1.1.1.1.1.1.3.1.cmml" xref="S3.E7.m1.1.1.1.1.1.3.1"></times><ci id="S3.E7.m1.1.1.1.1.1.3.2.cmml" xref="S3.E7.m1.1.1.1.1.1.3.2">𝑠</ci><ci id="S3.E7.m1.1.1.1.1.1.3.3.cmml" xref="S3.E7.m1.1.1.1.1.1.3.3">𝑝</ci><ci id="S3.E7.m1.1.1.1.1.1.3.4.cmml" xref="S3.E7.m1.1.1.1.1.1.3.4">𝑜</ci><ci id="S3.E7.m1.1.1.1.1.1.3.5.cmml" xref="S3.E7.m1.1.1.1.1.1.3.5">𝑜</ci><ci id="S3.E7.m1.1.1.1.1.1.3.6.cmml" xref="S3.E7.m1.1.1.1.1.1.3.6">𝑓</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.1c">\displaystyle P(\boldsymbol{e}_{spoof})</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.1d">italic_P ( bold_italic_e start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{1+\exp(-\boldsymbol{e}_{spoof}^{\top}\boldsymbol{\theta% }_{spoof})}," class="ltx_Math" display="inline" id="S3.E7.m2.3"><semantics id="S3.E7.m2.3a"><mrow id="S3.E7.m2.3.3.1" xref="S3.E7.m2.3.3.1.1.cmml"><mrow id="S3.E7.m2.3.3.1.1" xref="S3.E7.m2.3.3.1.1.cmml"><mi id="S3.E7.m2.3.3.1.1.2" xref="S3.E7.m2.3.3.1.1.2.cmml"></mi><mo id="S3.E7.m2.3.3.1.1.1" mathsize="90%" xref="S3.E7.m2.3.3.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E7.m2.2.2" xref="S3.E7.m2.2.2.cmml"><mfrac id="S3.E7.m2.2.2a" xref="S3.E7.m2.2.2.cmml"><mn id="S3.E7.m2.2.2.4" mathsize="90%" xref="S3.E7.m2.2.2.4.cmml">1</mn><mrow id="S3.E7.m2.2.2.2" xref="S3.E7.m2.2.2.2.cmml"><mn id="S3.E7.m2.2.2.2.4" mathsize="90%" 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xref="S3.E7.m2.2.2.2.2.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.E7.m2.2.2.2.2.1.1.1.2.3.1.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3">subscript</csymbol><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.2.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.2">𝜽</ci><apply id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3"><times id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.1.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.1"></times><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.2.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.2">𝑠</ci><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.3.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.3">𝑝</ci><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.4.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.4">𝑜</ci><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.5.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.5">𝑜</ci><ci id="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.6.cmml" xref="S3.E7.m2.2.2.2.2.1.1.1.2.3.3.6">𝑓</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m2.3c">\displaystyle=\frac{1}{1+\exp(-\boldsymbol{e}_{spoof}^{\top}\boldsymbol{\theta% }_{spoof})},</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m2.3d">= divide start_ARG 1 end_ARG start_ARG 1 + roman_exp ( - bold_italic_e start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_italic_θ start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT ) 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">(7)</td> </tr></tbody> <tbody id="S3.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{spoof}" class="ltx_Math" display="inline" id="S3.Ex1.m1.1"><semantics id="S3.Ex1.m1.1a"><msub id="S3.Ex1.m1.1.1" xref="S3.Ex1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex1.m1.1.1.2" mathsize="90%" xref="S3.Ex1.m1.1.1.2.cmml">ℒ</mi><mrow id="S3.Ex1.m1.1.1.3" xref="S3.Ex1.m1.1.1.3.cmml"><mi id="S3.Ex1.m1.1.1.3.2" mathsize="90%" xref="S3.Ex1.m1.1.1.3.2.cmml">s</mi><mo id="S3.Ex1.m1.1.1.3.1" xref="S3.Ex1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3" mathsize="90%" xref="S3.Ex1.m1.1.1.3.3.cmml">p</mi><mo id="S3.Ex1.m1.1.1.3.1a" xref="S3.Ex1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.4" mathsize="90%" xref="S3.Ex1.m1.1.1.3.4.cmml">o</mi><mo id="S3.Ex1.m1.1.1.3.1b" xref="S3.Ex1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.5" mathsize="90%" xref="S3.Ex1.m1.1.1.3.5.cmml">o</mi><mo id="S3.Ex1.m1.1.1.3.1c" xref="S3.Ex1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.6" mathsize="90%" xref="S3.Ex1.m1.1.1.3.6.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.1b"><apply id="S3.Ex1.m1.1.1.cmml" xref="S3.Ex1.m1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.cmml" xref="S3.Ex1.m1.1.1">subscript</csymbol><ci id="S3.Ex1.m1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.2">ℒ</ci><apply id="S3.Ex1.m1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.3"><times id="S3.Ex1.m1.1.1.3.1.cmml" xref="S3.Ex1.m1.1.1.3.1"></times><ci id="S3.Ex1.m1.1.1.3.2.cmml" xref="S3.Ex1.m1.1.1.3.2">𝑠</ci><ci id="S3.Ex1.m1.1.1.3.3.cmml" xref="S3.Ex1.m1.1.1.3.3">𝑝</ci><ci id="S3.Ex1.m1.1.1.3.4.cmml" xref="S3.Ex1.m1.1.1.3.4">𝑜</ci><ci id="S3.Ex1.m1.1.1.3.5.cmml" xref="S3.Ex1.m1.1.1.3.5">𝑜</ci><ci id="S3.Ex1.m1.1.1.3.6.cmml" xref="S3.Ex1.m1.1.1.3.6">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.1c">\displaystyle\mathcal{L}_{spoof}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.1d">caligraphic_L start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{M}\sum_{m}^{M}[\mathcal{I}(y^{m}_{spoof}=1)\log P(% \boldsymbol{e}^{m}_{spoof})" class="ltx_math_unparsed" display="inline" id="S3.Ex1.m2.1"><semantics id="S3.Ex1.m2.1a"><mrow id="S3.Ex1.m2.1b"><mo id="S3.Ex1.m2.1.1" mathsize="90%" rspace="0em">=</mo><mo id="S3.Ex1.m2.1.2" lspace="0em" mathsize="90%">−</mo><mstyle displaystyle="true" id="S3.Ex1.m2.1.3"><mfrac id="S3.Ex1.m2.1.3a"><mn id="S3.Ex1.m2.1.3.2" mathsize="90%">1</mn><mi id="S3.Ex1.m2.1.3.3" mathsize="90%">M</mi></mfrac></mstyle><mstyle displaystyle="true" id="S3.Ex1.m2.1.4"><munderover id="S3.Ex1.m2.1.4a"><mo id="S3.Ex1.m2.1.4.2.2" maxsize="90%" minsize="90%" movablelimits="false" stretchy="true">∑</mo><mi id="S3.Ex1.m2.1.4.2.3" mathsize="90%">m</mi><mi id="S3.Ex1.m2.1.4.3" mathsize="90%">M</mi></munderover></mstyle><mrow id="S3.Ex1.m2.1.5"><mo id="S3.Ex1.m2.1.5.1" maxsize="90%" minsize="90%">[</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex1.m2.1.5.2" mathsize="90%">ℐ</mi><mrow id="S3.Ex1.m2.1.5.3"><mo id="S3.Ex1.m2.1.5.3.1" maxsize="90%" minsize="90%">(</mo><msubsup id="S3.Ex1.m2.1.5.3.2"><mi id="S3.Ex1.m2.1.5.3.2.2.2" mathsize="90%">y</mi><mrow id="S3.Ex1.m2.1.5.3.2.3"><mi id="S3.Ex1.m2.1.5.3.2.3.2" mathsize="90%">s</mi><mo id="S3.Ex1.m2.1.5.3.2.3.1">⁢</mo><mi id="S3.Ex1.m2.1.5.3.2.3.3" mathsize="90%">p</mi><mo id="S3.Ex1.m2.1.5.3.2.3.1a">⁢</mo><mi id="S3.Ex1.m2.1.5.3.2.3.4" mathsize="90%">o</mi><mo id="S3.Ex1.m2.1.5.3.2.3.1b">⁢</mo><mi id="S3.Ex1.m2.1.5.3.2.3.5" mathsize="90%">o</mi><mo id="S3.Ex1.m2.1.5.3.2.3.1c">⁢</mo><mi id="S3.Ex1.m2.1.5.3.2.3.6" mathsize="90%">f</mi></mrow><mi id="S3.Ex1.m2.1.5.3.2.2.3" mathsize="90%">m</mi></msubsup><mo id="S3.Ex1.m2.1.5.3.3" mathsize="90%">=</mo><mn id="S3.Ex1.m2.1.5.3.4" mathsize="90%">1</mn><mo id="S3.Ex1.m2.1.5.3.5" maxsize="90%" minsize="90%" rspace="0.167em">)</mo></mrow><mi id="S3.Ex1.m2.1.5.4" mathsize="90%">log</mi><mi id="S3.Ex1.m2.1.5.5" mathsize="90%">P</mi><mrow id="S3.Ex1.m2.1.5.6"><mo id="S3.Ex1.m2.1.5.6.1" maxsize="90%" minsize="90%">(</mo><msubsup id="S3.Ex1.m2.1.5.6.2"><mi id="S3.Ex1.m2.1.5.6.2.2.2" mathsize="90%">𝒆</mi><mrow id="S3.Ex1.m2.1.5.6.2.3"><mi id="S3.Ex1.m2.1.5.6.2.3.2" mathsize="90%">s</mi><mo id="S3.Ex1.m2.1.5.6.2.3.1">⁢</mo><mi id="S3.Ex1.m2.1.5.6.2.3.3" mathsize="90%">p</mi><mo id="S3.Ex1.m2.1.5.6.2.3.1a">⁢</mo><mi id="S3.Ex1.m2.1.5.6.2.3.4" mathsize="90%">o</mi><mo id="S3.Ex1.m2.1.5.6.2.3.1b">⁢</mo><mi id="S3.Ex1.m2.1.5.6.2.3.5" mathsize="90%">o</mi><mo id="S3.Ex1.m2.1.5.6.2.3.1c">⁢</mo><mi id="S3.Ex1.m2.1.5.6.2.3.6" mathsize="90%">f</mi></mrow><mi id="S3.Ex1.m2.1.5.6.2.2.3" mathsize="90%">m</mi></msubsup><mo id="S3.Ex1.m2.1.5.6.3" maxsize="90%" minsize="90%">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S3.Ex1.m2.1c">\displaystyle=-\frac{1}{M}\sum_{m}^{M}[\mathcal{I}(y^{m}_{spoof}=1)\log P(% \boldsymbol{e}^{m}_{spoof})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m2.1d">= - divide start_ARG 1 end_ARG start_ARG italic_M end_ARG ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT [ caligraphic_I ( italic_y start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT = 1 ) roman_log italic_P ( bold_italic_e start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\mathcal{I}(y^{m}_{spoof}\neq 1)\log(1-P(\boldsymbol{e}^{m}_{% spoof}))]," class="ltx_math_unparsed" display="inline" id="S3.E8.m1.1"><semantics id="S3.E8.m1.1a"><mrow id="S3.E8.m1.1b"><mo id="S3.E8.m1.1.2" mathsize="90%">+</mo><mi class="ltx_font_mathcaligraphic" id="S3.E8.m1.1.3" mathsize="90%">ℐ</mi><mrow id="S3.E8.m1.1.4"><mo id="S3.E8.m1.1.4.1" maxsize="90%" minsize="90%">(</mo><msubsup id="S3.E8.m1.1.4.2"><mi id="S3.E8.m1.1.4.2.2.2" mathsize="90%">y</mi><mrow id="S3.E8.m1.1.4.2.3"><mi id="S3.E8.m1.1.4.2.3.2" mathsize="90%">s</mi><mo id="S3.E8.m1.1.4.2.3.1">⁢</mo><mi id="S3.E8.m1.1.4.2.3.3" mathsize="90%">p</mi><mo id="S3.E8.m1.1.4.2.3.1a">⁢</mo><mi id="S3.E8.m1.1.4.2.3.4" mathsize="90%">o</mi><mo id="S3.E8.m1.1.4.2.3.1b">⁢</mo><mi id="S3.E8.m1.1.4.2.3.5" mathsize="90%">o</mi><mo id="S3.E8.m1.1.4.2.3.1c">⁢</mo><mi id="S3.E8.m1.1.4.2.3.6" mathsize="90%">f</mi></mrow><mi id="S3.E8.m1.1.4.2.2.3" mathsize="90%">m</mi></msubsup><mo id="S3.E8.m1.1.4.3" mathsize="90%">≠</mo><mn id="S3.E8.m1.1.4.4" mathsize="90%">1</mn><mo id="S3.E8.m1.1.4.5" maxsize="90%" minsize="90%" rspace="0.167em">)</mo></mrow><mi id="S3.E8.m1.1.1" mathsize="90%">log</mi><mrow id="S3.E8.m1.1.5"><mo id="S3.E8.m1.1.5.1" maxsize="90%" minsize="90%">(</mo><mn id="S3.E8.m1.1.5.2" mathsize="90%">1</mn><mo id="S3.E8.m1.1.5.3" mathsize="90%">−</mo><mi id="S3.E8.m1.1.5.4" mathsize="90%">P</mi><mrow id="S3.E8.m1.1.5.5"><mo id="S3.E8.m1.1.5.5.1" maxsize="90%" minsize="90%">(</mo><msubsup id="S3.E8.m1.1.5.5.2"><mi id="S3.E8.m1.1.5.5.2.2.2" mathsize="90%">𝒆</mi><mrow id="S3.E8.m1.1.5.5.2.3"><mi id="S3.E8.m1.1.5.5.2.3.2" mathsize="90%">s</mi><mo id="S3.E8.m1.1.5.5.2.3.1">⁢</mo><mi id="S3.E8.m1.1.5.5.2.3.3" mathsize="90%">p</mi><mo id="S3.E8.m1.1.5.5.2.3.1a">⁢</mo><mi id="S3.E8.m1.1.5.5.2.3.4" mathsize="90%">o</mi><mo id="S3.E8.m1.1.5.5.2.3.1b">⁢</mo><mi id="S3.E8.m1.1.5.5.2.3.5" mathsize="90%">o</mi><mo id="S3.E8.m1.1.5.5.2.3.1c">⁢</mo><mi id="S3.E8.m1.1.5.5.2.3.6" mathsize="90%">f</mi></mrow><mi id="S3.E8.m1.1.5.5.2.2.3" mathsize="90%">m</mi></msubsup><mo id="S3.E8.m1.1.5.5.3" maxsize="90%" minsize="90%">)</mo></mrow><mo id="S3.E8.m1.1.5.6" maxsize="90%" minsize="90%">)</mo></mrow><mo id="S3.E8.m1.1.6" maxsize="90%" minsize="90%">]</mo><mo id="S3.E8.m1.1.7" mathsize="90%">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E8.m1.1c">\displaystyle+\mathcal{I}(y^{m}_{spoof}\neq 1)\log(1-P(\boldsymbol{e}^{m}_{% spoof}))],</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.1d">+ caligraphic_I ( italic_y start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT ≠ 1 ) roman_log ( 1 - italic_P ( bold_italic_e start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f 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">(8)</td> </tr></tbody> <tbody id="S3.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle P(\boldsymbol{e}_{asv},s_{i})" class="ltx_Math" display="inline" id="S3.E9.m1.2"><semantics id="S3.E9.m1.2a"><mrow id="S3.E9.m1.2.2" xref="S3.E9.m1.2.2.cmml"><mi id="S3.E9.m1.2.2.4" mathsize="90%" xref="S3.E9.m1.2.2.4.cmml">P</mi><mo id="S3.E9.m1.2.2.3" xref="S3.E9.m1.2.2.3.cmml">⁢</mo><mrow id="S3.E9.m1.2.2.2.2" xref="S3.E9.m1.2.2.2.3.cmml"><mo id="S3.E9.m1.2.2.2.2.3" maxsize="90%" minsize="90%" xref="S3.E9.m1.2.2.2.3.cmml">(</mo><msub id="S3.E9.m1.1.1.1.1.1" xref="S3.E9.m1.1.1.1.1.1.cmml"><mi id="S3.E9.m1.1.1.1.1.1.2" mathsize="90%" xref="S3.E9.m1.1.1.1.1.1.2.cmml">𝒆</mi><mrow id="S3.E9.m1.1.1.1.1.1.3" xref="S3.E9.m1.1.1.1.1.1.3.cmml"><mi id="S3.E9.m1.1.1.1.1.1.3.2" mathsize="90%" xref="S3.E9.m1.1.1.1.1.1.3.2.cmml">a</mi><mo id="S3.E9.m1.1.1.1.1.1.3.1" xref="S3.E9.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E9.m1.1.1.1.1.1.3.3" mathsize="90%" xref="S3.E9.m1.1.1.1.1.1.3.3.cmml">s</mi><mo id="S3.E9.m1.1.1.1.1.1.3.1a" xref="S3.E9.m1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.E9.m1.1.1.1.1.1.3.4" mathsize="90%" xref="S3.E9.m1.1.1.1.1.1.3.4.cmml">v</mi></mrow></msub><mo id="S3.E9.m1.2.2.2.2.4" mathsize="90%" xref="S3.E9.m1.2.2.2.3.cmml">,</mo><msub id="S3.E9.m1.2.2.2.2.2" xref="S3.E9.m1.2.2.2.2.2.cmml"><mi id="S3.E9.m1.2.2.2.2.2.2" mathsize="90%" xref="S3.E9.m1.2.2.2.2.2.2.cmml">s</mi><mi id="S3.E9.m1.2.2.2.2.2.3" mathsize="90%" xref="S3.E9.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.E9.m1.2.2.2.2.5" maxsize="90%" minsize="90%" xref="S3.E9.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E9.m1.2b"><apply id="S3.E9.m1.2.2.cmml" xref="S3.E9.m1.2.2"><times id="S3.E9.m1.2.2.3.cmml" xref="S3.E9.m1.2.2.3"></times><ci id="S3.E9.m1.2.2.4.cmml" xref="S3.E9.m1.2.2.4">𝑃</ci><interval closure="open" id="S3.E9.m1.2.2.2.3.cmml" xref="S3.E9.m1.2.2.2.2"><apply id="S3.E9.m1.1.1.1.1.1.cmml" xref="S3.E9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E9.m1.1.1.1.1.1.1.cmml" xref="S3.E9.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.E9.m1.1.1.1.1.1.2.cmml" xref="S3.E9.m1.1.1.1.1.1.2">𝒆</ci><apply id="S3.E9.m1.1.1.1.1.1.3.cmml" xref="S3.E9.m1.1.1.1.1.1.3"><times id="S3.E9.m1.1.1.1.1.1.3.1.cmml" xref="S3.E9.m1.1.1.1.1.1.3.1"></times><ci id="S3.E9.m1.1.1.1.1.1.3.2.cmml" xref="S3.E9.m1.1.1.1.1.1.3.2">𝑎</ci><ci id="S3.E9.m1.1.1.1.1.1.3.3.cmml" xref="S3.E9.m1.1.1.1.1.1.3.3">𝑠</ci><ci id="S3.E9.m1.1.1.1.1.1.3.4.cmml" xref="S3.E9.m1.1.1.1.1.1.3.4">𝑣</ci></apply></apply><apply id="S3.E9.m1.2.2.2.2.2.cmml" xref="S3.E9.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.E9.m1.2.2.2.2.2.1.cmml" xref="S3.E9.m1.2.2.2.2.2">subscript</csymbol><ci id="S3.E9.m1.2.2.2.2.2.2.cmml" xref="S3.E9.m1.2.2.2.2.2.2">𝑠</ci><ci id="S3.E9.m1.2.2.2.2.2.3.cmml" xref="S3.E9.m1.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9.m1.2c">\displaystyle P(\boldsymbol{e}_{asv},s_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.E9.m1.2d">italic_P ( bold_italic_e start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\exp(h_{\boldsymbol{\theta}_{asv}^{s_{i}}}(\boldsymbol{e}_% {asv}))}{\sum_{j=1}^{S}\exp(h_{\boldsymbol{\theta}_{asv}^{s_{j}}}(\boldsymbol{% e}_{asv}))}," class="ltx_Math" display="inline" id="S3.E9.m2.5"><semantics id="S3.E9.m2.5a"><mrow id="S3.E9.m2.5.5.1" xref="S3.E9.m2.5.5.1.1.cmml"><mrow id="S3.E9.m2.5.5.1.1" xref="S3.E9.m2.5.5.1.1.cmml"><mi id="S3.E9.m2.5.5.1.1.2" xref="S3.E9.m2.5.5.1.1.2.cmml"></mi><mo id="S3.E9.m2.5.5.1.1.1" mathsize="90%" xref="S3.E9.m2.5.5.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E9.m2.4.4" xref="S3.E9.m2.4.4.cmml"><mfrac id="S3.E9.m2.4.4a" xref="S3.E9.m2.4.4.cmml"><mrow 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end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT roman_exp ( italic_h start_POSTSUBSCRIPT bold_italic_θ start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( bold_italic_e start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT ) ) 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">(9)</td> </tr></tbody> <tbody id="S3.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{asv}" class="ltx_Math" display="inline" id="S3.E10.m1.1"><semantics id="S3.E10.m1.1a"><msub id="S3.E10.m1.1.1" xref="S3.E10.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E10.m1.1.1.2" mathsize="90%" xref="S3.E10.m1.1.1.2.cmml">ℒ</mi><mrow id="S3.E10.m1.1.1.3" xref="S3.E10.m1.1.1.3.cmml"><mi id="S3.E10.m1.1.1.3.2" mathsize="90%" xref="S3.E10.m1.1.1.3.2.cmml">a</mi><mo id="S3.E10.m1.1.1.3.1" xref="S3.E10.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E10.m1.1.1.3.3" mathsize="90%" xref="S3.E10.m1.1.1.3.3.cmml">s</mi><mo id="S3.E10.m1.1.1.3.1a" xref="S3.E10.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E10.m1.1.1.3.4" mathsize="90%" xref="S3.E10.m1.1.1.3.4.cmml">v</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.E10.m1.1b"><apply id="S3.E10.m1.1.1.cmml" xref="S3.E10.m1.1.1"><csymbol cd="ambiguous" id="S3.E10.m1.1.1.1.cmml" xref="S3.E10.m1.1.1">subscript</csymbol><ci id="S3.E10.m1.1.1.2.cmml" xref="S3.E10.m1.1.1.2">ℒ</ci><apply id="S3.E10.m1.1.1.3.cmml" xref="S3.E10.m1.1.1.3"><times id="S3.E10.m1.1.1.3.1.cmml" xref="S3.E10.m1.1.1.3.1"></times><ci id="S3.E10.m1.1.1.3.2.cmml" xref="S3.E10.m1.1.1.3.2">𝑎</ci><ci id="S3.E10.m1.1.1.3.3.cmml" xref="S3.E10.m1.1.1.3.3">𝑠</ci><ci id="S3.E10.m1.1.1.3.4.cmml" xref="S3.E10.m1.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.1c">\displaystyle\mathcal{L}_{asv}</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.1d">caligraphic_L start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{M}\sum_{m}^{M}\log P(\boldsymbol{e}_{asv}^{m},s_{i}^{m% })," class="ltx_Math" display="inline" id="S3.E10.m2.1"><semantics id="S3.E10.m2.1a"><mrow id="S3.E10.m2.1.1.1" xref="S3.E10.m2.1.1.1.1.cmml"><mrow id="S3.E10.m2.1.1.1.1" xref="S3.E10.m2.1.1.1.1.cmml"><mi id="S3.E10.m2.1.1.1.1.4" xref="S3.E10.m2.1.1.1.1.4.cmml"></mi><mo id="S3.E10.m2.1.1.1.1.3" mathsize="90%" xref="S3.E10.m2.1.1.1.1.3.cmml">=</mo><mrow id="S3.E10.m2.1.1.1.1.2" 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xref="S3.E10.m2.1.1.1.1.2.2.2.2.2.2.2.3">𝑚</ci></apply></interval></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m2.1c">\displaystyle=-\frac{1}{M}\sum_{m}^{M}\log P(\boldsymbol{e}_{asv}^{m},s_{i}^{m% }),</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m2.1d">= - divide start_ARG 1 end_ARG start_ARG italic_M end_ARG ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT roman_log italic_P ( bold_italic_e start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT , italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m 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">(10)</td> </tr></tbody> </table> where <math alttext="P(\boldsymbol{e}_{spoof})" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.3.m1.1"><semantics id="S3.SS2.SSS2.p2.3.m1.1a"><mrow id="S3.SS2.SSS2.p2.3.m1.1.1" xref="S3.SS2.SSS2.p2.3.m1.1.1.cmml"><mi id="S3.SS2.SSS2.p2.3.m1.1.1.3" mathsize="90%" xref="S3.SS2.SSS2.p2.3.m1.1.1.3.cmml">P</mi><mo id="S3.SS2.SSS2.p2.3.m1.1.1.2" xref="S3.SS2.SSS2.p2.3.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.SSS2.p2.3.m1.1.1.1.1" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.cmml"><mo id="S3.SS2.SSS2.p2.3.m1.1.1.1.1.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.cmml"><mi id="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.3" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.3.m1.1.1.1.1.1.3.2.cmml">s</mi><mo 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xref="S3.SS2.SSS2.p2.4.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.SSS2.p2.4.m2.1.1.1.1" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.cmml"><mo id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.cmml"><mi id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.2.cmml">a</mi><mo id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.3.cmml">s</mi><mo id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1a" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.4.cmml">v</mi></mrow></msub><mo id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.3" maxsize="90%" minsize="90%" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.4.m2.1b"><apply id="S3.SS2.SSS2.p2.4.m2.1.1.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1"><times id="S3.SS2.SSS2.p2.4.m2.1.1.2.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.2"></times><ci id="S3.SS2.SSS2.p2.4.m2.1.1.3.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.3">𝑃</ci><apply id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.2.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.2">𝒆</ci><apply id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3"><times id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.1"></times><ci id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.2.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.2">𝑎</ci><ci id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.3.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.3">𝑠</ci><ci id="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.4.cmml" xref="S3.SS2.SSS2.p2.4.m2.1.1.1.1.1.3.4">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.4.m2.1c">P(\boldsymbol{e}_{asv})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.4.m2.1d">italic_P ( bold_italic_e start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT )</annotation></semantics></math> represent the probabilities for the anti-spoofing and ASV tasks, respectively. <math alttext="\mathcal{I}(\cdot)" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.5.m3.1"><semantics id="S3.SS2.SSS2.p2.5.m3.1a"><mrow id="S3.SS2.SSS2.p2.5.m3.1.2" xref="S3.SS2.SSS2.p2.5.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.SSS2.p2.5.m3.1.2.2" mathsize="90%" xref="S3.SS2.SSS2.p2.5.m3.1.2.2.cmml">ℐ</mi><mo id="S3.SS2.SSS2.p2.5.m3.1.2.1" xref="S3.SS2.SSS2.p2.5.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.SSS2.p2.5.m3.1.2.3.2" xref="S3.SS2.SSS2.p2.5.m3.1.2.cmml"><mo id="S3.SS2.SSS2.p2.5.m3.1.2.3.2.1" maxsize="90%" minsize="90%" xref="S3.SS2.SSS2.p2.5.m3.1.2.cmml">(</mo><mo id="S3.SS2.SSS2.p2.5.m3.1.1" lspace="0em" mathsize="90%" rspace="0em" xref="S3.SS2.SSS2.p2.5.m3.1.1.cmml">⋅</mo><mo id="S3.SS2.SSS2.p2.5.m3.1.2.3.2.2" maxsize="90%" minsize="90%" xref="S3.SS2.SSS2.p2.5.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.5.m3.1b"><apply id="S3.SS2.SSS2.p2.5.m3.1.2.cmml" xref="S3.SS2.SSS2.p2.5.m3.1.2"><times id="S3.SS2.SSS2.p2.5.m3.1.2.1.cmml" xref="S3.SS2.SSS2.p2.5.m3.1.2.1"></times><ci id="S3.SS2.SSS2.p2.5.m3.1.2.2.cmml" xref="S3.SS2.SSS2.p2.5.m3.1.2.2">ℐ</ci><ci id="S3.SS2.SSS2.p2.5.m3.1.1.cmml" xref="S3.SS2.SSS2.p2.5.m3.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.5.m3.1c">\mathcal{I}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.5.m3.1d">caligraphic_I ( ⋅ )</annotation></semantics></math> is an indicator function that returns 1 if the condition is true and 0 otherwise. The variable <math alttext="S" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.6.m4.1"><semantics id="S3.SS2.SSS2.p2.6.m4.1a"><mi id="S3.SS2.SSS2.p2.6.m4.1.1" mathsize="90%" xref="S3.SS2.SSS2.p2.6.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.6.m4.1b"><ci id="S3.SS2.SSS2.p2.6.m4.1.1.cmml" xref="S3.SS2.SSS2.p2.6.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.6.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.6.m4.1d">italic_S</annotation></semantics></math> denotes the number of speakers in the training dataset, while <math alttext="M" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.7.m5.1"><semantics id="S3.SS2.SSS2.p2.7.m5.1a"><mi id="S3.SS2.SSS2.p2.7.m5.1.1" mathsize="90%" xref="S3.SS2.SSS2.p2.7.m5.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.7.m5.1b"><ci id="S3.SS2.SSS2.p2.7.m5.1.1.cmml" xref="S3.SS2.SSS2.p2.7.m5.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.7.m5.1c">M</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.7.m5.1d">italic_M</annotation></semantics></math> represents the number of training samples. <math alttext="s_{i}^{m}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.8.m6.1"><semantics id="S3.SS2.SSS2.p2.8.m6.1a"><msubsup id="S3.SS2.SSS2.p2.8.m6.1.1" xref="S3.SS2.SSS2.p2.8.m6.1.1.cmml"><mi id="S3.SS2.SSS2.p2.8.m6.1.1.2.2" mathsize="90%" xref="S3.SS2.SSS2.p2.8.m6.1.1.2.2.cmml">s</mi><mi id="S3.SS2.SSS2.p2.8.m6.1.1.2.3" mathsize="90%" xref="S3.SS2.SSS2.p2.8.m6.1.1.2.3.cmml">i</mi><mi id="S3.SS2.SSS2.p2.8.m6.1.1.3" mathsize="90%" xref="S3.SS2.SSS2.p2.8.m6.1.1.3.cmml">m</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.8.m6.1b"><apply id="S3.SS2.SSS2.p2.8.m6.1.1.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.8.m6.1.1.1.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1">superscript</csymbol><apply id="S3.SS2.SSS2.p2.8.m6.1.1.2.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.8.m6.1.1.2.1.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.8.m6.1.1.2.2.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1.2.2">𝑠</ci><ci id="S3.SS2.SSS2.p2.8.m6.1.1.2.3.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1.2.3">𝑖</ci></apply><ci id="S3.SS2.SSS2.p2.8.m6.1.1.3.cmml" xref="S3.SS2.SSS2.p2.8.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.8.m6.1c">s_{i}^{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.8.m6.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> signifies the <math alttext="m" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.9.m7.1"><semantics id="S3.SS2.SSS2.p2.9.m7.1a"><mi id="S3.SS2.SSS2.p2.9.m7.1.1" mathsize="90%" xref="S3.SS2.SSS2.p2.9.m7.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.9.m7.1b"><ci id="S3.SS2.SSS2.p2.9.m7.1.1.cmml" xref="S3.SS2.SSS2.p2.9.m7.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.9.m7.1c">m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.9.m7.1d">italic_m</annotation></semantics></math>-th training sample with the <math alttext="i" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.10.m8.1"><semantics id="S3.SS2.SSS2.p2.10.m8.1a"><mi id="S3.SS2.SSS2.p2.10.m8.1.1" mathsize="90%" xref="S3.SS2.SSS2.p2.10.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.10.m8.1b"><ci id="S3.SS2.SSS2.p2.10.m8.1.1.cmml" xref="S3.SS2.SSS2.p2.10.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.10.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.10.m8.1d">italic_i</annotation></semantics></math>-th speaker in the training dataset. Furthermore, <math alttext="\boldsymbol{\theta}_{spoof}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.11.m9.1"><semantics id="S3.SS2.SSS2.p2.11.m9.1a"><msub id="S3.SS2.SSS2.p2.11.m9.1.1" xref="S3.SS2.SSS2.p2.11.m9.1.1.cmml"><mi id="S3.SS2.SSS2.p2.11.m9.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.2.cmml">𝜽</mi><mrow id="S3.SS2.SSS2.p2.11.m9.1.1.3" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.11.m9.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.2.cmml">s</mi><mo id="S3.SS2.SSS2.p2.11.m9.1.1.3.1" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.11.m9.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.3.cmml">p</mi><mo id="S3.SS2.SSS2.p2.11.m9.1.1.3.1a" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.11.m9.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.4.cmml">o</mi><mo id="S3.SS2.SSS2.p2.11.m9.1.1.3.1b" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.11.m9.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.5.cmml">o</mi><mo id="S3.SS2.SSS2.p2.11.m9.1.1.3.1c" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.11.m9.1.1.3.6" mathsize="90%" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.6.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.11.m9.1b"><apply id="S3.SS2.SSS2.p2.11.m9.1.1.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.11.m9.1.1.1.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.11.m9.1.1.2.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.2">𝜽</ci><apply id="S3.SS2.SSS2.p2.11.m9.1.1.3.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3"><times id="S3.SS2.SSS2.p2.11.m9.1.1.3.1.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.1"></times><ci id="S3.SS2.SSS2.p2.11.m9.1.1.3.2.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.2">𝑠</ci><ci id="S3.SS2.SSS2.p2.11.m9.1.1.3.3.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.3">𝑝</ci><ci id="S3.SS2.SSS2.p2.11.m9.1.1.3.4.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.4">𝑜</ci><ci id="S3.SS2.SSS2.p2.11.m9.1.1.3.5.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.5">𝑜</ci><ci id="S3.SS2.SSS2.p2.11.m9.1.1.3.6.cmml" xref="S3.SS2.SSS2.p2.11.m9.1.1.3.6">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.11.m9.1c">\boldsymbol{\theta}_{spoof}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.11.m9.1d">bold_italic_θ start_POSTSUBSCRIPT italic_s italic_p italic_o italic_o italic_f end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\boldsymbol{\theta}_{asv}" class="ltx_Math" display="inline" id="S3.SS2.SSS2.p2.12.m10.1"><semantics id="S3.SS2.SSS2.p2.12.m10.1a"><msub id="S3.SS2.SSS2.p2.12.m10.1.1" xref="S3.SS2.SSS2.p2.12.m10.1.1.cmml"><mi id="S3.SS2.SSS2.p2.12.m10.1.1.2" mathsize="90%" xref="S3.SS2.SSS2.p2.12.m10.1.1.2.cmml">𝜽</mi><mrow id="S3.SS2.SSS2.p2.12.m10.1.1.3" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.cmml"><mi id="S3.SS2.SSS2.p2.12.m10.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.2.cmml">a</mi><mo id="S3.SS2.SSS2.p2.12.m10.1.1.3.1" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.12.m10.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.3.cmml">s</mi><mo id="S3.SS2.SSS2.p2.12.m10.1.1.3.1a" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS2.p2.12.m10.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.4.cmml">v</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS2.p2.12.m10.1b"><apply id="S3.SS2.SSS2.p2.12.m10.1.1.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS2.p2.12.m10.1.1.1.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1">subscript</csymbol><ci id="S3.SS2.SSS2.p2.12.m10.1.1.2.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.2">𝜽</ci><apply id="S3.SS2.SSS2.p2.12.m10.1.1.3.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.3"><times id="S3.SS2.SSS2.p2.12.m10.1.1.3.1.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.1"></times><ci id="S3.SS2.SSS2.p2.12.m10.1.1.3.2.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.2">𝑎</ci><ci id="S3.SS2.SSS2.p2.12.m10.1.1.3.3.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.3">𝑠</ci><ci id="S3.SS2.SSS2.p2.12.m10.1.1.3.4.cmml" xref="S3.SS2.SSS2.p2.12.m10.1.1.3.4">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS2.p2.12.m10.1c">\boldsymbol{\theta}_{asv}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS2.p2.12.m10.1d">bold_italic_θ start_POSTSUBSCRIPT italic_a italic_s italic_v end_POSTSUBSCRIPT</annotation></semantics></math> correspond to the parameters of the output layers, respectively. </div> </section> <section class="ltx_subsubsection" id="S3.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 3.2.3 SASV binary classifier</h4> <div class="ltx_para" id="S3.SS2.SSS3.p1"> For the SASV binary classifier, the initial step involves aggregating the outputs <math alttext="\boldsymbol{H}_{fuse}" class="ltx_Math" display="inline" id="S3.SS2.SSS3.p1.1.m1.1"><semantics id="S3.SS2.SSS3.p1.1.m1.1a"><msub id="S3.SS2.SSS3.p1.1.m1.1.1" xref="S3.SS2.SSS3.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS3.p1.1.m1.1.1.2" mathsize="90%" xref="S3.SS2.SSS3.p1.1.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS3.p1.1.m1.1.1.3" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.cmml"><mi id="S3.SS2.SSS3.p1.1.m1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.2.cmml">f</mi><mo id="S3.SS2.SSS3.p1.1.m1.1.1.3.1" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.1.m1.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.3.cmml">u</mi><mo id="S3.SS2.SSS3.p1.1.m1.1.1.3.1a" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.1.m1.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.4.cmml">s</mi><mo id="S3.SS2.SSS3.p1.1.m1.1.1.3.1b" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.1.m1.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS3.p1.1.m1.1b"><apply id="S3.SS2.SSS3.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS3.p1.1.m1.1.1.1.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS3.p1.1.m1.1.1.2.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.2">𝑯</ci><apply id="S3.SS2.SSS3.p1.1.m1.1.1.3.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3"><times id="S3.SS2.SSS3.p1.1.m1.1.1.3.1.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.1"></times><ci id="S3.SS2.SSS3.p1.1.m1.1.1.3.2.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.2">𝑓</ci><ci id="S3.SS2.SSS3.p1.1.m1.1.1.3.3.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.3">𝑢</ci><ci id="S3.SS2.SSS3.p1.1.m1.1.1.3.4.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.4">𝑠</ci><ci id="S3.SS2.SSS3.p1.1.m1.1.1.3.5.cmml" xref="S3.SS2.SSS3.p1.1.m1.1.1.3.5">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS3.p1.1.m1.1c">\boldsymbol{H}_{fuse}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS3.p1.1.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT</annotation></semantics></math> from each block by summation. Subsequently, layer normalization is applied to the aggregation, as expressed in the following equation: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx5"> <tbody id="S3.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{H}_{fuse}" class="ltx_Math" display="inline" id="S3.E11.m1.1"><semantics id="S3.E11.m1.1a"><msub id="S3.E11.m1.1.1" xref="S3.E11.m1.1.1.cmml"><mi id="S3.E11.m1.1.1.2" mathsize="90%" xref="S3.E11.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.E11.m1.1.1.3" xref="S3.E11.m1.1.1.3.cmml"><mi id="S3.E11.m1.1.1.3.2" mathsize="90%" xref="S3.E11.m1.1.1.3.2.cmml">f</mi><mo id="S3.E11.m1.1.1.3.1" xref="S3.E11.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E11.m1.1.1.3.3" mathsize="90%" xref="S3.E11.m1.1.1.3.3.cmml">u</mi><mo id="S3.E11.m1.1.1.3.1a" xref="S3.E11.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E11.m1.1.1.3.4" mathsize="90%" xref="S3.E11.m1.1.1.3.4.cmml">s</mi><mo id="S3.E11.m1.1.1.3.1b" xref="S3.E11.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.E11.m1.1.1.3.5" mathsize="90%" xref="S3.E11.m1.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.E11.m1.1b"><apply id="S3.E11.m1.1.1.cmml" xref="S3.E11.m1.1.1"><csymbol cd="ambiguous" id="S3.E11.m1.1.1.1.cmml" xref="S3.E11.m1.1.1">subscript</csymbol><ci id="S3.E11.m1.1.1.2.cmml" xref="S3.E11.m1.1.1.2">𝑯</ci><apply id="S3.E11.m1.1.1.3.cmml" xref="S3.E11.m1.1.1.3"><times id="S3.E11.m1.1.1.3.1.cmml" xref="S3.E11.m1.1.1.3.1"></times><ci id="S3.E11.m1.1.1.3.2.cmml" xref="S3.E11.m1.1.1.3.2">𝑓</ci><ci id="S3.E11.m1.1.1.3.3.cmml" xref="S3.E11.m1.1.1.3.3">𝑢</ci><ci id="S3.E11.m1.1.1.3.4.cmml" xref="S3.E11.m1.1.1.3.4">𝑠</ci><ci id="S3.E11.m1.1.1.3.5.cmml" xref="S3.E11.m1.1.1.3.5">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m1.1c">\displaystyle\boldsymbol{H}_{fuse}</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\text{LN}(\sum_{n=1}^{N}\boldsymbol{H}_{final}^{n})," class="ltx_Math" display="inline" id="S3.E11.m2.1"><semantics id="S3.E11.m2.1a"><mrow id="S3.E11.m2.1.1.1" xref="S3.E11.m2.1.1.1.1.cmml"><mrow id="S3.E11.m2.1.1.1.1" xref="S3.E11.m2.1.1.1.1.cmml"><mi id="S3.E11.m2.1.1.1.1.3" xref="S3.E11.m2.1.1.1.1.3.cmml"></mi><mo id="S3.E11.m2.1.1.1.1.2" mathsize="90%" xref="S3.E11.m2.1.1.1.1.2.cmml">=</mo><mrow id="S3.E11.m2.1.1.1.1.1" xref="S3.E11.m2.1.1.1.1.1.cmml"><mtext id="S3.E11.m2.1.1.1.1.1.3" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.3a.cmml">LN</mtext><mo id="S3.E11.m2.1.1.1.1.1.2" xref="S3.E11.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E11.m2.1.1.1.1.1.1.1" xref="S3.E11.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E11.m2.1.1.1.1.1.1.1.2" maxsize="90%" minsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E11.m2.1.1.1.1.1.1.1.1" xref="S3.E11.m2.1.1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S3.E11.m2.1.1.1.1.1.1.1.1.1" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.cmml"><munderover id="S3.E11.m2.1.1.1.1.1.1.1.1.1a" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.2" maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.2" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.1" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.1.cmml">=</mo><mn id="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.3" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.1.3" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.1.3.cmml">N</mi></munderover></mstyle><msubsup id="S3.E11.m2.1.1.1.1.1.1.1.1.2" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.2" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.2.cmml">𝑯</mi><mrow id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.cmml"><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.2" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.2.cmml">f</mi><mo id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.3" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.3.cmml">i</mi><mo id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1a" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.4" mathsize="90%" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.4.cmml">n</mi><mo id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1b" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi 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id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.2.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.2">𝑓</ci><ci id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.3.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.3">𝑖</ci><ci id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.4.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.4">𝑛</ci><ci id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.5.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.5">𝑎</ci><ci id="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.6.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.2.3.6">𝑙</ci></apply></apply><ci id="S3.E11.m2.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.E11.m2.1.1.1.1.1.1.1.1.2.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E11.m2.1c">\displaystyle=\text{LN}(\sum_{n=1}^{N}\boldsymbol{H}_{final}^{n}),</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m2.1d">= LN ( ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT bold_italic_H start_POSTSUBSCRIPT italic_f italic_i italic_n italic_a italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n 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">(11)</td> </tr></tbody> </table> Here, <math alttext="\boldsymbol{H}_{fuse}" class="ltx_Math" display="inline" id="S3.SS2.SSS3.p1.2.m1.1"><semantics id="S3.SS2.SSS3.p1.2.m1.1a"><msub id="S3.SS2.SSS3.p1.2.m1.1.1" xref="S3.SS2.SSS3.p1.2.m1.1.1.cmml"><mi id="S3.SS2.SSS3.p1.2.m1.1.1.2" mathsize="90%" xref="S3.SS2.SSS3.p1.2.m1.1.1.2.cmml">𝑯</mi><mrow id="S3.SS2.SSS3.p1.2.m1.1.1.3" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.cmml"><mi id="S3.SS2.SSS3.p1.2.m1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.2.cmml">f</mi><mo id="S3.SS2.SSS3.p1.2.m1.1.1.3.1" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.2.m1.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.3.cmml">u</mi><mo id="S3.SS2.SSS3.p1.2.m1.1.1.3.1a" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.2.m1.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.4.cmml">s</mi><mo id="S3.SS2.SSS3.p1.2.m1.1.1.3.1b" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.2.m1.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS3.p1.2.m1.1b"><apply id="S3.SS2.SSS3.p1.2.m1.1.1.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS3.p1.2.m1.1.1.1.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS3.p1.2.m1.1.1.2.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.2">𝑯</ci><apply id="S3.SS2.SSS3.p1.2.m1.1.1.3.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3"><times id="S3.SS2.SSS3.p1.2.m1.1.1.3.1.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.1"></times><ci id="S3.SS2.SSS3.p1.2.m1.1.1.3.2.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.2">𝑓</ci><ci id="S3.SS2.SSS3.p1.2.m1.1.1.3.3.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.3">𝑢</ci><ci id="S3.SS2.SSS3.p1.2.m1.1.1.3.4.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.4">𝑠</ci><ci id="S3.SS2.SSS3.p1.2.m1.1.1.3.5.cmml" xref="S3.SS2.SSS3.p1.2.m1.1.1.3.5">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS3.p1.2.m1.1c">\boldsymbol{H}_{fuse}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS3.p1.2.m1.1d">bold_italic_H start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT</annotation></semantics></math> represents the output of the aggregation. Following this, the aggregated features undergo transformation through a one-dimensional convolutional layer and a pooling layer, generating an embedding denoted as <math alttext="\boldsymbol{e}_{fuse}" class="ltx_Math" display="inline" id="S3.SS2.SSS3.p1.3.m2.1"><semantics id="S3.SS2.SSS3.p1.3.m2.1a"><msub id="S3.SS2.SSS3.p1.3.m2.1.1" xref="S3.SS2.SSS3.p1.3.m2.1.1.cmml"><mi id="S3.SS2.SSS3.p1.3.m2.1.1.2" mathsize="90%" xref="S3.SS2.SSS3.p1.3.m2.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS3.p1.3.m2.1.1.3" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.cmml"><mi id="S3.SS2.SSS3.p1.3.m2.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.2.cmml">f</mi><mo id="S3.SS2.SSS3.p1.3.m2.1.1.3.1" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.3.m2.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.3.cmml">u</mi><mo id="S3.SS2.SSS3.p1.3.m2.1.1.3.1a" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.3.m2.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.4.cmml">s</mi><mo id="S3.SS2.SSS3.p1.3.m2.1.1.3.1b" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p1.3.m2.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS3.p1.3.m2.1b"><apply id="S3.SS2.SSS3.p1.3.m2.1.1.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS3.p1.3.m2.1.1.1.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1">subscript</csymbol><ci id="S3.SS2.SSS3.p1.3.m2.1.1.2.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.2">𝒆</ci><apply id="S3.SS2.SSS3.p1.3.m2.1.1.3.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3"><times id="S3.SS2.SSS3.p1.3.m2.1.1.3.1.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.1"></times><ci id="S3.SS2.SSS3.p1.3.m2.1.1.3.2.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.2">𝑓</ci><ci id="S3.SS2.SSS3.p1.3.m2.1.1.3.3.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.3">𝑢</ci><ci id="S3.SS2.SSS3.p1.3.m2.1.1.3.4.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.4">𝑠</ci><ci id="S3.SS2.SSS3.p1.3.m2.1.1.3.5.cmml" xref="S3.SS2.SSS3.p1.3.m2.1.1.3.5">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS3.p1.3.m2.1c">\boldsymbol{e}_{fuse}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS3.p1.3.m2.1d">bold_italic_e start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_para" id="S3.SS2.SSS3.p2"> The architecture employed for the sub-module labeled “SASV back-end” in Figure <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.F2" style="font-size:90%;" title="Figure 2 ‣ 3.1 Overview ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2</a> is identical to that outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>]</cite>. However, our approach is distinct due to the assertion that the informative embedding <math alttext="\boldsymbol{e}_{fuse}" class="ltx_Math" display="inline" id="S3.SS2.SSS3.p2.1.m1.1"><semantics id="S3.SS2.SSS3.p2.1.m1.1a"><msub id="S3.SS2.SSS3.p2.1.m1.1.1" xref="S3.SS2.SSS3.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS3.p2.1.m1.1.1.2" mathsize="90%" xref="S3.SS2.SSS3.p2.1.m1.1.1.2.cmml">𝒆</mi><mrow id="S3.SS2.SSS3.p2.1.m1.1.1.3" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.cmml"><mi id="S3.SS2.SSS3.p2.1.m1.1.1.3.2" mathsize="90%" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.2.cmml">f</mi><mo id="S3.SS2.SSS3.p2.1.m1.1.1.3.1" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p2.1.m1.1.1.3.3" mathsize="90%" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.3.cmml">u</mi><mo id="S3.SS2.SSS3.p2.1.m1.1.1.3.1a" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p2.1.m1.1.1.3.4" mathsize="90%" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.4.cmml">s</mi><mo id="S3.SS2.SSS3.p2.1.m1.1.1.3.1b" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.SSS3.p2.1.m1.1.1.3.5" mathsize="90%" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.5.cmml">e</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS3.p2.1.m1.1b"><apply id="S3.SS2.SSS3.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS3.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS3.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.2">𝒆</ci><apply id="S3.SS2.SSS3.p2.1.m1.1.1.3.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3"><times id="S3.SS2.SSS3.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.1"></times><ci id="S3.SS2.SSS3.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.2">𝑓</ci><ci id="S3.SS2.SSS3.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.3">𝑢</ci><ci id="S3.SS2.SSS3.p2.1.m1.1.1.3.4.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.4">𝑠</ci><ci id="S3.SS2.SSS3.p2.1.m1.1.1.3.5.cmml" xref="S3.SS2.SSS3.p2.1.m1.1.1.3.5">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS3.p2.1.m1.1c">\boldsymbol{e}_{fuse}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS3.p2.1.m1.1d">bold_italic_e start_POSTSUBSCRIPT italic_f italic_u italic_s italic_e end_POSTSUBSCRIPT</annotation></semantics></math> can serve both the anti-spoofing and ASV tasks. Consequently, this embedding is utilized for both CM decision and ASV decision, which diverges from the input used in the SASV back-end, as detailed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>]</cite>. </div> </section> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 3.3 Learning paradigm</h3> <div class="ltx_para" id="S3.SS3.p1"> To confer resilience upon the proposed model, we integrate the pair-wise learning paradigm, simulation of spoofing attacks, and meta-learning paradigm. As elucidated in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib31" title="">31</a>]</cite>, meta-learning can be understood as a process of bilevel optimization involving a hierarchical optimization problem where one optimization contains another as a constraint <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib46" title="">46</a>]</cite>. This bilevel optimization is divided into two loops, the outer and inner, which effectively simulate domain mismatch during the training of machine learning models. It can be formalized as follows: <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx6"> <tbody id="S3.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\theta_{i+1}" class="ltx_Math" display="inline" id="S3.Ex2.m1.1"><semantics id="S3.Ex2.m1.1a"><msub id="S3.Ex2.m1.1.1" xref="S3.Ex2.m1.1.1.cmml"><mi id="S3.Ex2.m1.1.1.2" mathsize="90%" xref="S3.Ex2.m1.1.1.2.cmml">θ</mi><mrow id="S3.Ex2.m1.1.1.3" xref="S3.Ex2.m1.1.1.3.cmml"><mi id="S3.Ex2.m1.1.1.3.2" mathsize="90%" xref="S3.Ex2.m1.1.1.3.2.cmml">i</mi><mo id="S3.Ex2.m1.1.1.3.1" mathsize="90%" xref="S3.Ex2.m1.1.1.3.1.cmml">+</mo><mn id="S3.Ex2.m1.1.1.3.3" mathsize="90%" xref="S3.Ex2.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.1b"><apply id="S3.Ex2.m1.1.1.cmml" xref="S3.Ex2.m1.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.1.1.1.cmml" 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roman_argmin start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT caligraphic_L start_POSTSUPERSCRIPT italic_o italic_u italic_t italic_e italic_r end_POSTSUPERSCRIPT ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_D start_POSTSUBSCRIPT italic_s italic_o italic_u italic_r italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_e italic_t italic_a - italic_t italic_e italic_s italic_t ( italic_i ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S3.E12.m1.1.1.1.1">𝑖</ci></apply></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.3c">\displaystyle\;\;\;\;+\mathcal{L}^{inner}(\theta_{i},\mathcal{D}_{source}^{% meta-train(i)})</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.3d">+ caligraphic_L start_POSTSUPERSCRIPT italic_i italic_n italic_n italic_e italic_r end_POSTSUPERSCRIPT ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , caligraphic_D start_POSTSUBSCRIPT italic_s italic_o italic_u italic_r italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_e italic_t italic_a - italic_t italic_r italic_a italic_i italic_n ( 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">(12)</td> </tr></tbody> <tbody id="S3.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\text{s}.\text{t}.\quad\theta_{i}^{*}" class="ltx_Math" display="inline" id="S3.E13.m1.3"><semantics id="S3.E13.m1.3a"><mrow id="S3.E13.m1.3.3.1" xref="S3.E13.m1.3.3.2.cmml"><mtext id="S3.E13.m1.1.1" mathsize="90%" xref="S3.E13.m1.1.1a.cmml">s</mtext><mo id="S3.E13.m1.3.3.1.2" lspace="0em" mathsize="90%" rspace="0.167em" xref="S3.E13.m1.3.3.2a.cmml">.</mo><mtext id="S3.E13.m1.2.2" mathsize="90%" xref="S3.E13.m1.2.2a.cmml">t</mtext><mo id="S3.E13.m1.3.3.1.3" lspace="0em" mathsize="90%" rspace="1.067em" xref="S3.E13.m1.3.3.2a.cmml">.</mo><msubsup id="S3.E13.m1.3.3.1.1" xref="S3.E13.m1.3.3.1.1.cmml"><mi id="S3.E13.m1.3.3.1.1.2.2" mathsize="90%" xref="S3.E13.m1.3.3.1.1.2.2.cmml">θ</mi><mi id="S3.E13.m1.3.3.1.1.2.3" mathsize="90%" xref="S3.E13.m1.3.3.1.1.2.3.cmml">i</mi><mo id="S3.E13.m1.3.3.1.1.3" mathsize="90%" xref="S3.E13.m1.3.3.1.1.3.cmml">∗</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.E13.m1.3b"><apply id="S3.E13.m1.3.3.2.cmml" xref="S3.E13.m1.3.3.1"><csymbol cd="ambiguous" id="S3.E13.m1.3.3.2a.cmml" xref="S3.E13.m1.3.3.1.2">formulae-sequence</csymbol><ci id="S3.E13.m1.1.1a.cmml" xref="S3.E13.m1.1.1"><mtext id="S3.E13.m1.1.1.cmml" mathsize="90%" xref="S3.E13.m1.1.1">s</mtext></ci><ci id="S3.E13.m1.2.2a.cmml" xref="S3.E13.m1.2.2"><mtext id="S3.E13.m1.2.2.cmml" mathsize="90%" xref="S3.E13.m1.2.2">t</mtext></ci><apply id="S3.E13.m1.3.3.1.1.cmml" xref="S3.E13.m1.3.3.1.1"><csymbol cd="ambiguous" id="S3.E13.m1.3.3.1.1.1.cmml" xref="S3.E13.m1.3.3.1.1">superscript</csymbol><apply id="S3.E13.m1.3.3.1.1.2.cmml" xref="S3.E13.m1.3.3.1.1"><csymbol cd="ambiguous" id="S3.E13.m1.3.3.1.1.2.1.cmml" xref="S3.E13.m1.3.3.1.1">subscript</csymbol><ci id="S3.E13.m1.3.3.1.1.2.2.cmml" xref="S3.E13.m1.3.3.1.1.2.2">𝜃</ci><ci id="S3.E13.m1.3.3.1.1.2.3.cmml" xref="S3.E13.m1.3.3.1.1.2.3">𝑖</ci></apply><times id="S3.E13.m1.3.3.1.1.3.cmml" xref="S3.E13.m1.3.3.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.3c">\displaystyle\text{s}.\text{t}.\quad\theta_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.3d">s . t . italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname*{argmin}_{\theta}\mathcal{L}^{inner}(\theta_{i},% \mathcal{D}_{source}^{meta-train(i)})," class="ltx_Math" display="inline" id="S3.E13.m2.2"><semantics id="S3.E13.m2.2a"><mrow id="S3.E13.m2.2.2.1" xref="S3.E13.m2.2.2.1.1.cmml"><mrow id="S3.E13.m2.2.2.1.1" xref="S3.E13.m2.2.2.1.1.cmml"><mi id="S3.E13.m2.2.2.1.1.4" xref="S3.E13.m2.2.2.1.1.4.cmml"></mi><mo id="S3.E13.m2.2.2.1.1.3" mathsize="90%" rspace="0.1389em" 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id="S3.E13.m2.1.1.1.4.4.cmml" xref="S3.E13.m2.1.1.1.4.4">𝑎</ci><ci id="S3.E13.m2.1.1.1.4.5.cmml" xref="S3.E13.m2.1.1.1.4.5">𝑖</ci><ci id="S3.E13.m2.1.1.1.4.6.cmml" xref="S3.E13.m2.1.1.1.4.6">𝑛</ci><ci id="S3.E13.m2.1.1.1.1.cmml" xref="S3.E13.m2.1.1.1.1">𝑖</ci></apply></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m2.2c">\displaystyle=\operatorname*{argmin}_{\theta}\mathcal{L}^{inner}(\theta_{i},% \mathcal{D}_{source}^{meta-train(i)}),</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m2.2d">= roman_argmin start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT caligraphic_L start_POSTSUPERSCRIPT italic_i italic_n italic_n italic_e italic_r end_POSTSUPERSCRIPT ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , caligraphic_D start_POSTSUBSCRIPT italic_s italic_o italic_u italic_r italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_e italic_t italic_a - italic_t italic_r italic_a italic_i italic_n ( 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">(13)</td> </tr></tbody> </table> where <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p1.1.m1.1.1" mathsize="90%" xref="S3.SS3.p1.1.m1.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><ci id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">caligraphic_D</annotation></semantics></math> represents the training dataset from the source domain, which is divided into a meta-train dataset denoted as <math 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xref="S3.SS3.p1.2.m2.1.1.3.3.2">𝑡</ci><ci id="S3.SS3.p1.2.m2.1.1.3.3.3.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3.3">𝑟</ci><ci id="S3.SS3.p1.2.m2.1.1.3.3.4.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3.4">𝑎</ci><ci id="S3.SS3.p1.2.m2.1.1.3.3.5.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3.5">𝑖</ci><ci id="S3.SS3.p1.2.m2.1.1.3.3.6.cmml" xref="S3.SS3.p1.2.m2.1.1.3.3.6">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.1c">\mathcal{D}_{source}^{meta-train}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.1d">caligraphic_D start_POSTSUBSCRIPT italic_s italic_o italic_u italic_r italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_e italic_t italic_a - italic_t italic_r italic_a italic_i italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> and a meta-test dataset denoted as <math alttext="\mathcal{D}_{source}^{meta-test}" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m3.1"><semantics 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xref="S3.SS3.p1.3.m3.1.1.3.3.5">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m3.1c">\mathcal{D}_{source}^{meta-test}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m3.1d">caligraphic_D start_POSTSUBSCRIPT italic_s italic_o italic_u italic_r italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m italic_e italic_t italic_a - italic_t italic_e italic_s italic_t end_POSTSUPERSCRIPT</annotation></semantics></math>. The combination of a meta-train and a meta-test dataset constitutes a meta-task, indexed by <math alttext="i" class="ltx_Math" display="inline" id="S3.SS3.p1.4.m4.1"><semantics id="S3.SS3.p1.4.m4.1a"><mi id="S3.SS3.p1.4.m4.1.1" mathsize="90%" xref="S3.SS3.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.4.m4.1b"><ci id="S3.SS3.p1.4.m4.1.1.cmml" xref="S3.SS3.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.4.m4.1d">italic_i</annotation></semantics></math>, which means the <math alttext="i" class="ltx_Math" display="inline" id="S3.SS3.p1.5.m5.1"><semantics id="S3.SS3.p1.5.m5.1a"><mi id="S3.SS3.p1.5.m5.1.1" mathsize="90%" xref="S3.SS3.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.5.m5.1b"><ci id="S3.SS3.p1.5.m5.1.1.cmml" xref="S3.SS3.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.5.m5.1d">italic_i</annotation></semantics></math>-th one in total <math alttext="T" class="ltx_Math" display="inline" id="S3.SS3.p1.6.m6.1"><semantics id="S3.SS3.p1.6.m6.1a"><mi id="S3.SS3.p1.6.m6.1.1" mathsize="90%" xref="S3.SS3.p1.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.6.m6.1b"><ci id="S3.SS3.p1.6.m6.1.1.cmml" xref="S3.SS3.p1.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.6.m6.1d">italic_T</annotation></semantics></math> meta-tasks. <math alttext="\theta_{i}" class="ltx_Math" display="inline" id="S3.SS3.p1.7.m7.1"><semantics id="S3.SS3.p1.7.m7.1a"><msub id="S3.SS3.p1.7.m7.1.1" xref="S3.SS3.p1.7.m7.1.1.cmml"><mi id="S3.SS3.p1.7.m7.1.1.2" mathsize="90%" xref="S3.SS3.p1.7.m7.1.1.2.cmml">θ</mi><mi id="S3.SS3.p1.7.m7.1.1.3" mathsize="90%" xref="S3.SS3.p1.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.7.m7.1b"><apply id="S3.SS3.p1.7.m7.1.1.cmml" xref="S3.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.7.m7.1.1.1.cmml" xref="S3.SS3.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS3.p1.7.m7.1.1.2.cmml" xref="S3.SS3.p1.7.m7.1.1.2">𝜃</ci><ci id="S3.SS3.p1.7.m7.1.1.3.cmml" xref="S3.SS3.p1.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.7.m7.1c">\theta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.7.m7.1d">italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\theta_{i+1}" class="ltx_Math" display="inline" id="S3.SS3.p1.8.m8.1"><semantics id="S3.SS3.p1.8.m8.1a"><msub id="S3.SS3.p1.8.m8.1.1" xref="S3.SS3.p1.8.m8.1.1.cmml"><mi id="S3.SS3.p1.8.m8.1.1.2" mathsize="90%" xref="S3.SS3.p1.8.m8.1.1.2.cmml">θ</mi><mrow id="S3.SS3.p1.8.m8.1.1.3" xref="S3.SS3.p1.8.m8.1.1.3.cmml"><mi id="S3.SS3.p1.8.m8.1.1.3.2" mathsize="90%" xref="S3.SS3.p1.8.m8.1.1.3.2.cmml">i</mi><mo id="S3.SS3.p1.8.m8.1.1.3.1" mathsize="90%" xref="S3.SS3.p1.8.m8.1.1.3.1.cmml">+</mo><mn id="S3.SS3.p1.8.m8.1.1.3.3" mathsize="90%" xref="S3.SS3.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.8.m8.1b"><apply id="S3.SS3.p1.8.m8.1.1.cmml" xref="S3.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.8.m8.1.1.1.cmml" xref="S3.SS3.p1.8.m8.1.1">subscript</csymbol><ci id="S3.SS3.p1.8.m8.1.1.2.cmml" xref="S3.SS3.p1.8.m8.1.1.2">𝜃</ci><apply id="S3.SS3.p1.8.m8.1.1.3.cmml" xref="S3.SS3.p1.8.m8.1.1.3"><plus id="S3.SS3.p1.8.m8.1.1.3.1.cmml" xref="S3.SS3.p1.8.m8.1.1.3.1"></plus><ci id="S3.SS3.p1.8.m8.1.1.3.2.cmml" xref="S3.SS3.p1.8.m8.1.1.3.2">𝑖</ci><cn id="S3.SS3.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S3.SS3.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.8.m8.1c">\theta_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.8.m8.1d">italic_θ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> correspond to the parameters of the inner and outer loops, respectively. Note that there exists a leader-follower asymmetry between the outer and inner loops. </div> <figure class="ltx_table" id="S3.T4"> <figcaption class="ltx_caption" style="font-size:90%;">Table 4: Differences in EER between the proposed approach and the baseline. The first row of the table denotes genres of testing utterances, and the first column of the table represents genres of enrollment utterances. The blue number means the proposed approach outperforms the baseline. The red number means the baseline outperforms the proposed approach.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T4.11"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T4.11.1.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S3.T4.11.1.1.1" style="padding-left:3.5pt;padding-right:3.5pt;"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.2" style="padding-left:3.5pt;padding-right:3.5pt;">dr</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.3" style="padding-left:3.5pt;padding-right:3.5pt;">en</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.4" style="padding-left:3.5pt;padding-right:3.5pt;">in</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.5" style="padding-left:3.5pt;padding-right:3.5pt;">lb</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.6" style="padding-left:3.5pt;padding-right:3.5pt;">re</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.7" style="padding-left:3.5pt;padding-right:3.5pt;">si</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.8" style="padding-left:3.5pt;padding-right:3.5pt;">sp</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T4.11.1.1.9" style="padding-left:3.5pt;padding-right:3.5pt;">vl</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T4.11.2.1"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_t" id="S3.T4.11.2.1.1" style="padding-left:3.5pt;padding-right:3.5pt;">dr</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.2" style="padding-left:3.5pt;padding-right:3.5pt;">+1.65</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.3" style="padding-left:3.5pt;padding-right:3.5pt;">-1.23</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.4" style="padding-left:3.5pt;padding-right:3.5pt;">+1.72</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.5" style="padding-left:3.5pt;padding-right:3.5pt;">+5.97</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.6" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.7" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.8" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T4.11.2.1.9" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> </tr> <tr class="ltx_tr" id="S3.T4.11.3.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S3.T4.11.3.2.1" style="padding-left:3.5pt;padding-right:3.5pt;">en</th> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.2" style="padding-left:3.5pt;padding-right:3.5pt;">+2.71</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.3" style="padding-left:3.5pt;padding-right:3.5pt;">+3.06</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.4" style="padding-left:3.5pt;padding-right:3.5pt;">-0.02</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.5" style="padding-left:3.5pt;padding-right:3.5pt;">-1.04</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.6" style="padding-left:3.5pt;padding-right:3.5pt;">-7.35</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.7" style="padding-left:3.5pt;padding-right:3.5pt;">+3.83</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.8" style="padding-left:3.5pt;padding-right:3.5pt;">+1.20</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.3.2.9" style="padding-left:3.5pt;padding-right:3.5pt;">-1.06</td> </tr> <tr class="ltx_tr" id="S3.T4.11.4.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S3.T4.11.4.3.1" style="padding-left:3.5pt;padding-right:3.5pt;">in</th> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.2" style="padding-left:3.5pt;padding-right:3.5pt;">+4.32</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.3" style="padding-left:3.5pt;padding-right:3.5pt;">+2.13</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.4" style="padding-left:3.5pt;padding-right:3.5pt;">+2.49</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.5" style="padding-left:3.5pt;padding-right:3.5pt;">-0.64</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.6" style="padding-left:3.5pt;padding-right:3.5pt;">+2.70</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.7" style="padding-left:3.5pt;padding-right:3.5pt;">+2.72</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.8" style="padding-left:3.5pt;padding-right:3.5pt;">+3.94</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.4.3.9" style="padding-left:3.5pt;padding-right:3.5pt;">+14.06</td> </tr> <tr class="ltx_tr" id="S3.T4.11.5.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S3.T4.11.5.4.1" style="padding-left:3.5pt;padding-right:3.5pt;">lb</th> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.2" style="padding-left:3.5pt;padding-right:3.5pt;">-1.74</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.3" style="padding-left:3.5pt;padding-right:3.5pt;">-1.13</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.4" style="padding-left:3.5pt;padding-right:3.5pt;">-2.91</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.5" style="padding-left:3.5pt;padding-right:3.5pt;">+0.02</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.6" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.7" style="padding-left:3.5pt;padding-right:3.5pt;">+7.73</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.8" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.5.4.9" style="padding-left:3.5pt;padding-right:3.5pt;">+6.71</td> </tr> <tr class="ltx_tr" id="S3.T4.11.6.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row" id="S3.T4.11.6.5.1" style="padding-left:3.5pt;padding-right:3.5pt;">sp</th> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.2" style="padding-left:3.5pt;padding-right:3.5pt;">+2.98</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.3" style="padding-left:3.5pt;padding-right:3.5pt;">+0.36</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.4" style="padding-left:3.5pt;padding-right:3.5pt;">+3.34</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.5" style="padding-left:3.5pt;padding-right:3.5pt;">+5.17</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.6" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.7" style="padding-left:3.5pt;padding-right:3.5pt;">+2.07</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.8" style="padding-left:3.5pt;padding-right:3.5pt;">-0.79</td> <td class="ltx_td ltx_align_center" id="S3.T4.11.6.5.9" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> </tr> <tr class="ltx_tr" id="S3.T4.11.7.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_bb" id="S3.T4.11.7.6.1" style="padding-left:3.5pt;padding-right:3.5pt;">vl</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.2" style="padding-left:3.5pt;padding-right:3.5pt;">-6.52</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.3" style="padding-left:3.5pt;padding-right:3.5pt;">+0.51</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.4" style="padding-left:3.5pt;padding-right:3.5pt;">+4.75</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.5" style="padding-left:3.5pt;padding-right:3.5pt;">+2.63</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.6" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.7" style="padding-left:3.5pt;padding-right:3.5pt;">-2.92</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.8" style="padding-left:3.5pt;padding-right:3.5pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T4.11.7.6.9" style="padding-left:3.5pt;padding-right:3.5pt;">+2.01</td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="S3.T5"> <figcaption class="ltx_caption" style="font-size:80%;">Table 5: Results of the proposed system on CNCeleb.Eval and CNComplex testing datasets. The performance is evaluated under the metrics of SV-EER and SASV-EER metrics, respectively.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T5.5"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T5.5.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="S3.T5.5.1.1.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">Protocol</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S3.T5.5.1.1.2" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">Training dataset</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S3.T5.5.1.1.3" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb.Eval</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S3.T5.5.1.1.4" style="padding-left:2.3pt;padding-right:2.3pt;">CNComplex</th> </tr> <tr class="ltx_tr" id="S3.T5.5.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S3.T5.5.2.2.1" style="padding-left:2.3pt;padding-right:2.3pt;">SV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S3.T5.5.2.2.2" style="padding-left:2.3pt;padding-right:2.3pt;">SASV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S3.T5.5.2.2.3" style="padding-left:2.3pt;padding-right:2.3pt;">SV-EER</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S3.T5.5.2.2.4" style="padding-left:2.3pt;padding-right:2.3pt;">SASV-EER</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T5.5.3.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="S3.T5.5.3.1.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP I</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T5.5.3.1.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T5.5.3.1.3" style="padding-left:2.3pt;padding-right:2.3pt;">7.96</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T5.5.3.1.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T5.5.3.1.5" style="padding-left:2.3pt;padding-right:2.3pt;">8.52</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T5.5.3.1.6" style="padding-left:2.3pt;padding-right:2.3pt;">7.37</td> </tr> <tr class="ltx_tr" id="S3.T5.5.4.2"> <td class="ltx_td ltx_align_center" id="S3.T5.5.4.2.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.4.2.2" style="padding-left:2.3pt;padding-right:2.3pt;">7.79</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.4.2.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.4.2.4" style="padding-left:2.3pt;padding-right:2.3pt;">7.56</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.4.2.5" style="padding-left:2.3pt;padding-right:2.3pt;">7.25</td> </tr> <tr class="ltx_tr" id="S3.T5.5.5.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S3.T5.5.5.3.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP II</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.5.3.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.5.3.3" style="padding-left:2.3pt;padding-right:2.3pt;">8.24</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.5.3.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.5.3.5" style="padding-left:2.3pt;padding-right:2.3pt;">8.85</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.5.3.6" style="padding-left:2.3pt;padding-right:2.3pt;">8.57</td> </tr> <tr class="ltx_tr" id="S3.T5.5.6.4"> <td class="ltx_td ltx_align_center" id="S3.T5.5.6.4.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.6.4.2" style="padding-left:2.3pt;padding-right:2.3pt;">7.96</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.6.4.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.6.4.4" style="padding-left:2.3pt;padding-right:2.3pt;">8.34</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.6.4.5" style="padding-left:2.3pt;padding-right:2.3pt;">8.47</td> </tr> <tr class="ltx_tr" id="S3.T5.5.7.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S3.T5.5.7.5.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP III</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.7.5.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.7.5.3" style="padding-left:2.3pt;padding-right:2.3pt;">8.43</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.7.5.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.7.5.5" style="padding-left:2.3pt;padding-right:2.3pt;">9.07</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.7.5.6" style="padding-left:2.3pt;padding-right:2.3pt;">8.52</td> </tr> <tr class="ltx_tr" id="S3.T5.5.8.6"> <td class="ltx_td ltx_align_center" id="S3.T5.5.8.6.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.8.6.2" style="padding-left:2.3pt;padding-right:2.3pt;">8.19</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.8.6.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.8.6.4" style="padding-left:2.3pt;padding-right:2.3pt;">8.82</td> <td class="ltx_td ltx_align_center" id="S3.T5.5.8.6.5" style="padding-left:2.3pt;padding-right:2.3pt;">8.25</td> </tr> <tr class="ltx_tr" id="S3.T5.5.9.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_tt" id="S3.T5.5.9.7.1" rowspan="2" style="padding-left:2.3pt;padding-right:2.3pt;">CGP IV</th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.9.7.2" style="padding-left:2.3pt;padding-right:2.3pt;">CNCeleb 1&2</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.9.7.3" style="padding-left:2.3pt;padding-right:2.3pt;">8.23</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.9.7.4" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.9.7.5" style="padding-left:2.3pt;padding-right:2.3pt;">8.68</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T5.5.9.7.6" style="padding-left:2.3pt;padding-right:2.3pt;">7.73</td> </tr> <tr class="ltx_tr" id="S3.T5.5.10.8"> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T5.5.10.8.1" style="padding-left:2.3pt;padding-right:2.3pt;">Combination</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T5.5.10.8.2" style="padding-left:2.3pt;padding-right:2.3pt;">8.13</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T5.5.10.8.3" style="padding-left:2.3pt;padding-right:2.3pt;">-</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T5.5.10.8.4" style="padding-left:2.3pt;padding-right:2.3pt;">8.45</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S3.T5.5.10.8.5" style="padding-left:2.3pt;padding-right:2.3pt;">7.48</td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="S3.T6"> <figcaption class="ltx_caption" style="font-size:90%;">Table 6: EER (%) of experimental results on CNCeleb.Eval testing dataset for the scenario of domain mismatch. For each protocol, the baseline system is established using the proposed model trained through the straightforward supervised learning paradigm. A bold number indicates the highest performance of this genre. The presence of an unseen group is indicated within brackets</figcaption> . Protocol System Overall Group I Group II Group III Group IV dr vl sp en in pl lb mo si re CGP I Baseline 21.31 23.15 20.77 15.86 21.61 22.99 27.35 17.50 29.32 28.63 14.58 (Group IV) proposed approach 17.42 22.66 16.44 14.46 20.95 19.47 20.93 16.31 24.57 22.33 13.89 CGP II Baseline 22.10 22.66 20.64 16.41 23.56 24.08 27.29 19.43 31.55 25.25 13.17 (Group III) proposed approach 18.48 18.04 18.08 13.84 19.11 20.10 19.71 17.84 27.16 24.16 13.91 CGP III Baseline 23.46 25.42 23.89 17.58 26.38 25.34 29.25 21.72 31.14 27.74 13.10 (Group II) proposed approach 20.02 23.27 20.70 16.79 22.08 22.83 22.96 19.54 28.53 22.92 13.55 CGP IV Baseline 22.59 24.13 23.47 16.29 19.50 22.95 27.81 19.99 28.12 24.87 11.33 (Group I) proposed approach 19.98 22.35 19.44 14.58 21.34 19.62 20.87 17.83 24.24 23.53 10.55 </figure> <section class="ltx_subsubsection" id="S3.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 3.3.1 Outer loop</h4> <div class="ltx_para" id="S3.SS3.SSS1.p1"> In the outer loop, the genre sampling strategy is employed to simulate the domain mismatch between the meta-train and meta-test data. Specifically, we configure the number of genres in the meta-train dataset (<math alttext="G_{mtr}" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.1.m1.1"><semantics id="S3.SS3.SSS1.p1.1.m1.1a"><msub id="S3.SS3.SSS1.p1.1.m1.1.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.2" mathsize="90%" xref="S3.SS3.SSS1.p1.1.m1.1.1.2.cmml">G</mi><mrow id="S3.SS3.SSS1.p1.1.m1.1.1.3" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.2" mathsize="90%" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.2.cmml">m</mi><mo id="S3.SS3.SSS1.p1.1.m1.1.1.3.1" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.3" mathsize="90%" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3.cmml">t</mi><mo id="S3.SS3.SSS1.p1.1.m1.1.1.3.1a" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS3.SSS1.p1.1.m1.1.1.3.4" mathsize="90%" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.4.cmml">r</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.1.m1.1b"><apply id="S3.SS3.SSS1.p1.1.m1.1.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.SSS1.p1.1.m1.1.1.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.SSS1.p1.1.m1.1.1.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.2">𝐺</ci><apply id="S3.SS3.SSS1.p1.1.m1.1.1.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3"><times id="S3.SS3.SSS1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.1"></times><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.2">𝑚</ci><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.3">𝑡</ci><ci id="S3.SS3.SSS1.p1.1.m1.1.1.3.4.cmml" xref="S3.SS3.SSS1.p1.1.m1.1.1.3.4">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.1.m1.1c">G_{mtr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.1.m1.1d">italic_G start_POSTSUBSCRIPT italic_m italic_t italic_r end_POSTSUBSCRIPT</annotation></semantics></math>) as <math alttext="G-2" class="ltx_Math" display="inline" id="S3.SS3.SSS1.p1.2.m2.1"><semantics id="S3.SS3.SSS1.p1.2.m2.1a"><mrow id="S3.SS3.SSS1.p1.2.m2.1.1" xref="S3.SS3.SSS1.p1.2.m2.1.1.cmml"><mi id="S3.SS3.SSS1.p1.2.m2.1.1.2" mathsize="90%" xref="S3.SS3.SSS1.p1.2.m2.1.1.2.cmml">G</mi><mo id="S3.SS3.SSS1.p1.2.m2.1.1.1" mathsize="90%" xref="S3.SS3.SSS1.p1.2.m2.1.1.1.cmml">−</mo><mn id="S3.SS3.SSS1.p1.2.m2.1.1.3" mathsize="90%" xref="S3.SS3.SSS1.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.SSS1.p1.2.m2.1b"><apply id="S3.SS3.SSS1.p1.2.m2.1.1.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1"><minus id="S3.SS3.SSS1.p1.2.m2.1.1.1.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1.1"></minus><ci id="S3.SS3.SSS1.p1.2.m2.1.1.2.cmml" xref="S3.SS3.SSS1.p1.2.m2.1.1.2">𝐺</ci><cn id="S3.SS3.SSS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS3.SSS1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.SSS1.p1.2.m2.1c">G-2</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.SSS1.p1.2.m2.1d">italic_G - 2</annotation></semantics></math>, with the remaining genre reserved for the meta-test dataset in all experiments in this paper. This approach is consistently applied across different training protocols. </div> </section> <section class="ltx_subsubsection" id="S3.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 3.3.2 Inner loop</h4> <div class="ltx_para" id="S3.SS3.SSS2.p1"> Within the inner loop, we employ the training trial sampling method, as outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib27" title="">27</a>]</cite>, to simulate both spoofing attacks and channel mismatch scenarios concurrently. Given that the training dataset contains both genuine and copy-synthesized utterances, with each utterance being associated with a specific genre indicating the acoustic environment, it becomes possible to leverage the pair-wise learning paradigm for generating a substantial number of training trials from the training dataset. These training trials effectively simulate various distinct scenarios involving channel mismatch and spoofing attacks, allowing the proposed model to acquire robustness-related knowledge through the inner loop training process. </div> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 4 Experimental results</h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 4.1 Dataset</h3> <div class="ltx_para" id="S4.SS1.p1"> In the experiments, we adhere to the cross-genre training protocols outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib31" title="">31</a>]</cite> to segregate the fusion of the CNSpoof and CNCeleb 1&2 datasets. Under each protocol, we exclude two or three genres, treating them as unseen genres found in the testing dataset. The resulting combination encompasses over 1.2 million utterances spanning eleven distinct genres. Furthermore, the diversity of the training data is enhanced by incorporating the MUSAN <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib47" title="">47</a>]</cite> and RIRs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib48" title="">48</a>]</cite> datasets. </div> <div class="ltx_para" id="S4.SS1.p2"> CNCeleb.Eval and CNComplex are both employed as the testing datasets, along with their respective evaluation protocols, to assess the performance of the proposed model. Both datasets feature 196 speakers for enrollment and consist of 17,777 utterances in the testing data, resulting in a total of 3,484,292 testing trials for each evaluation protocol. </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 4.2 Experimental settings</h3> <div class="ltx_para" id="S4.SS2.p1"> To facilitate the implementation of the meta-learning paradigm, a subset of 64 samples was randomly selected from the dataset. Within this subset, one genre was chosen randomly to constitute the meta-test dataset for the outer loop, while the remaining samples were utilized as the meta-train dataset for the inner loop. All systems underwent training for 80 epochs using an Adam optimizer <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib49" title="">49</a>]</cite>. Training commenced with an initial learning rate of 0.01, <math alttext="\beta_{1}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" mathsize="90%" xref="S4.SS2.p1.1.m1.1.1.2.cmml">β</mi><mn id="S4.SS2.p1.1.m1.1.1.3" mathsize="90%" xref="S4.SS2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝛽</ci><cn id="S4.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\beta_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> of 0.9, and <math alttext="\beta_{2}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><msub id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" mathsize="90%" xref="S4.SS2.p1.2.m2.1.1.2.cmml">β</mi><mn id="S4.SS2.p1.2.m2.1.1.3" mathsize="90%" xref="S4.SS2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝛽</ci><cn id="S4.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\beta_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> of 0.98. A warm-up strategy was employed, gradually increasing the learning rate from 0 to 0.001 within the first 5,000 steps and subsequently reducing the learning rate using an exponential scheduler. </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection" style="font-size:90%;"> 4.3 Results and analysis</h3> <div class="ltx_para" id="S4.SS3.p1"> To illustrate the promising outcomes of the proposed model, which integrates a comprehensive learning paradigm, we conduct independent evaluations of the system across various scenarios encompassing channel mismatch, spoofing attacks, and domain mismatch, respectively. </div> <section class="ltx_subsubsection" id="S4.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 4.3.1 Result for channel mismatch scenario</h4> <div class="ltx_para" id="S4.SS3.SSS1.p1"> To assess the performance of the proposed approach in scenarios involving channel mismatch, we utilize a protocol based on the CNCeleb.Eval testing dataset similar to the one outlined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#bib.bib11" title="">11</a>]</cite>, which encompasses various mismatch conditions. In this experimental evaluation, the baseline system is established using the proposed model trained through the straightforward supervised learning paradigm without the meta-learning paradigm. The differences in EER between the system and the baseline system within the context of channel mismatch are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.T4" style="font-size:90%;" title="Table 4 ‣ 3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">4</a>. Each cell in the table indicates the EER difference. </div> <div class="ltx_para" id="S4.SS3.SSS1.p2"> The table reveals that the proposed approach exhibits significantly improved performance in scenarios characterized by channel mismatch, which refers to the genre mismatch between the enrollment and testing utterances in the case. Positive differences are notable when testing utterances from the “dr,” “in,” “lb,” “sp,” and “vl” genres are enrolled from the “dr,” “en,” “in,” “sp,” and “vl” genres, respectively. This analysis demonstrates that the proposed approach is particularly effective in mitigating the impact of channel mismatch, emphasizing its capability to adapt to varied mismatch conditions during channel variability scenarios. </div> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 4.3.2 Result for spoofing attack scenario</h4> <div class="ltx_para" id="S4.SS3.SSS2.p1"> Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.T5" style="font-size:90%;" title="Table 5 ‣ 3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">5</a> provides a comprehensive overview of the performance evaluation results for the proposed system considering various training protocols and two separate testing datasets. The evaluation is conducted using the SV-EER and SASV-EER metrics. </div> <div class="ltx_para" id="S4.SS3.SSS2.p2"> First and foremost, comparing the number of SV-EERs on the CNCeleb.Eval dataset of two different training datasets for all training protocols, it is evident that the inclusion of the CNSpoof dataset in the training data consistently improves the performance of the traditional ASV task, aligning with the results presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S2.SS2" style="font-size:90%;" title="2.2 Preliminary experimental result of contemporary ASV model ‣ 2 Vulnerability of contemporary ASV model to simultaneous threats ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">2.2</a>. In all training protocols, models trained with the CNSpoof data consistently outperform those trained without it, as evidenced by improvements in all SV-EER numbers. </div> <div class="ltx_para" id="S4.SS3.SSS2.p3"> Furthermore, it is noteworthy that the SASV-EER values for the same system demonstrate comparable results across different protocols when compared to the SV-EER. This observation underscores the system’s capacity for maintaining stability in addressing spoofing attacks, which is a crucial attribute for practical deployment scenarios. </div> </section> <section class="ltx_subsubsection" id="S4.SS3.SSS3"> <h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;"> 4.3.3 Result for domain mismatch scenario</h4> <div class="ltx_para" id="S4.SS3.SSS3.p1"> Table <a class="ltx_ref" href="https://arxiv.org/html/2409.06327v1#S3.T6" style="font-size:90%;" title="Table 6 ‣ 3.3 Learning paradigm ‣ 3 Proposed integrated approach ‣ Spoofing-Aware Speaker Verification Robust Against Domain and Channel Mismatches">6</a> shows the EER results for the scenario of domain mismatch under various protocols. Notably, regardless of the training protocols employed, the proposed model, trained using the integrated learning paradigm, exhibits a substantial overall improvement in performance, as indicated by the third column in the table. </div> <div class="ltx_para" id="S4.SS3.SSS3.p2"> Within each protocol, the presence of an unseen group is indicated within brackets. The proposed approach greatly outperformed in terms of EER in comparison to the baseline system, particularly in the “si” genre of CGP I and the “pl” genre of CGP III. This observation underscores the effectiveness of the proposed model, trained using the integrated learning paradigm, in mitigating the challenges posed by domain mismatch in the context of the ASV task. </div> <div class="ltx_para" id="S4.SS3.SSS3.p3"> Furthermore, the EER numbers for seen genres are also enhanced through the use of the integrated learning paradigm. This suggests that the gradients derived from the meta-test optimization process can effectively serve as a regularization term, correcting the biases inherently introduced by the supervised learning paradigm. </div> <div class="ltx_para" id="S4.SS3.SSS3.p4"> In summary, the results presented in the table highlight that the proposed approach consistently outperforms the baseline system in mitigating domain mismatch, resulting in improved overall EER and EER reductions across various genres in each training protocol. </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 5 Conclusion</h2> <div class="ltx_para" id="S5.p1"> In this paper, we proposed an asymmetric dual-path model and an integrated framework unifying pair-wise learning, spoofing simulation, and meta-learning for robustness against channel mismatch, spoofing attacks, and domain shifts. The bilevel optimization structure makes it possible to these multifaceted issues concurrently in an end-to-end model. </div> <div class="ltx_para" id="S5.p2"> To demonstrate the advantages of the proposed model and learning paradigm, a series of experiments were conducted to show the performance improvement independently under scenarios of channel mismatch, spoofing attacks, and domain mismatch, respectively. The experiments demonstrated consistent improvements over baseline systems trained in a supervised learning paradigm. The proposed approach provides a blueprint for developing production ASV systems that are dependable under diverse channel mismatch, spoofing attacks, and domain mismatch. </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section" style="font-size:90%;"> 6 ACKNOWLEDGMENTS</h2> <div class="ltx_para" id="S6.p1"> This study is partially supported by JST CREST Grants (JPMJCR18A6, JPMJCR20D3) and JST AIP Acceleration Research (JPMJCR24U3). This research was also partially funded by the project to develop and demonstrate countermeasures against disinformation and misinformation on the Internet with the Ministry of Internal Affairs and Communications of Japan. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography" style="font-size:90%;">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> [1] Douglas A Reynolds, Thomas F Quatieri, and Robert B Dunn, “Speaker Verification Using Adapted Gaussian Mixture Models,” Digital Signal Processing, vol. 10, no. 1-3, pp. 19–41, 2000. </li> <li class="ltx_bibitem" id="bib.bib2"> [2] Najim Dehak, Patrick J Kenny, Réda Dehak, Pierre Dumouchel, and Pierre Ouellet, “Front-end factor analysis for speaker verification,” IEEE Transactions on Audio, Speech, and Language Processing, vol. 19, no. 4, pp. 788–798, 2010. </li> <li class="ltx_bibitem" id="bib.bib3"> [3] David Snyder, Daniel Garcia-Romero, Daniel Povey, and Sanjeev Khudanpur, “Deep Neural Network Embeddings for Text-Independent Speaker Verification.,” in Proc. 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