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<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> <meta content=" Beam management, transfer learning, beam prediction, millimeter-wave " lang="en" name="keywords"/> <base href="/html/2503.14287v1/"/></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/2503.14287v1#S1" title="In Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2" title="In Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">System Model </span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.SS1" title="In II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-A</span> </span><span class="ltx_text ltx_font_italic">Network Model</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.SS2" title="In II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-B</span> </span><span class="ltx_text ltx_font_italic">Channel Model</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3" title="In Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Beam Prediction Problem Formulation & Proposed Transfer Learning Solution</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3.SS1" title="In III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic">Beam Prediction Problem Formulation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3.SS2" title="In III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic">Proposed Location-Based Transfer Learning Solution</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4" title="In Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Results</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS1" title="In IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">Intra-City Transfer Learning Performance</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS2" title="In IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Inter-City Transfer Learning Performance</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS3" title="In IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-C</span> </span><span class="ltx_text ltx_font_italic">Fine-Tuning Effect</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S5" title="In Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Conclusions</span></span></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">Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks <br class="ltx_break"/><span class="ltx_note ltx_role_thanks" id="id1.id1"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span>This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038.</span></span></span> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Enrico Tosi2, Panwei Hu2, Aleksandar Ichkov2, Marina Petrova12, Ljiljana Simić2 </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"> 2 <span class="ltx_text ltx_font_italic" id="id2.1.id1">Institute for Networked Systems, RWTH Aachen University</span> </span> <span class="ltx_contact ltx_role_affiliation"> 1 <span class="ltx_text ltx_font_italic" id="id3.2.id1">Mobile Communications and Computing, RWTH Aachen University</span> </span> <span class="ltx_contact ltx_role_affiliation"> 2 {eto, pwu, aic, lsi}@inets.rwth-aachen.de 1 petrova@mcc.rwth-aachen.de </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id4.id1">Millimeter-wave (mm-wave) communications require beamforming and consequent precise beam alignment between the gNodeB (gNB) and the user equipment (UE) to overcome high propagation losses. This beam alignment needs to be constantly updated for different UE locations based on beam-sweeping radio frequency measurements, leading to significant beam management overhead. One potential solution involves using machine learning (ML) beam prediction algorithms that leverage UE position information to select the serving beam without the overhead of beam sweeping. However, the highly site-specific nature of mm-wave propagation means that ML models require training from scratch for each scenario, which is inefficient in practice. In this paper, we propose a robust cross-environment transfer learning solution for location-aided beam prediction, whereby the ML model trained on a reference gNB is transferred to a target gNB by fine-tuning with a limited dataset. Extensive simulation results based on ray-tracing in two urban environments show the effectiveness of our solution for both inter- and intra-city model transfer. Our results show that by training the model on a reference gNB and transferring the model by fine-tuning with only 5% of the target gNB dataset, we can achieve 80% accuracy in predicting the best beam for the target gNB. Importantly, our approach improves the poor generalization accuracy of transferring the model to new environments without fine-tuning by around 75 percentage points. This demonstrates that transfer learning enables high prediction accuracy while reducing the computational and training dataset collection burden of ML-based beam prediction, making it practical for 5G-and-beyond deployments.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> Beam management, transfer learning, beam prediction, millimeter-wave </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">The rapid evolution of wireless communication technologies has led to the exploration of millimeter-wave (mm-wave) spectrum, which provides substantial bandwidth and enables ultra-high data rates and low latency essential for next-generation cellular networks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib1" title="">1</a>]</cite>. However, mm-wave frequencies require directional beamforming to overcome high propagation loss. In turn, the sparse nature of the mm-wave channel necessitates precise beam alignment between the gNodeB (gNB) and user equipment (UE). Since the feasible line-of-sight (LoS) and non-LoS (NLoS) propagation paths depend on the urban geometry of the network site, this beam alignment must be constantly updated for different UE locations, resulting in high beam management overhead <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib2" title="">2</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Machine learning (ML) algorithms have been proposed as a solution for predicting the best beam-pair-link (BPL), thus minimizing the associated overheads (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib3" title="">3</a>]</cite> and references therein). One promising approach leverages UE location information as training data, capitalizing on the inherent directionality of mm-wave to improve beam prediction accuracy, as explored in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib5" title="">5</a>]</cite>. However, the ability of these location-aided beam prediction algorithms to generalize across different scenarios has not been investigated. Namely, the prior works on ML-based location-aided beam prediction showed high accuracy when trained and tested on the same gNB, but the generalization accuracy of these models, trained on one gNB but tested on another, is expected to be poor given the site-specific nature of mm-wave propagation. This is a crucial aspect, as training a new ML model from scratch for every scenario is highly inefficient in practice, requires datasets from up to several thousand locations which are costly to acquire, and poses significant computational overhead <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib4" title="">4</a>]</cite>. Transfer learning has emerged as a powerful technique that allows ML models trained for one task to be adapted for another related task through fine-tuning with minimal retraining <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib6" title="">6</a>]</cite>. While some prior works in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib8" title="">8</a>]</cite> demonstrated the benefit of transfer learning in enabling fast model readaptation in the context of mm-wave beam management, they focus on transferring knowledge between related tasks within the same gNB. Consequently, they do not explore the transfer capabilities of their algorithms across different physical scenarios, i.e., different gNBs or network environments, leaving open the question of whether ML-based beam management knowledge can be transferred.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">To address this important gap, we propose a robust cross-environment transfer learning solution based on a common fully connected neural network (NN) model that inputs UE location information and predicts the best BPL for serving that location. To the best of our knowledge, we are the first to tackle the challenge of transferring the acquired location-aided beam prediction knowledge between gNBs in different physical environments, both within the same city or across cities with varying degrees of urbanization. We consider training the ML model on a reference gNB and then transferring it to a target gNB by fine-tuning the model with a much more limited dataset on the target gNB. We present extensive simulation results, based on ray-tracing in mm-wave networks of 27 gNBs in the real urban environments of Frankfurt and Seoul, to show the excellent transfer learning performance both for inter- and intra-city model transfer. Specifically, our results show that by training the model on a reference gNB and then transferring the model by fine-tuning with only the 5% of the target gNB dataset, we achieve up to 80% accuracy in correctly predicting the best BPL for the target gNB. We note that our approach improves the very poor generalization accuracy of simply transferring the model to new environments without fine-tuning by around 75 percentage points. This demonstrates that our proposed transfer learning solution enables high beam prediction accuracy while reducing the computational and training dataset collection burden of ML-based beam prediction, thus making it suitable for practical 5G-and-beyond network deployments.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">The rest of the paper is organized as follows. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2" title="II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">II</span></a> presents the system model. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3" title="III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">III</span></a> formulates the BPL prediction problem and presents our transfer learning solution. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4" title="IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">IV</span></a> presents simulation results to evaluate the BPL prediction accuracy. Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S5" title="V Conclusions ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">V</span></a> concludes the paper.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">System Model </span> </h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS1.4.1.1">II-A</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS1.5.2">Network Model</span> </h3> <figure class="ltx_figure" id="S2.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S2.F1.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="804" id="S2.F1.sf1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F1.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S2.F1.sf1.3.2" style="font-size:90%;">Seoul.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S2.F1.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="802" id="S2.F1.sf2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F1.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S2.F1.sf2.3.2" style="font-size:90%;">Frankfurt. </span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S2.F1.3.2" style="font-size:90%;">Building layout (grey) and gNB locations (green) in our two urban network study areas.</span></figcaption> </figure> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.10">We consider an urban downlink mm-wave cellular network consisting of 27 gNBs placed at the building corners at a height of <math alttext="10" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mn id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><cn id="S2.SS1.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS1.p1.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">10</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">10</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S2.SS1.p1.10.1">m</span>, and distributed in a roughly uniform manner in a study area of <math alttext="500" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mn id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><cn id="S2.SS1.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p1.3.m3.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">500</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">500</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S2.SS1.p1.10.2">m</span> <math alttext="\times" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mo id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><times id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">×</annotation></semantics></math> <math alttext="500" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mn id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><cn id="S2.SS1.p1.6.m6.1.1.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">500</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">500</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S2.SS1.p1.10.3">m</span>. We assume a carrier frequency <math alttext="f_{c}=28" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mrow id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml"><msub id="S2.SS1.p1.8.m8.1.1.2" xref="S2.SS1.p1.8.m8.1.1.2.cmml"><mi id="S2.SS1.p1.8.m8.1.1.2.2" xref="S2.SS1.p1.8.m8.1.1.2.2.cmml">f</mi><mi id="S2.SS1.p1.8.m8.1.1.2.3" xref="S2.SS1.p1.8.m8.1.1.2.3.cmml">c</mi></msub><mo id="S2.SS1.p1.8.m8.1.1.1" xref="S2.SS1.p1.8.m8.1.1.1.cmml">=</mo><mn id="S2.SS1.p1.8.m8.1.1.3" xref="S2.SS1.p1.8.m8.1.1.3.cmml">28</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><apply id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1"><eq id="S2.SS1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1.1"></eq><apply id="S2.SS1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.1.1.2.1.cmml" xref="S2.SS1.p1.8.m8.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.8.m8.1.1.2.2.cmml" xref="S2.SS1.p1.8.m8.1.1.2.2">𝑓</ci><ci id="S2.SS1.p1.8.m8.1.1.2.3.cmml" xref="S2.SS1.p1.8.m8.1.1.2.3">𝑐</ci></apply><cn id="S2.SS1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p1.8.m8.1.1.3">28</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">f_{c}=28</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">italic_f start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 28</annotation></semantics></math> GHz and a nominal gNB transmit power of <math alttext="P_{\text{tx}}=20" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><mrow id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml"><msub id="S2.SS1.p1.9.m9.1.1.2" xref="S2.SS1.p1.9.m9.1.1.2.cmml"><mi id="S2.SS1.p1.9.m9.1.1.2.2" xref="S2.SS1.p1.9.m9.1.1.2.2.cmml">P</mi><mtext id="S2.SS1.p1.9.m9.1.1.2.3" xref="S2.SS1.p1.9.m9.1.1.2.3a.cmml">tx</mtext></msub><mo id="S2.SS1.p1.9.m9.1.1.1" xref="S2.SS1.p1.9.m9.1.1.1.cmml">=</mo><mn id="S2.SS1.p1.9.m9.1.1.3" xref="S2.SS1.p1.9.m9.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><apply id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1"><eq id="S2.SS1.p1.9.m9.1.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1.1"></eq><apply id="S2.SS1.p1.9.m9.1.1.2.cmml" xref="S2.SS1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.1.1.2.1.cmml" xref="S2.SS1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.9.m9.1.1.2.2.cmml" xref="S2.SS1.p1.9.m9.1.1.2.2">𝑃</ci><ci id="S2.SS1.p1.9.m9.1.1.2.3a.cmml" xref="S2.SS1.p1.9.m9.1.1.2.3"><mtext id="S2.SS1.p1.9.m9.1.1.2.3.cmml" mathsize="70%" xref="S2.SS1.p1.9.m9.1.1.2.3">tx</mtext></ci></apply><cn id="S2.SS1.p1.9.m9.1.1.3.cmml" type="integer" xref="S2.SS1.p1.9.m9.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">P_{\text{tx}}=20</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">italic_P start_POSTSUBSCRIPT tx end_POSTSUBSCRIPT = 20</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S2.SS1.p1.10.4">dBm</span>. We select two study areas with different levels of urbanization: an open space commercial district in Seoul, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.F1.sf1" title="In Figure 1 ‣ II-A Network Model ‣ II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1(a)</span></a>, and a building-dense area around the central station in Frankfurt, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.F1.sf2" title="In Figure 1 ‣ II-A Network Model ‣ II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1(b)</span></a>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.16">We assume phased antenna arrays of size <math alttext="8" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mn id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><cn id="S2.SS1.p2.1.m1.1.1.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">8</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">8</annotation></semantics></math> <math alttext="\times" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mo id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><times id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">×</annotation></semantics></math> <math alttext="8" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mn id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><cn id="S2.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p2.3.m3.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">8</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">8</annotation></semantics></math> at the gNB and <math alttext="4" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mn id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><cn id="S2.SS1.p2.4.m4.1.1.cmml" type="integer" xref="S2.SS1.p2.4.m4.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">4</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">4</annotation></semantics></math> <math alttext="\times" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mo id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><times id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">×</annotation></semantics></math> <math alttext="4" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><mn id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.1b"><cn id="S2.SS1.p2.6.m6.1.1.cmml" type="integer" xref="S2.SS1.p2.6.m6.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">4</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">4</annotation></semantics></math> at the UE, employing codebook-based analog beamforming to cover the complete angular space. We assume a codebook of size <math alttext="N_{gNB}=64" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mrow id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml"><msub id="S2.SS1.p2.7.m7.1.1.2" xref="S2.SS1.p2.7.m7.1.1.2.cmml"><mi id="S2.SS1.p2.7.m7.1.1.2.2" xref="S2.SS1.p2.7.m7.1.1.2.2.cmml">N</mi><mrow id="S2.SS1.p2.7.m7.1.1.2.3" xref="S2.SS1.p2.7.m7.1.1.2.3.cmml"><mi id="S2.SS1.p2.7.m7.1.1.2.3.2" xref="S2.SS1.p2.7.m7.1.1.2.3.2.cmml">g</mi><mo id="S2.SS1.p2.7.m7.1.1.2.3.1" xref="S2.SS1.p2.7.m7.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.7.m7.1.1.2.3.3" xref="S2.SS1.p2.7.m7.1.1.2.3.3.cmml">N</mi><mo id="S2.SS1.p2.7.m7.1.1.2.3.1a" xref="S2.SS1.p2.7.m7.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.7.m7.1.1.2.3.4" xref="S2.SS1.p2.7.m7.1.1.2.3.4.cmml">B</mi></mrow></msub><mo id="S2.SS1.p2.7.m7.1.1.1" xref="S2.SS1.p2.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS1.p2.7.m7.1.1.3" xref="S2.SS1.p2.7.m7.1.1.3.cmml">64</mn></mrow><annotation-xml 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id="S2.SS1.p2.7.m7.1c">N_{gNB}=64</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">italic_N start_POSTSUBSCRIPT italic_g italic_N italic_B end_POSTSUBSCRIPT = 64</annotation></semantics></math> at the gNB and <math alttext="N_{UE}=16" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><mrow id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml"><msub id="S2.SS1.p2.8.m8.1.1.2" xref="S2.SS1.p2.8.m8.1.1.2.cmml"><mi id="S2.SS1.p2.8.m8.1.1.2.2" xref="S2.SS1.p2.8.m8.1.1.2.2.cmml">N</mi><mrow id="S2.SS1.p2.8.m8.1.1.2.3" xref="S2.SS1.p2.8.m8.1.1.2.3.cmml"><mi id="S2.SS1.p2.8.m8.1.1.2.3.2" xref="S2.SS1.p2.8.m8.1.1.2.3.2.cmml">U</mi><mo id="S2.SS1.p2.8.m8.1.1.2.3.1" xref="S2.SS1.p2.8.m8.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.8.m8.1.1.2.3.3" xref="S2.SS1.p2.8.m8.1.1.2.3.3.cmml">E</mi></mrow></msub><mo id="S2.SS1.p2.8.m8.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.cmml">=</mo><mn id="S2.SS1.p2.8.m8.1.1.3" xref="S2.SS1.p2.8.m8.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><apply id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1"><eq id="S2.SS1.p2.8.m8.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1.1"></eq><apply id="S2.SS1.p2.8.m8.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.8.m8.1.1.2.1.cmml" xref="S2.SS1.p2.8.m8.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.8.m8.1.1.2.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2.2">𝑁</ci><apply id="S2.SS1.p2.8.m8.1.1.2.3.cmml" xref="S2.SS1.p2.8.m8.1.1.2.3"><times id="S2.SS1.p2.8.m8.1.1.2.3.1.cmml" xref="S2.SS1.p2.8.m8.1.1.2.3.1"></times><ci id="S2.SS1.p2.8.m8.1.1.2.3.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2.3.2">𝑈</ci><ci id="S2.SS1.p2.8.m8.1.1.2.3.3.cmml" xref="S2.SS1.p2.8.m8.1.1.2.3.3">𝐸</ci></apply></apply><cn id="S2.SS1.p2.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p2.8.m8.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">N_{UE}=16</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_N start_POSTSUBSCRIPT italic_U italic_E end_POSTSUBSCRIPT = 16</annotation></semantics></math> at the UE, corresponding to a beamwidth of <math alttext="5.6^{\circ}" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><msup id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml"><mn id="S2.SS1.p2.9.m9.1.1.2" xref="S2.SS1.p2.9.m9.1.1.2.cmml">5.6</mn><mo id="S2.SS1.p2.9.m9.1.1.3" xref="S2.SS1.p2.9.m9.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><apply id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.9.m9.1.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1">superscript</csymbol><cn id="S2.SS1.p2.9.m9.1.1.2.cmml" type="float" xref="S2.SS1.p2.9.m9.1.1.2">5.6</cn><compose id="S2.SS1.p2.9.m9.1.1.3.cmml" xref="S2.SS1.p2.9.m9.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">5.6^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">5.6 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="22.5^{\circ}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.1"><semantics id="S2.SS1.p2.10.m10.1a"><msup id="S2.SS1.p2.10.m10.1.1" xref="S2.SS1.p2.10.m10.1.1.cmml"><mn id="S2.SS1.p2.10.m10.1.1.2" xref="S2.SS1.p2.10.m10.1.1.2.cmml">22.5</mn><mo id="S2.SS1.p2.10.m10.1.1.3" xref="S2.SS1.p2.10.m10.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.1b"><apply id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.10.m10.1.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">superscript</csymbol><cn id="S2.SS1.p2.10.m10.1.1.2.cmml" type="float" xref="S2.SS1.p2.10.m10.1.1.2">22.5</cn><compose id="S2.SS1.p2.10.m10.1.1.3.cmml" xref="S2.SS1.p2.10.m10.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.1c">22.5^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.1d">22.5 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively. We define a directional BPL between a given gNB/UE pair as <math alttext="l_{i,j}" class="ltx_Math" display="inline" id="S2.SS1.p2.11.m11.2"><semantics id="S2.SS1.p2.11.m11.2a"><msub id="S2.SS1.p2.11.m11.2.3" xref="S2.SS1.p2.11.m11.2.3.cmml"><mi id="S2.SS1.p2.11.m11.2.3.2" xref="S2.SS1.p2.11.m11.2.3.2.cmml">l</mi><mrow id="S2.SS1.p2.11.m11.2.2.2.4" xref="S2.SS1.p2.11.m11.2.2.2.3.cmml"><mi id="S2.SS1.p2.11.m11.1.1.1.1" xref="S2.SS1.p2.11.m11.1.1.1.1.cmml">i</mi><mo id="S2.SS1.p2.11.m11.2.2.2.4.1" xref="S2.SS1.p2.11.m11.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p2.11.m11.2.2.2.2" xref="S2.SS1.p2.11.m11.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.11.m11.2b"><apply id="S2.SS1.p2.11.m11.2.3.cmml" xref="S2.SS1.p2.11.m11.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.11.m11.2.3.1.cmml" xref="S2.SS1.p2.11.m11.2.3">subscript</csymbol><ci id="S2.SS1.p2.11.m11.2.3.2.cmml" xref="S2.SS1.p2.11.m11.2.3.2">𝑙</ci><list id="S2.SS1.p2.11.m11.2.2.2.3.cmml" xref="S2.SS1.p2.11.m11.2.2.2.4"><ci id="S2.SS1.p2.11.m11.1.1.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1.1.1">𝑖</ci><ci id="S2.SS1.p2.11.m11.2.2.2.2.cmml" xref="S2.SS1.p2.11.m11.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.11.m11.2c">l_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.11.m11.2d">italic_l start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p2.12.m12.1"><semantics id="S2.SS1.p2.12.m12.1a"><mi id="S2.SS1.p2.12.m12.1.1" xref="S2.SS1.p2.12.m12.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.12.m12.1b"><ci id="S2.SS1.p2.12.m12.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.12.m12.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.12.m12.1d">italic_i</annotation></semantics></math> denotes the gNB beam ID and <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p2.13.m13.1"><semantics id="S2.SS1.p2.13.m13.1a"><mi id="S2.SS1.p2.13.m13.1.1" xref="S2.SS1.p2.13.m13.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.13.m13.1b"><ci id="S2.SS1.p2.13.m13.1.1.cmml" xref="S2.SS1.p2.13.m13.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.13.m13.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.13.m13.1d">italic_j</annotation></semantics></math> denotes the UE beam ID. In total, there are <math alttext="64" class="ltx_Math" display="inline" id="S2.SS1.p2.14.m14.1"><semantics id="S2.SS1.p2.14.m14.1a"><mn id="S2.SS1.p2.14.m14.1.1" xref="S2.SS1.p2.14.m14.1.1.cmml">64</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.14.m14.1b"><cn id="S2.SS1.p2.14.m14.1.1.cmml" type="integer" xref="S2.SS1.p2.14.m14.1.1">64</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.14.m14.1c">64</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.14.m14.1d">64</annotation></semantics></math> <math alttext="\times" class="ltx_Math" display="inline" id="S2.SS1.p2.15.m15.1"><semantics id="S2.SS1.p2.15.m15.1a"><mo id="S2.SS1.p2.15.m15.1.1" xref="S2.SS1.p2.15.m15.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.15.m15.1b"><times id="S2.SS1.p2.15.m15.1.1.cmml" xref="S2.SS1.p2.15.m15.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.15.m15.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.15.m15.1d">×</annotation></semantics></math> <math alttext="16" class="ltx_Math" display="inline" id="S2.SS1.p2.16.m16.1"><semantics id="S2.SS1.p2.16.m16.1a"><mn id="S2.SS1.p2.16.m16.1.1" xref="S2.SS1.p2.16.m16.1.1.cmml">16</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.16.m16.1b"><cn id="S2.SS1.p2.16.m16.1.1.cmml" type="integer" xref="S2.SS1.p2.16.m16.1.1">16</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.16.m16.1c">16</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.16.m16.1d">16</annotation></semantics></math> potential candidate BPLs. We use real 3D antenna patterns obtained via lab measurements with commercially available mm-wave phased antenna arrays <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib9" title="">9</a>]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS2.4.1.1">II-B</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS2.5.2">Channel Model</span> </h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.4">We use the commercial ray-tracing simulator Wireless InSite <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib10" title="">10</a>]</cite> to obtain site-specific channel data based on publicly available 3D building models shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.F1.sf1" title="In Figure 1 ‣ II-A Network Model ‣ II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1(a)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.F1.sf2" title="In Figure 1 ‣ II-A Network Model ‣ II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1(b)</span></a> for Seoul and Frankfurt, respectively. Channel information is collected with a grid resolution of <math alttext="1\text{m}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mrow id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mn id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">1</mn><mo id="S2.SS2.p1.1.m1.1.1.1" xref="S2.SS2.p1.1.m1.1.1.1.cmml">⁢</mo><mtext id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3a.cmml">m</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"><times id="S2.SS2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1.1"></times><cn id="S2.SS2.p1.1.m1.1.1.2.cmml" type="integer" xref="S2.SS2.p1.1.m1.1.1.2">1</cn><ci id="S2.SS2.p1.1.m1.1.1.3a.cmml" xref="S2.SS2.p1.1.m1.1.1.3"><mtext id="S2.SS2.p1.1.m1.1.1.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3">m</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">1\text{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">1 m</annotation></semantics></math> <math alttext="\times" class="ltx_Math" display="inline" id="S2.SS2.p1.2.m2.1"><semantics id="S2.SS2.p1.2.m2.1a"><mo id="S2.SS2.p1.2.m2.1.1" xref="S2.SS2.p1.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.2.m2.1b"><times id="S2.SS2.p1.2.m2.1.1.cmml" xref="S2.SS2.p1.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.1c">\times</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.1d">×</annotation></semantics></math> <math alttext="1\text{m}" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mrow id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml"><mn id="S2.SS2.p1.3.m3.1.1.2" xref="S2.SS2.p1.3.m3.1.1.2.cmml">1</mn><mo id="S2.SS2.p1.3.m3.1.1.1" xref="S2.SS2.p1.3.m3.1.1.1.cmml">⁢</mo><mtext id="S2.SS2.p1.3.m3.1.1.3" xref="S2.SS2.p1.3.m3.1.1.3a.cmml">m</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><apply id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1"><times id="S2.SS2.p1.3.m3.1.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1.1"></times><cn id="S2.SS2.p1.3.m3.1.1.2.cmml" type="integer" xref="S2.SS2.p1.3.m3.1.1.2">1</cn><ci id="S2.SS2.p1.3.m3.1.1.3a.cmml" xref="S2.SS2.p1.3.m3.1.1.3"><mtext id="S2.SS2.p1.3.m3.1.1.3.cmml" xref="S2.SS2.p1.3.m3.1.1.3">m</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">1\text{m}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">1 m</annotation></semantics></math> with a ray-launching granularity of <math alttext="0.1^{\circ}" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.1"><semantics id="S2.SS2.p1.4.m4.1a"><msup id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><mn id="S2.SS2.p1.4.m4.1.1.2" xref="S2.SS2.p1.4.m4.1.1.2.cmml">0.1</mn><mo id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.1b"><apply id="S2.SS2.p1.4.m4.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1">superscript</csymbol><cn id="S2.SS2.p1.4.m4.1.1.2.cmml" type="float" xref="S2.SS2.p1.4.m4.1.1.2">0.1</cn><compose id="S2.SS2.p1.4.m4.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.1c">0.1^{\circ}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.1d">0.1 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math>. Given the dominant propagation characteristics of mm-wave, diffraction and scattering are neglected, while the maximum number of reflections is set to four <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib11" title="">11</a>]</cite>. We model the buildings using the standardized international telecommunication union (ITU) glass as the material.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.5">For each considered gNB, the ray-tracing output consists of all the propagation paths from that gNB to each potential UE location within the area. 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departure (AoD), <math alttext="\{\phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k}\}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.2"><semantics id="S2.SS2.p2.3.m3.2a"><mrow id="S2.SS2.p2.3.m3.2.2.2" xref="S2.SS2.p2.3.m3.2.2.3.cmml"><mo id="S2.SS2.p2.3.m3.2.2.2.3" stretchy="false" xref="S2.SS2.p2.3.m3.2.2.3.cmml">{</mo><msubsup id="S2.SS2.p2.3.m3.1.1.1.1" xref="S2.SS2.p2.3.m3.1.1.1.1.cmml"><mi id="S2.SS2.p2.3.m3.1.1.1.1.2.2" xref="S2.SS2.p2.3.m3.1.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.SS2.p2.3.m3.1.1.1.1.3" xref="S2.SS2.p2.3.m3.1.1.1.1.3.cmml">k</mi><mtext id="S2.SS2.p2.3.m3.1.1.1.1.2.3" xref="S2.SS2.p2.3.m3.1.1.1.1.2.3a.cmml">gNB</mtext></msubsup><mo id="S2.SS2.p2.3.m3.2.2.2.4" xref="S2.SS2.p2.3.m3.2.2.3.cmml">,</mo><msubsup id="S2.SS2.p2.3.m3.2.2.2.2" xref="S2.SS2.p2.3.m3.2.2.2.2.cmml"><mi id="S2.SS2.p2.3.m3.2.2.2.2.2.2" xref="S2.SS2.p2.3.m3.2.2.2.2.2.2.cmml">θ</mi><mi id="S2.SS2.p2.3.m3.2.2.2.2.3" xref="S2.SS2.p2.3.m3.2.2.2.2.3.cmml">k</mi><mtext id="S2.SS2.p2.3.m3.2.2.2.2.2.3" 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xref="S2.SS2.p2.3.m3.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS2.p2.3.m3.2.2.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m3.2.2.2.2.1.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p2.3.m3.2.2.2.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m3.2.2.2.2.2.1.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p2.3.m3.2.2.2.2.2.2.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2.2.2">𝜃</ci><ci id="S2.SS2.p2.3.m3.2.2.2.2.2.3a.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2.2.3"><mtext id="S2.SS2.p2.3.m3.2.2.2.2.2.3.cmml" mathsize="70%" xref="S2.SS2.p2.3.m3.2.2.2.2.2.3">gNB</mtext></ci></apply><ci id="S2.SS2.p2.3.m3.2.2.2.2.3.cmml" xref="S2.SS2.p2.3.m3.2.2.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.2c">\{\phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.2d">{ italic_ϕ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> in the azimuth and elevation respectively, and (<span class="ltx_text ltx_font_italic" id="S2.SS2.p2.5.3">iii</span>) total path loss considering free-space path loss plus any reflection loss. To derive the directional RSS values from this omnidirectional output, we align the antenna patterns of gNB and UE by applying different antenna codebook entries for each steering angle combination. Specifically, the RSS associated with a specific gNB/UE BPL combination <math alttext="\{i,j\}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.2"><semantics id="S2.SS2.p2.4.m4.2a"><mrow id="S2.SS2.p2.4.m4.2.3.2" xref="S2.SS2.p2.4.m4.2.3.1.cmml"><mo id="S2.SS2.p2.4.m4.2.3.2.1" stretchy="false" xref="S2.SS2.p2.4.m4.2.3.1.cmml">{</mo><mi id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">i</mi><mo id="S2.SS2.p2.4.m4.2.3.2.2" xref="S2.SS2.p2.4.m4.2.3.1.cmml">,</mo><mi id="S2.SS2.p2.4.m4.2.2" xref="S2.SS2.p2.4.m4.2.2.cmml">j</mi><mo id="S2.SS2.p2.4.m4.2.3.2.3" stretchy="false" xref="S2.SS2.p2.4.m4.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.2b"><set id="S2.SS2.p2.4.m4.2.3.1.cmml" xref="S2.SS2.p2.4.m4.2.3.2"><ci id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">𝑖</ci><ci id="S2.SS2.p2.4.m4.2.2.cmml" xref="S2.SS2.p2.4.m4.2.2">𝑗</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.2c">\{i,j\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.2d">{ italic_i , italic_j }</annotation></semantics></math> is computed by considering the contribution of all the propagation paths <math alttext="K" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m5.1"><semantics id="S2.SS2.p2.5.m5.1a"><mi id="S2.SS2.p2.5.m5.1.1" xref="S2.SS2.p2.5.m5.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m5.1b"><ci id="S2.SS2.p2.5.m5.1.1.cmml" xref="S2.SS2.p2.5.m5.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m5.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m5.1d">italic_K</annotation></semantics></math> for an arbitrary UE location, as given by:</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\text{RSS}_{i,j}=P_{\text{tx}}\left(\sum_{k=1}^{K}PL_{k}G^{\text{gNB}}_{i}(% \phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k})G^{\text{UE}}_{j}(\phi^{\text{UE% }}_{k},\theta^{\text{UE}}_{k})\right)" class="ltx_Math" display="block" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml"><msub id="S2.E1.m1.3.3.3" xref="S2.E1.m1.3.3.3.cmml"><mtext id="S2.E1.m1.3.3.3.2" xref="S2.E1.m1.3.3.3.2a.cmml">RSS</mtext><mrow id="S2.E1.m1.2.2.2.4" xref="S2.E1.m1.2.2.2.3.cmml"><mi id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml">i</mi><mo id="S2.E1.m1.2.2.2.4.1" xref="S2.E1.m1.2.2.2.3.cmml">,</mo><mi id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S2.E1.m1.3.3.2" xref="S2.E1.m1.3.3.2.cmml">=</mo><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.cmml"><msub id="S2.E1.m1.3.3.1.3" xref="S2.E1.m1.3.3.1.3.cmml"><mi id="S2.E1.m1.3.3.1.3.2" 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}}_{k},\theta^{\text{UE}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">RSS start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_P start_POSTSUBSCRIPT tx end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT italic_P italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_G start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.9">where <math alttext="PL_{k}" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><mrow id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><mi id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml">P</mi><mo id="S2.SS2.p4.1.m1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.cmml">⁢</mo><msub id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml"><mi id="S2.SS2.p4.1.m1.1.1.3.2" xref="S2.SS2.p4.1.m1.1.1.3.2.cmml">L</mi><mi id="S2.SS2.p4.1.m1.1.1.3.3" xref="S2.SS2.p4.1.m1.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><times id="S2.SS2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1.1"></times><ci id="S2.SS2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2">𝑃</ci><apply id="S2.SS2.p4.1.m1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.3.1.cmml" xref="S2.SS2.p4.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.1.m1.1.1.3.2.cmml" xref="S2.SS2.p4.1.m1.1.1.3.2">𝐿</ci><ci id="S2.SS2.p4.1.m1.1.1.3.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">PL_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">italic_P italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> denotes the total path loss for path <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mi id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><ci id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">italic_k</annotation></semantics></math>, <math alttext="G^{\text{gNB}}_{i}(\phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k})" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.2"><semantics id="S2.SS2.p4.3.m3.2a"><mrow id="S2.SS2.p4.3.m3.2.2" xref="S2.SS2.p4.3.m3.2.2.cmml"><msubsup id="S2.SS2.p4.3.m3.2.2.4" xref="S2.SS2.p4.3.m3.2.2.4.cmml"><mi id="S2.SS2.p4.3.m3.2.2.4.2.2" xref="S2.SS2.p4.3.m3.2.2.4.2.2.cmml">G</mi><mi id="S2.SS2.p4.3.m3.2.2.4.3" xref="S2.SS2.p4.3.m3.2.2.4.3.cmml">i</mi><mtext id="S2.SS2.p4.3.m3.2.2.4.2.3" xref="S2.SS2.p4.3.m3.2.2.4.2.3a.cmml">gNB</mtext></msubsup><mo id="S2.SS2.p4.3.m3.2.2.3" xref="S2.SS2.p4.3.m3.2.2.3.cmml">⁢</mo><mrow 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xref="S2.SS2.p4.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.2b"><apply id="S2.SS2.p4.3.m3.2.2.cmml" xref="S2.SS2.p4.3.m3.2.2"><times id="S2.SS2.p4.3.m3.2.2.3.cmml" xref="S2.SS2.p4.3.m3.2.2.3"></times><apply id="S2.SS2.p4.3.m3.2.2.4.cmml" xref="S2.SS2.p4.3.m3.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.2.2.4.1.cmml" xref="S2.SS2.p4.3.m3.2.2.4">subscript</csymbol><apply id="S2.SS2.p4.3.m3.2.2.4.2.cmml" xref="S2.SS2.p4.3.m3.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.2.2.4.2.1.cmml" xref="S2.SS2.p4.3.m3.2.2.4">superscript</csymbol><ci id="S2.SS2.p4.3.m3.2.2.4.2.2.cmml" xref="S2.SS2.p4.3.m3.2.2.4.2.2">𝐺</ci><ci id="S2.SS2.p4.3.m3.2.2.4.2.3a.cmml" xref="S2.SS2.p4.3.m3.2.2.4.2.3"><mtext id="S2.SS2.p4.3.m3.2.2.4.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.3.m3.2.2.4.2.3">gNB</mtext></ci></apply><ci id="S2.SS2.p4.3.m3.2.2.4.3.cmml" xref="S2.SS2.p4.3.m3.2.2.4.3">𝑖</ci></apply><interval closure="open" id="S2.SS2.p4.3.m3.2.2.2.3.cmml" xref="S2.SS2.p4.3.m3.2.2.2.2"><apply id="S2.SS2.p4.3.m3.1.1.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1.2.2">italic-ϕ</ci><ci id="S2.SS2.p4.3.m3.1.1.1.1.1.2.3a.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1.2.3"><mtext id="S2.SS2.p4.3.m3.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.3.m3.1.1.1.1.1.2.3">gNB</mtext></ci></apply><ci id="S2.SS2.p4.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.3.m3.1.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS2.p4.3.m3.2.2.2.2.2.cmml" xref="S2.SS2.p4.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.2.2.2.2.2.1.cmml" 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gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is the gNB antenna gain for the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mi id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><ci id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_i</annotation></semantics></math>-th codebook beam ID in the direction <math alttext="\{\phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k}\}" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.2"><semantics id="S2.SS2.p4.5.m5.2a"><mrow id="S2.SS2.p4.5.m5.2.2.2" 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id="S2.SS2.p4.5.m5.2b"><set id="S2.SS2.p4.5.m5.2.2.3.cmml" xref="S2.SS2.p4.5.m5.2.2.2"><apply id="S2.SS2.p4.5.m5.1.1.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.5.m5.1.1.1.1.2.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.5.m5.1.1.1.1.2.2.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1.2.2">italic-ϕ</ci><ci id="S2.SS2.p4.5.m5.1.1.1.1.2.3a.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1.2.3"><mtext id="S2.SS2.p4.5.m5.1.1.1.1.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.5.m5.1.1.1.1.2.3">gNB</mtext></ci></apply><ci id="S2.SS2.p4.5.m5.1.1.1.1.3.cmml" xref="S2.SS2.p4.5.m5.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS2.p4.5.m5.2.2.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.2.2.2.2.1.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p4.5.m5.2.2.2.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.2.2.2.2.2.1.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p4.5.m5.2.2.2.2.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2.2.2">𝜃</ci><ci id="S2.SS2.p4.5.m5.2.2.2.2.2.3a.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2.2.3"><mtext id="S2.SS2.p4.5.m5.2.2.2.2.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.5.m5.2.2.2.2.2.3">gNB</mtext></ci></apply><ci id="S2.SS2.p4.5.m5.2.2.2.2.3.cmml" xref="S2.SS2.p4.5.m5.2.2.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.2c">\{\phi^{\text{gNB}}_{k},\theta^{\text{gNB}}_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.2d">{ italic_ϕ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT gNB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> corresponding to the AoD of propagation path <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.1"><semantics id="S2.SS2.p4.6.m6.1a"><mi id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.1b"><ci id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.1d">italic_k</annotation></semantics></math>, and <math alttext="G^{\text{UE}}_{j}(\phi^{\text{UE}}_{k},\theta^{\text{UE}}_{k})" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.2"><semantics id="S2.SS2.p4.7.m7.2a"><mrow id="S2.SS2.p4.7.m7.2.2" xref="S2.SS2.p4.7.m7.2.2.cmml"><msubsup id="S2.SS2.p4.7.m7.2.2.4" xref="S2.SS2.p4.7.m7.2.2.4.cmml"><mi id="S2.SS2.p4.7.m7.2.2.4.2.2" xref="S2.SS2.p4.7.m7.2.2.4.2.2.cmml">G</mi><mi id="S2.SS2.p4.7.m7.2.2.4.3" xref="S2.SS2.p4.7.m7.2.2.4.3.cmml">j</mi><mtext id="S2.SS2.p4.7.m7.2.2.4.2.3" xref="S2.SS2.p4.7.m7.2.2.4.2.3a.cmml">UE</mtext></msubsup><mo id="S2.SS2.p4.7.m7.2.2.3" xref="S2.SS2.p4.7.m7.2.2.3.cmml">⁢</mo><mrow id="S2.SS2.p4.7.m7.2.2.2.2" xref="S2.SS2.p4.7.m7.2.2.2.3.cmml"><mo id="S2.SS2.p4.7.m7.2.2.2.2.3" stretchy="false" xref="S2.SS2.p4.7.m7.2.2.2.3.cmml">(</mo><msubsup id="S2.SS2.p4.7.m7.1.1.1.1.1" xref="S2.SS2.p4.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.7.m7.1.1.1.1.1.2.2" xref="S2.SS2.p4.7.m7.1.1.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.SS2.p4.7.m7.1.1.1.1.1.3" xref="S2.SS2.p4.7.m7.1.1.1.1.1.3.cmml">k</mi><mtext id="S2.SS2.p4.7.m7.1.1.1.1.1.2.3" xref="S2.SS2.p4.7.m7.1.1.1.1.1.2.3a.cmml">UE</mtext></msubsup><mo id="S2.SS2.p4.7.m7.2.2.2.2.4" xref="S2.SS2.p4.7.m7.2.2.2.3.cmml">,</mo><msubsup id="S2.SS2.p4.7.m7.2.2.2.2.2" xref="S2.SS2.p4.7.m7.2.2.2.2.2.cmml"><mi id="S2.SS2.p4.7.m7.2.2.2.2.2.2.2" xref="S2.SS2.p4.7.m7.2.2.2.2.2.2.2.cmml">θ</mi><mi id="S2.SS2.p4.7.m7.2.2.2.2.2.3" xref="S2.SS2.p4.7.m7.2.2.2.2.2.3.cmml">k</mi><mtext id="S2.SS2.p4.7.m7.2.2.2.2.2.2.3" xref="S2.SS2.p4.7.m7.2.2.2.2.2.2.3a.cmml">UE</mtext></msubsup><mo id="S2.SS2.p4.7.m7.2.2.2.2.5" stretchy="false" xref="S2.SS2.p4.7.m7.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.2b"><apply id="S2.SS2.p4.7.m7.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2"><times id="S2.SS2.p4.7.m7.2.2.3.cmml" xref="S2.SS2.p4.7.m7.2.2.3"></times><apply id="S2.SS2.p4.7.m7.2.2.4.cmml" xref="S2.SS2.p4.7.m7.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.2.2.4.1.cmml" xref="S2.SS2.p4.7.m7.2.2.4">subscript</csymbol><apply id="S2.SS2.p4.7.m7.2.2.4.2.cmml" xref="S2.SS2.p4.7.m7.2.2.4"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.2.2.4.2.1.cmml" xref="S2.SS2.p4.7.m7.2.2.4">superscript</csymbol><ci id="S2.SS2.p4.7.m7.2.2.4.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2.4.2.2">𝐺</ci><ci id="S2.SS2.p4.7.m7.2.2.4.2.3a.cmml" xref="S2.SS2.p4.7.m7.2.2.4.2.3"><mtext id="S2.SS2.p4.7.m7.2.2.4.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.7.m7.2.2.4.2.3">UE</mtext></ci></apply><ci id="S2.SS2.p4.7.m7.2.2.4.3.cmml" xref="S2.SS2.p4.7.m7.2.2.4.3">𝑗</ci></apply><interval closure="open" id="S2.SS2.p4.7.m7.2.2.2.3.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2"><apply id="S2.SS2.p4.7.m7.1.1.1.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.7.m7.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1.2.2">italic-ϕ</ci><ci id="S2.SS2.p4.7.m7.1.1.1.1.1.2.3a.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1.2.3"><mtext id="S2.SS2.p4.7.m7.1.1.1.1.1.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.7.m7.1.1.1.1.1.2.3">UE</mtext></ci></apply><ci id="S2.SS2.p4.7.m7.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.7.m7.1.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS2.p4.7.m7.2.2.2.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.2.2.2.2.2.1.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p4.7.m7.2.2.2.2.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.2.2.2.2.2.2.1.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p4.7.m7.2.2.2.2.2.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2.2.2">𝜃</ci><ci id="S2.SS2.p4.7.m7.2.2.2.2.2.2.3a.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2.2.3"><mtext id="S2.SS2.p4.7.m7.2.2.2.2.2.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.7.m7.2.2.2.2.2.2.3">UE</mtext></ci></apply><ci id="S2.SS2.p4.7.m7.2.2.2.2.2.3.cmml" xref="S2.SS2.p4.7.m7.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.2c">G^{\text{UE}}_{j}(\phi^{\text{UE}}_{k},\theta^{\text{UE}}_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.2d">italic_G start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is the UE antenna gain for the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m8.1"><semantics id="S2.SS2.p4.8.m8.1a"><mi id="S2.SS2.p4.8.m8.1.1" xref="S2.SS2.p4.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m8.1b"><ci id="S2.SS2.p4.8.m8.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.1d">italic_j</annotation></semantics></math>-th codebook ID in the direction <math alttext="\{\phi^{\text{UE}}_{k},\theta^{\text{UE}}_{k}\}" class="ltx_Math" display="inline" id="S2.SS2.p4.9.m9.2"><semantics id="S2.SS2.p4.9.m9.2a"><mrow id="S2.SS2.p4.9.m9.2.2.2" xref="S2.SS2.p4.9.m9.2.2.3.cmml"><mo id="S2.SS2.p4.9.m9.2.2.2.3" stretchy="false" xref="S2.SS2.p4.9.m9.2.2.3.cmml">{</mo><msubsup id="S2.SS2.p4.9.m9.1.1.1.1" xref="S2.SS2.p4.9.m9.1.1.1.1.cmml"><mi id="S2.SS2.p4.9.m9.1.1.1.1.2.2" xref="S2.SS2.p4.9.m9.1.1.1.1.2.2.cmml">ϕ</mi><mi id="S2.SS2.p4.9.m9.1.1.1.1.3" xref="S2.SS2.p4.9.m9.1.1.1.1.3.cmml">k</mi><mtext id="S2.SS2.p4.9.m9.1.1.1.1.2.3" xref="S2.SS2.p4.9.m9.1.1.1.1.2.3a.cmml">UE</mtext></msubsup><mo id="S2.SS2.p4.9.m9.2.2.2.4" xref="S2.SS2.p4.9.m9.2.2.3.cmml">,</mo><msubsup id="S2.SS2.p4.9.m9.2.2.2.2" xref="S2.SS2.p4.9.m9.2.2.2.2.cmml"><mi id="S2.SS2.p4.9.m9.2.2.2.2.2.2" xref="S2.SS2.p4.9.m9.2.2.2.2.2.2.cmml">θ</mi><mi id="S2.SS2.p4.9.m9.2.2.2.2.3" xref="S2.SS2.p4.9.m9.2.2.2.2.3.cmml">k</mi><mtext id="S2.SS2.p4.9.m9.2.2.2.2.2.3" xref="S2.SS2.p4.9.m9.2.2.2.2.2.3a.cmml">UE</mtext></msubsup><mo id="S2.SS2.p4.9.m9.2.2.2.5" stretchy="false" xref="S2.SS2.p4.9.m9.2.2.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m9.2b"><set id="S2.SS2.p4.9.m9.2.2.3.cmml" xref="S2.SS2.p4.9.m9.2.2.2"><apply id="S2.SS2.p4.9.m9.1.1.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m9.1.1.1.1.1.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1">subscript</csymbol><apply id="S2.SS2.p4.9.m9.1.1.1.1.2.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m9.1.1.1.1.2.1.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.9.m9.1.1.1.1.2.2.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1.2.2">italic-ϕ</ci><ci id="S2.SS2.p4.9.m9.1.1.1.1.2.3a.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1.2.3"><mtext id="S2.SS2.p4.9.m9.1.1.1.1.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.9.m9.1.1.1.1.2.3">UE</mtext></ci></apply><ci id="S2.SS2.p4.9.m9.1.1.1.1.3.cmml" xref="S2.SS2.p4.9.m9.1.1.1.1.3">𝑘</ci></apply><apply id="S2.SS2.p4.9.m9.2.2.2.2.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m9.2.2.2.2.1.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p4.9.m9.2.2.2.2.2.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.9.m9.2.2.2.2.2.1.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p4.9.m9.2.2.2.2.2.2.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2.2.2">𝜃</ci><ci id="S2.SS2.p4.9.m9.2.2.2.2.2.3a.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2.2.3"><mtext id="S2.SS2.p4.9.m9.2.2.2.2.2.3.cmml" mathsize="70%" xref="S2.SS2.p4.9.m9.2.2.2.2.2.3">UE</mtext></ci></apply><ci id="S2.SS2.p4.9.m9.2.2.2.2.3.cmml" xref="S2.SS2.p4.9.m9.2.2.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m9.2c">\{\phi^{\text{UE}}_{k},\theta^{\text{UE}}_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m9.2d">{ italic_ϕ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT UE end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> corresponding to the AoA of the same propagation path.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.6">Using (<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S2.E1" title="In II-B Channel Model ‣ II System Model ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1</span></a>), we build the BPL matrix of size <math alttext="N_{gNB}\times N_{UE}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><mrow id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><msub id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml"><mi id="S2.SS2.p5.1.m1.1.1.2.2" xref="S2.SS2.p5.1.m1.1.1.2.2.cmml">N</mi><mrow id="S2.SS2.p5.1.m1.1.1.2.3" xref="S2.SS2.p5.1.m1.1.1.2.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.2.3.2" xref="S2.SS2.p5.1.m1.1.1.2.3.2.cmml">g</mi><mo id="S2.SS2.p5.1.m1.1.1.2.3.1" xref="S2.SS2.p5.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.1.m1.1.1.2.3.3" xref="S2.SS2.p5.1.m1.1.1.2.3.3.cmml">N</mi><mo id="S2.SS2.p5.1.m1.1.1.2.3.1a" xref="S2.SS2.p5.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.1.m1.1.1.2.3.4" xref="S2.SS2.p5.1.m1.1.1.2.3.4.cmml">B</mi></mrow></msub><mo id="S2.SS2.p5.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p5.1.m1.1.1.1.cmml">×</mo><msub id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.3.2" xref="S2.SS2.p5.1.m1.1.1.3.2.cmml">N</mi><mrow id="S2.SS2.p5.1.m1.1.1.3.3" xref="S2.SS2.p5.1.m1.1.1.3.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1.3.3.2" xref="S2.SS2.p5.1.m1.1.1.3.3.2.cmml">U</mi><mo id="S2.SS2.p5.1.m1.1.1.3.3.1" xref="S2.SS2.p5.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.1.m1.1.1.3.3.3" xref="S2.SS2.p5.1.m1.1.1.3.3.3.cmml">E</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><times id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1.1"></times><apply id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.2.1.cmml" xref="S2.SS2.p5.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.2.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2.2">𝑁</ci><apply id="S2.SS2.p5.1.m1.1.1.2.3.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3"><times id="S2.SS2.p5.1.m1.1.1.2.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3.1"></times><ci id="S2.SS2.p5.1.m1.1.1.2.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3.2">𝑔</ci><ci id="S2.SS2.p5.1.m1.1.1.2.3.3.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3.3">𝑁</ci><ci id="S2.SS2.p5.1.m1.1.1.2.3.4.cmml" xref="S2.SS2.p5.1.m1.1.1.2.3.4">𝐵</ci></apply></apply><apply id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.3">subscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.3.2">𝑁</ci><apply id="S2.SS2.p5.1.m1.1.1.3.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3.3"><times id="S2.SS2.p5.1.m1.1.1.3.3.1.cmml" xref="S2.SS2.p5.1.m1.1.1.3.3.1"></times><ci id="S2.SS2.p5.1.m1.1.1.3.3.2.cmml" xref="S2.SS2.p5.1.m1.1.1.3.3.2">𝑈</ci><ci id="S2.SS2.p5.1.m1.1.1.3.3.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3.3.3">𝐸</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">N_{gNB}\times N_{UE}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">italic_N start_POSTSUBSCRIPT italic_g italic_N italic_B end_POSTSUBSCRIPT × italic_N start_POSTSUBSCRIPT italic_U italic_E end_POSTSUBSCRIPT</annotation></semantics></math> containing the RSS value for every candidate BPL <math alttext="l_{i,j}" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.2"><semantics id="S2.SS2.p5.2.m2.2a"><msub id="S2.SS2.p5.2.m2.2.3" xref="S2.SS2.p5.2.m2.2.3.cmml"><mi id="S2.SS2.p5.2.m2.2.3.2" xref="S2.SS2.p5.2.m2.2.3.2.cmml">l</mi><mrow id="S2.SS2.p5.2.m2.2.2.2.4" xref="S2.SS2.p5.2.m2.2.2.2.3.cmml"><mi id="S2.SS2.p5.2.m2.1.1.1.1" xref="S2.SS2.p5.2.m2.1.1.1.1.cmml">i</mi><mo id="S2.SS2.p5.2.m2.2.2.2.4.1" xref="S2.SS2.p5.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p5.2.m2.2.2.2.2" xref="S2.SS2.p5.2.m2.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.2b"><apply id="S2.SS2.p5.2.m2.2.3.cmml" xref="S2.SS2.p5.2.m2.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.2.m2.2.3.1.cmml" xref="S2.SS2.p5.2.m2.2.3">subscript</csymbol><ci id="S2.SS2.p5.2.m2.2.3.2.cmml" xref="S2.SS2.p5.2.m2.2.3.2">𝑙</ci><list id="S2.SS2.p5.2.m2.2.2.2.3.cmml" xref="S2.SS2.p5.2.m2.2.2.2.4"><ci id="S2.SS2.p5.2.m2.1.1.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1.1.1">𝑖</ci><ci id="S2.SS2.p5.2.m2.2.2.2.2.cmml" xref="S2.SS2.p5.2.m2.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.2c">l_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.2d">italic_l start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. We extract the top-5 best BPLs from the BPL matrix, i.e., the gNB/UE codebook entry pairs corresponding to the five BPLs having the highest RSS values. Each top-5 set, tagged with the corresponding UE location in the network area covered by an arbitrary <math alttext="\text{gNB}_{x}" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><msub id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml"><mtext id="S2.SS2.p5.3.m3.1.1.2" xref="S2.SS2.p5.3.m3.1.1.2a.cmml">gNB</mtext><mi id="S2.SS2.p5.3.m3.1.1.3" xref="S2.SS2.p5.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><apply id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.3.m3.1.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.p5.3.m3.1.1.2a.cmml" xref="S2.SS2.p5.3.m3.1.1.2"><mtext id="S2.SS2.p5.3.m3.1.1.2.cmml" xref="S2.SS2.p5.3.m3.1.1.2">gNB</mtext></ci><ci id="S2.SS2.p5.3.m3.1.1.3.cmml" xref="S2.SS2.p5.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">\text{gNB}_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">gNB start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, constitutes the BPL dataset <math alttext="D_{gNB_{x}}" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><msub id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml"><mi id="S2.SS2.p5.4.m4.1.1.2" xref="S2.SS2.p5.4.m4.1.1.2.cmml">D</mi><mrow id="S2.SS2.p5.4.m4.1.1.3" xref="S2.SS2.p5.4.m4.1.1.3.cmml"><mi id="S2.SS2.p5.4.m4.1.1.3.2" xref="S2.SS2.p5.4.m4.1.1.3.2.cmml">g</mi><mo id="S2.SS2.p5.4.m4.1.1.3.1" xref="S2.SS2.p5.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.4.m4.1.1.3.3" xref="S2.SS2.p5.4.m4.1.1.3.3.cmml">N</mi><mo id="S2.SS2.p5.4.m4.1.1.3.1a" xref="S2.SS2.p5.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S2.SS2.p5.4.m4.1.1.3.4" xref="S2.SS2.p5.4.m4.1.1.3.4.cmml"><mi id="S2.SS2.p5.4.m4.1.1.3.4.2" xref="S2.SS2.p5.4.m4.1.1.3.4.2.cmml">B</mi><mi id="S2.SS2.p5.4.m4.1.1.3.4.3" xref="S2.SS2.p5.4.m4.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><apply id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.1.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p5.4.m4.1.1.2.cmml" xref="S2.SS2.p5.4.m4.1.1.2">𝐷</ci><apply id="S2.SS2.p5.4.m4.1.1.3.cmml" xref="S2.SS2.p5.4.m4.1.1.3"><times id="S2.SS2.p5.4.m4.1.1.3.1.cmml" xref="S2.SS2.p5.4.m4.1.1.3.1"></times><ci id="S2.SS2.p5.4.m4.1.1.3.2.cmml" xref="S2.SS2.p5.4.m4.1.1.3.2">𝑔</ci><ci id="S2.SS2.p5.4.m4.1.1.3.3.cmml" xref="S2.SS2.p5.4.m4.1.1.3.3">𝑁</ci><apply id="S2.SS2.p5.4.m4.1.1.3.4.cmml" xref="S2.SS2.p5.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.1.1.3.4.1.cmml" xref="S2.SS2.p5.4.m4.1.1.3.4">subscript</csymbol><ci id="S2.SS2.p5.4.m4.1.1.3.4.2.cmml" xref="S2.SS2.p5.4.m4.1.1.3.4.2">𝐵</ci><ci id="S2.SS2.p5.4.m4.1.1.3.4.3.cmml" xref="S2.SS2.p5.4.m4.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">D_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for that gNB. We note that we consider locations to be covered only if the maximum RSS value in the corresponding BPL matrix is greater than a minimum RSS threshold (<math alttext="\text{RSS}_{thresh}=-174" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><mrow id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml"><msub id="S2.SS2.p5.5.m5.1.1.2" xref="S2.SS2.p5.5.m5.1.1.2.cmml"><mtext id="S2.SS2.p5.5.m5.1.1.2.2" xref="S2.SS2.p5.5.m5.1.1.2.2a.cmml">RSS</mtext><mrow id="S2.SS2.p5.5.m5.1.1.2.3" xref="S2.SS2.p5.5.m5.1.1.2.3.cmml"><mi id="S2.SS2.p5.5.m5.1.1.2.3.2" xref="S2.SS2.p5.5.m5.1.1.2.3.2.cmml">t</mi><mo id="S2.SS2.p5.5.m5.1.1.2.3.1" xref="S2.SS2.p5.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.1.1.2.3.3" xref="S2.SS2.p5.5.m5.1.1.2.3.3.cmml">h</mi><mo id="S2.SS2.p5.5.m5.1.1.2.3.1a" xref="S2.SS2.p5.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.1.1.2.3.4" xref="S2.SS2.p5.5.m5.1.1.2.3.4.cmml">r</mi><mo id="S2.SS2.p5.5.m5.1.1.2.3.1b" xref="S2.SS2.p5.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.1.1.2.3.5" xref="S2.SS2.p5.5.m5.1.1.2.3.5.cmml">e</mi><mo id="S2.SS2.p5.5.m5.1.1.2.3.1c" xref="S2.SS2.p5.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.1.1.2.3.6" xref="S2.SS2.p5.5.m5.1.1.2.3.6.cmml">s</mi><mo id="S2.SS2.p5.5.m5.1.1.2.3.1d" xref="S2.SS2.p5.5.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.5.m5.1.1.2.3.7" xref="S2.SS2.p5.5.m5.1.1.2.3.7.cmml">h</mi></mrow></msub><mo id="S2.SS2.p5.5.m5.1.1.1" xref="S2.SS2.p5.5.m5.1.1.1.cmml">=</mo><mrow id="S2.SS2.p5.5.m5.1.1.3" xref="S2.SS2.p5.5.m5.1.1.3.cmml"><mo id="S2.SS2.p5.5.m5.1.1.3a" xref="S2.SS2.p5.5.m5.1.1.3.cmml">−</mo><mn id="S2.SS2.p5.5.m5.1.1.3.2" xref="S2.SS2.p5.5.m5.1.1.3.2.cmml">174</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><apply id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1"><eq id="S2.SS2.p5.5.m5.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1.1"></eq><apply id="S2.SS2.p5.5.m5.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.1.2.1.cmml" xref="S2.SS2.p5.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS2.p5.5.m5.1.1.2.2a.cmml" xref="S2.SS2.p5.5.m5.1.1.2.2"><mtext id="S2.SS2.p5.5.m5.1.1.2.2.cmml" xref="S2.SS2.p5.5.m5.1.1.2.2">RSS</mtext></ci><apply id="S2.SS2.p5.5.m5.1.1.2.3.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3"><times id="S2.SS2.p5.5.m5.1.1.2.3.1.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.1"></times><ci id="S2.SS2.p5.5.m5.1.1.2.3.2.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.2">𝑡</ci><ci id="S2.SS2.p5.5.m5.1.1.2.3.3.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.3">ℎ</ci><ci id="S2.SS2.p5.5.m5.1.1.2.3.4.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.4">𝑟</ci><ci id="S2.SS2.p5.5.m5.1.1.2.3.5.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.5">𝑒</ci><ci id="S2.SS2.p5.5.m5.1.1.2.3.6.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.6">𝑠</ci><ci id="S2.SS2.p5.5.m5.1.1.2.3.7.cmml" xref="S2.SS2.p5.5.m5.1.1.2.3.7">ℎ</ci></apply></apply><apply id="S2.SS2.p5.5.m5.1.1.3.cmml" xref="S2.SS2.p5.5.m5.1.1.3"><minus id="S2.SS2.p5.5.m5.1.1.3.1.cmml" xref="S2.SS2.p5.5.m5.1.1.3"></minus><cn id="S2.SS2.p5.5.m5.1.1.3.2.cmml" type="integer" xref="S2.SS2.p5.5.m5.1.1.3.2">174</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">\text{RSS}_{thresh}=-174</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">RSS start_POSTSUBSCRIPT italic_t italic_h italic_r italic_e italic_s italic_h end_POSTSUBSCRIPT = - 174</annotation></semantics></math> dBm); otherwise, they are considered to be locations in outage and excluded from the dataset <math alttext="D_{gNB_{x}}" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><msub id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml"><mi id="S2.SS2.p5.6.m6.1.1.2" xref="S2.SS2.p5.6.m6.1.1.2.cmml">D</mi><mrow id="S2.SS2.p5.6.m6.1.1.3" xref="S2.SS2.p5.6.m6.1.1.3.cmml"><mi id="S2.SS2.p5.6.m6.1.1.3.2" xref="S2.SS2.p5.6.m6.1.1.3.2.cmml">g</mi><mo id="S2.SS2.p5.6.m6.1.1.3.1" xref="S2.SS2.p5.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p5.6.m6.1.1.3.3" xref="S2.SS2.p5.6.m6.1.1.3.3.cmml">N</mi><mo id="S2.SS2.p5.6.m6.1.1.3.1a" xref="S2.SS2.p5.6.m6.1.1.3.1.cmml">⁢</mo><msub id="S2.SS2.p5.6.m6.1.1.3.4" xref="S2.SS2.p5.6.m6.1.1.3.4.cmml"><mi id="S2.SS2.p5.6.m6.1.1.3.4.2" xref="S2.SS2.p5.6.m6.1.1.3.4.2.cmml">B</mi><mi id="S2.SS2.p5.6.m6.1.1.3.4.3" xref="S2.SS2.p5.6.m6.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><apply id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.1.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.p5.6.m6.1.1.2.cmml" xref="S2.SS2.p5.6.m6.1.1.2">𝐷</ci><apply id="S2.SS2.p5.6.m6.1.1.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3"><times id="S2.SS2.p5.6.m6.1.1.3.1.cmml" xref="S2.SS2.p5.6.m6.1.1.3.1"></times><ci id="S2.SS2.p5.6.m6.1.1.3.2.cmml" xref="S2.SS2.p5.6.m6.1.1.3.2">𝑔</ci><ci id="S2.SS2.p5.6.m6.1.1.3.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3.3">𝑁</ci><apply id="S2.SS2.p5.6.m6.1.1.3.4.cmml" xref="S2.SS2.p5.6.m6.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.1.1.3.4.1.cmml" xref="S2.SS2.p5.6.m6.1.1.3.4">subscript</csymbol><ci id="S2.SS2.p5.6.m6.1.1.3.4.2.cmml" xref="S2.SS2.p5.6.m6.1.1.3.4.2">𝐵</ci><ci id="S2.SS2.p5.6.m6.1.1.3.4.3.cmml" xref="S2.SS2.p5.6.m6.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">D_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Consequently, the sizes of these datasets vary for different gNBs, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf2" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(b)</span></a> for Frankfurt and Seoul gNBs, respectively.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Beam Prediction Problem Formulation & Proposed Transfer Learning Solution</span> </h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.4.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.5.2">Beam Prediction Problem Formulation</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.9">In our communication system, both the gNB and the UE have beamsteering capability using predefined codebooks. Let <math alttext="i" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mi id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_i</annotation></semantics></math> be the codebook entry at the gNB, selected from the codebook <math alttext="\mathcal{F}=\{i_{m}\}_{m=1}^{N_{gNB}}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">ℱ</mi><mo id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">=</mo><msubsup id="S3.SS1.p1.2.m2.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.cmml"><mrow id="S3.SS1.p1.2.m2.1.1.1.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.1.1.2.cmml"><mo id="S3.SS1.p1.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p1.2.m2.1.1.1.1.1.2.cmml">{</mo><msub id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.2" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.2.cmml">i</mi><mi id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.3" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S3.SS1.p1.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p1.2.m2.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.p1.2.m2.1.1.1.1.3" xref="S3.SS1.p1.2.m2.1.1.1.1.3.cmml"><mi id="S3.SS1.p1.2.m2.1.1.1.1.3.2" xref="S3.SS1.p1.2.m2.1.1.1.1.3.2.cmml">m</mi><mo id="S3.SS1.p1.2.m2.1.1.1.1.3.1" xref="S3.SS1.p1.2.m2.1.1.1.1.3.1.cmml">=</mo><mn id="S3.SS1.p1.2.m2.1.1.1.1.3.3" xref="S3.SS1.p1.2.m2.1.1.1.1.3.3.cmml">1</mn></mrow><msub id="S3.SS1.p1.2.m2.1.1.1.3" xref="S3.SS1.p1.2.m2.1.1.1.3.cmml"><mi id="S3.SS1.p1.2.m2.1.1.1.3.2" xref="S3.SS1.p1.2.m2.1.1.1.3.2.cmml">N</mi><mrow id="S3.SS1.p1.2.m2.1.1.1.3.3" xref="S3.SS1.p1.2.m2.1.1.1.3.3.cmml"><mi id="S3.SS1.p1.2.m2.1.1.1.3.3.2" xref="S3.SS1.p1.2.m2.1.1.1.3.3.2.cmml">g</mi><mo id="S3.SS1.p1.2.m2.1.1.1.3.3.1" xref="S3.SS1.p1.2.m2.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p1.2.m2.1.1.1.3.3.3" xref="S3.SS1.p1.2.m2.1.1.1.3.3.3.cmml">N</mi><mo id="S3.SS1.p1.2.m2.1.1.1.3.3.1a" xref="S3.SS1.p1.2.m2.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p1.2.m2.1.1.1.3.3.4" xref="S3.SS1.p1.2.m2.1.1.1.3.3.4.cmml">B</mi></mrow></msub></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><eq id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2"></eq><ci id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3">ℱ</ci><apply id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1">superscript</csymbol><apply id="S3.SS1.p1.2.m2.1.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1">subscript</csymbol><set id="S3.SS1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1"><apply id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.2">𝑖</ci><ci id="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.1.1.1.3">𝑚</ci></apply></set><apply id="S3.SS1.p1.2.m2.1.1.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.3"><eq id="S3.SS1.p1.2.m2.1.1.1.1.3.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.3.1"></eq><ci id="S3.SS1.p1.2.m2.1.1.1.1.3.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1.1.3.2">𝑚</ci><cn id="S3.SS1.p1.2.m2.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.SS1.p1.2.m2.1.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.3.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.1.3.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.2">𝑁</ci><apply id="S3.SS1.p1.2.m2.1.1.1.3.3.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.3"><times id="S3.SS1.p1.2.m2.1.1.1.3.3.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.3.1"></times><ci id="S3.SS1.p1.2.m2.1.1.1.3.3.2.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.3.2">𝑔</ci><ci id="S3.SS1.p1.2.m2.1.1.1.3.3.3.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.3.3">𝑁</ci><ci id="S3.SS1.p1.2.m2.1.1.1.3.3.4.cmml" xref="S3.SS1.p1.2.m2.1.1.1.3.3.4">𝐵</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\mathcal{F}=\{i_{m}\}_{m=1}^{N_{gNB}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">caligraphic_F = { italic_i start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_m = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_g italic_N italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="j" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_j</annotation></semantics></math> be the codebook entry at the UE, from the codebook <math alttext="\mathcal{W}=\{j_{n}\}_{n=1}^{N_{UE}}" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.4.m4.1.1.3" xref="S3.SS1.p1.4.m4.1.1.3.cmml">𝒲</mi><mo id="S3.SS1.p1.4.m4.1.1.2" xref="S3.SS1.p1.4.m4.1.1.2.cmml">=</mo><msubsup id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml"><mrow id="S3.SS1.p1.4.m4.1.1.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.1.2.cmml"><mo id="S3.SS1.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p1.4.m4.1.1.1.1.1.2.cmml">{</mo><msub id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.2" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.2.cmml">j</mi><mi id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.3" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.SS1.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p1.4.m4.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.p1.4.m4.1.1.1.1.3" xref="S3.SS1.p1.4.m4.1.1.1.1.3.cmml"><mi id="S3.SS1.p1.4.m4.1.1.1.1.3.2" xref="S3.SS1.p1.4.m4.1.1.1.1.3.2.cmml">n</mi><mo id="S3.SS1.p1.4.m4.1.1.1.1.3.1" xref="S3.SS1.p1.4.m4.1.1.1.1.3.1.cmml">=</mo><mn id="S3.SS1.p1.4.m4.1.1.1.1.3.3" xref="S3.SS1.p1.4.m4.1.1.1.1.3.3.cmml">1</mn></mrow><msub id="S3.SS1.p1.4.m4.1.1.1.3" xref="S3.SS1.p1.4.m4.1.1.1.3.cmml"><mi id="S3.SS1.p1.4.m4.1.1.1.3.2" xref="S3.SS1.p1.4.m4.1.1.1.3.2.cmml">N</mi><mrow id="S3.SS1.p1.4.m4.1.1.1.3.3" xref="S3.SS1.p1.4.m4.1.1.1.3.3.cmml"><mi id="S3.SS1.p1.4.m4.1.1.1.3.3.2" xref="S3.SS1.p1.4.m4.1.1.1.3.3.2.cmml">U</mi><mo id="S3.SS1.p1.4.m4.1.1.1.3.3.1" xref="S3.SS1.p1.4.m4.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.SS1.p1.4.m4.1.1.1.3.3.3" xref="S3.SS1.p1.4.m4.1.1.1.3.3.3.cmml">E</mi></mrow></msub></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><eq id="S3.SS1.p1.4.m4.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2"></eq><ci id="S3.SS1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3">𝒲</ci><apply id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1">superscript</csymbol><apply id="S3.SS1.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1">subscript</csymbol><set id="S3.SS1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1"><apply id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.2">𝑗</ci><ci id="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.1.1.1.3">𝑛</ci></apply></set><apply id="S3.SS1.p1.4.m4.1.1.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.3"><eq id="S3.SS1.p1.4.m4.1.1.1.1.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.3.1"></eq><ci id="S3.SS1.p1.4.m4.1.1.1.1.3.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1.3.2">𝑛</ci><cn id="S3.SS1.p1.4.m4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.SS1.p1.4.m4.1.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.1.1.1.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.4.m4.1.1.1.3.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.2">𝑁</ci><apply id="S3.SS1.p1.4.m4.1.1.1.3.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.3"><times id="S3.SS1.p1.4.m4.1.1.1.3.3.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.3.1"></times><ci id="S3.SS1.p1.4.m4.1.1.1.3.3.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.3.2">𝑈</ci><ci id="S3.SS1.p1.4.m4.1.1.1.3.3.3.cmml" xref="S3.SS1.p1.4.m4.1.1.1.3.3.3">𝐸</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">\mathcal{W}=\{j_{n}\}_{n=1}^{N_{UE}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">caligraphic_W = { italic_j start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_U italic_E end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. The beam selection aims to identify the optimal BPL <math alttext="l_{i^{*},j^{*}}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.2"><semantics id="S3.SS1.p1.5.m5.2a"><msub id="S3.SS1.p1.5.m5.2.3" xref="S3.SS1.p1.5.m5.2.3.cmml"><mi id="S3.SS1.p1.5.m5.2.3.2" xref="S3.SS1.p1.5.m5.2.3.2.cmml">l</mi><mrow id="S3.SS1.p1.5.m5.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.3.cmml"><msup id="S3.SS1.p1.5.m5.1.1.1.1.1" xref="S3.SS1.p1.5.m5.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.5.m5.1.1.1.1.1.2" xref="S3.SS1.p1.5.m5.1.1.1.1.1.2.cmml">i</mi><mo id="S3.SS1.p1.5.m5.1.1.1.1.1.3" xref="S3.SS1.p1.5.m5.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.5.m5.2.2.2.2.3" xref="S3.SS1.p1.5.m5.2.2.2.3.cmml">,</mo><msup id="S3.SS1.p1.5.m5.2.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.2.cmml"><mi id="S3.SS1.p1.5.m5.2.2.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.cmml">j</mi><mo id="S3.SS1.p1.5.m5.2.2.2.2.2.3" xref="S3.SS1.p1.5.m5.2.2.2.2.2.3.cmml">∗</mo></msup></mrow></msub><annotation-xml encoding="MathML-Content" 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xref="S3.SS1.p1.5.m5.2.2.2.2.2.3"></times></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.2c">l_{i^{*},j^{*}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.2d">italic_l start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> consisting of the gNB and UE codebook entry pair <math alttext="(i^{*},j^{*})" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.2"><semantics id="S3.SS1.p1.6.m6.2a"><mrow id="S3.SS1.p1.6.m6.2.2.2" xref="S3.SS1.p1.6.m6.2.2.3.cmml"><mo id="S3.SS1.p1.6.m6.2.2.2.3" stretchy="false" xref="S3.SS1.p1.6.m6.2.2.3.cmml">(</mo><msup id="S3.SS1.p1.6.m6.1.1.1.1" xref="S3.SS1.p1.6.m6.1.1.1.1.cmml"><mi id="S3.SS1.p1.6.m6.1.1.1.1.2" xref="S3.SS1.p1.6.m6.1.1.1.1.2.cmml">i</mi><mo id="S3.SS1.p1.6.m6.1.1.1.1.3" xref="S3.SS1.p1.6.m6.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.6.m6.2.2.2.4" xref="S3.SS1.p1.6.m6.2.2.3.cmml">,</mo><msup id="S3.SS1.p1.6.m6.2.2.2.2" xref="S3.SS1.p1.6.m6.2.2.2.2.cmml"><mi id="S3.SS1.p1.6.m6.2.2.2.2.2" xref="S3.SS1.p1.6.m6.2.2.2.2.2.cmml">j</mi><mo id="S3.SS1.p1.6.m6.2.2.2.2.3" xref="S3.SS1.p1.6.m6.2.2.2.2.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.6.m6.2.2.2.5" stretchy="false" xref="S3.SS1.p1.6.m6.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.2b"><interval closure="open" id="S3.SS1.p1.6.m6.2.2.3.cmml" xref="S3.SS1.p1.6.m6.2.2.2"><apply id="S3.SS1.p1.6.m6.1.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.1.1.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p1.6.m6.1.1.1.1.2.cmml" xref="S3.SS1.p1.6.m6.1.1.1.1.2">𝑖</ci><times id="S3.SS1.p1.6.m6.1.1.1.1.3.cmml" xref="S3.SS1.p1.6.m6.1.1.1.1.3"></times></apply><apply id="S3.SS1.p1.6.m6.2.2.2.2.cmml" xref="S3.SS1.p1.6.m6.2.2.2.2"><csymbol cd="ambiguous" 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16KISK038."><span class="ltx_text ltx_ref_tag">1</span></a>). However, rather than explicitly calculating the optimal beam codebook pair <math alttext="(i^{*},j^{*})" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m7.2"><semantics id="S3.SS1.p1.7.m7.2a"><mrow id="S3.SS1.p1.7.m7.2.2.2" xref="S3.SS1.p1.7.m7.2.2.3.cmml"><mo id="S3.SS1.p1.7.m7.2.2.2.3" stretchy="false" xref="S3.SS1.p1.7.m7.2.2.3.cmml">(</mo><msup id="S3.SS1.p1.7.m7.1.1.1.1" xref="S3.SS1.p1.7.m7.1.1.1.1.cmml"><mi id="S3.SS1.p1.7.m7.1.1.1.1.2" xref="S3.SS1.p1.7.m7.1.1.1.1.2.cmml">i</mi><mo id="S3.SS1.p1.7.m7.1.1.1.1.3" xref="S3.SS1.p1.7.m7.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.7.m7.2.2.2.4" xref="S3.SS1.p1.7.m7.2.2.3.cmml">,</mo><msup id="S3.SS1.p1.7.m7.2.2.2.2" xref="S3.SS1.p1.7.m7.2.2.2.2.cmml"><mi id="S3.SS1.p1.7.m7.2.2.2.2.2" xref="S3.SS1.p1.7.m7.2.2.2.2.2.cmml">j</mi><mo id="S3.SS1.p1.7.m7.2.2.2.2.3" xref="S3.SS1.p1.7.m7.2.2.2.2.3.cmml">∗</mo></msup><mo id="S3.SS1.p1.7.m7.2.2.2.5" stretchy="false" xref="S3.SS1.p1.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.2b"><interval closure="open" id="S3.SS1.p1.7.m7.2.2.3.cmml" xref="S3.SS1.p1.7.m7.2.2.2"><apply id="S3.SS1.p1.7.m7.1.1.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m7.1.1.1.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p1.7.m7.1.1.1.1.2.cmml" xref="S3.SS1.p1.7.m7.1.1.1.1.2">𝑖</ci><times id="S3.SS1.p1.7.m7.1.1.1.1.3.cmml" xref="S3.SS1.p1.7.m7.1.1.1.1.3"></times></apply><apply id="S3.SS1.p1.7.m7.2.2.2.2.cmml" xref="S3.SS1.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m7.2.2.2.2.1.cmml" xref="S3.SS1.p1.7.m7.2.2.2.2">superscript</csymbol><ci id="S3.SS1.p1.7.m7.2.2.2.2.2.cmml" xref="S3.SS1.p1.7.m7.2.2.2.2.2">𝑗</ci><times id="S3.SS1.p1.7.m7.2.2.2.2.3.cmml" xref="S3.SS1.p1.7.m7.2.2.2.2.3"></times></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.2c">(i^{*},j^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m7.2d">( italic_i start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_j start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> using exhaustive beam sweeping measurements as per FR2 initial access procedure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib2" title="">2</a>]</cite> which incur in a high overhead, we aim to estimate the beam codebook pair <math alttext="(\hat{i},\hat{j})" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m8.2"><semantics id="S3.SS1.p1.8.m8.2a"><mrow id="S3.SS1.p1.8.m8.2.3.2" xref="S3.SS1.p1.8.m8.2.3.1.cmml"><mo id="S3.SS1.p1.8.m8.2.3.2.1" stretchy="false" xref="S3.SS1.p1.8.m8.2.3.1.cmml">(</mo><mover accent="true" id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml"><mi id="S3.SS1.p1.8.m8.1.1.2" xref="S3.SS1.p1.8.m8.1.1.2.cmml">i</mi><mo id="S3.SS1.p1.8.m8.1.1.1" xref="S3.SS1.p1.8.m8.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.p1.8.m8.2.3.2.2" xref="S3.SS1.p1.8.m8.2.3.1.cmml">,</mo><mover accent="true" id="S3.SS1.p1.8.m8.2.2" xref="S3.SS1.p1.8.m8.2.2.cmml"><mi id="S3.SS1.p1.8.m8.2.2.2" xref="S3.SS1.p1.8.m8.2.2.2.cmml">j</mi><mo id="S3.SS1.p1.8.m8.2.2.1" xref="S3.SS1.p1.8.m8.2.2.1.cmml">^</mo></mover><mo id="S3.SS1.p1.8.m8.2.3.2.3" stretchy="false" xref="S3.SS1.p1.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.2b"><interval closure="open" id="S3.SS1.p1.8.m8.2.3.1.cmml" xref="S3.SS1.p1.8.m8.2.3.2"><apply id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1"><ci id="S3.SS1.p1.8.m8.1.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1.1">^</ci><ci id="S3.SS1.p1.8.m8.1.1.2.cmml" xref="S3.SS1.p1.8.m8.1.1.2">𝑖</ci></apply><apply id="S3.SS1.p1.8.m8.2.2.cmml" xref="S3.SS1.p1.8.m8.2.2"><ci id="S3.SS1.p1.8.m8.2.2.1.cmml" xref="S3.SS1.p1.8.m8.2.2.1">^</ci><ci id="S3.SS1.p1.8.m8.2.2.2.cmml" xref="S3.SS1.p1.8.m8.2.2.2">𝑗</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.2c">(\hat{i},\hat{j})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m8.2d">( over^ start_ARG italic_i end_ARG , over^ start_ARG italic_j end_ARG )</annotation></semantics></math> that maximizes the probability of selecting the optimal codebook pair based solely on the UE position <math alttext="\{x,y\}" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m9.2"><semantics id="S3.SS1.p1.9.m9.2a"><mrow id="S3.SS1.p1.9.m9.2.3.2" xref="S3.SS1.p1.9.m9.2.3.1.cmml"><mo id="S3.SS1.p1.9.m9.2.3.2.1" stretchy="false" xref="S3.SS1.p1.9.m9.2.3.1.cmml">{</mo><mi id="S3.SS1.p1.9.m9.1.1" xref="S3.SS1.p1.9.m9.1.1.cmml">x</mi><mo id="S3.SS1.p1.9.m9.2.3.2.2" xref="S3.SS1.p1.9.m9.2.3.1.cmml">,</mo><mi id="S3.SS1.p1.9.m9.2.2" xref="S3.SS1.p1.9.m9.2.2.cmml">y</mi><mo id="S3.SS1.p1.9.m9.2.3.2.3" stretchy="false" xref="S3.SS1.p1.9.m9.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.2b"><set id="S3.SS1.p1.9.m9.2.3.1.cmml" 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class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.4.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.5.2">Proposed Location-Based Transfer Learning Solution</span> </h3> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> Cross-Environment Transfer Learning</figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" 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end_POSTSUBSCRIPT</annotation></semantics></math>: BPL dataset for <math alttext="\text{gNB}_{x}" class="ltx_Math" display="inline" id="alg1.l1.m2.1"><semantics id="alg1.l1.m2.1a"><msub id="alg1.l1.m2.1.1" xref="alg1.l1.m2.1.1.cmml"><mtext id="alg1.l1.m2.1.1.2" xref="alg1.l1.m2.1.1.2a.cmml">gNB</mtext><mi id="alg1.l1.m2.1.1.3" xref="alg1.l1.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.1b"><apply id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1">subscript</csymbol><ci id="alg1.l1.m2.1.1.2a.cmml" xref="alg1.l1.m2.1.1.2"><mtext id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2">gNB</mtext></ci><ci id="alg1.l1.m2.1.1.3.cmml" xref="alg1.l1.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">\text{gNB}_{x}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">gNB start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="D_{gNB_{y}}" class="ltx_Math" display="inline" id="alg1.l1.m3.1"><semantics id="alg1.l1.m3.1a"><msub id="alg1.l1.m3.1.1" xref="alg1.l1.m3.1.1.cmml"><mi id="alg1.l1.m3.1.1.2" xref="alg1.l1.m3.1.1.2.cmml">D</mi><mrow id="alg1.l1.m3.1.1.3" xref="alg1.l1.m3.1.1.3.cmml"><mi id="alg1.l1.m3.1.1.3.2" xref="alg1.l1.m3.1.1.3.2.cmml">g</mi><mo id="alg1.l1.m3.1.1.3.1" xref="alg1.l1.m3.1.1.3.1.cmml">⁢</mo><mi id="alg1.l1.m3.1.1.3.3" xref="alg1.l1.m3.1.1.3.3.cmml">N</mi><mo id="alg1.l1.m3.1.1.3.1a" xref="alg1.l1.m3.1.1.3.1.cmml">⁢</mo><msub id="alg1.l1.m3.1.1.3.4" xref="alg1.l1.m3.1.1.3.4.cmml"><mi id="alg1.l1.m3.1.1.3.4.2" xref="alg1.l1.m3.1.1.3.4.2.cmml">B</mi><mi id="alg1.l1.m3.1.1.3.4.3" xref="alg1.l1.m3.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l1.m3.1b"><apply id="alg1.l1.m3.1.1.cmml" xref="alg1.l1.m3.1.1"><csymbol cd="ambiguous" id="alg1.l1.m3.1.1.1.cmml" xref="alg1.l1.m3.1.1">subscript</csymbol><ci id="alg1.l1.m3.1.1.2.cmml" xref="alg1.l1.m3.1.1.2">𝐷</ci><apply id="alg1.l1.m3.1.1.3.cmml" xref="alg1.l1.m3.1.1.3"><times id="alg1.l1.m3.1.1.3.1.cmml" xref="alg1.l1.m3.1.1.3.1"></times><ci id="alg1.l1.m3.1.1.3.2.cmml" xref="alg1.l1.m3.1.1.3.2">𝑔</ci><ci id="alg1.l1.m3.1.1.3.3.cmml" xref="alg1.l1.m3.1.1.3.3">𝑁</ci><apply id="alg1.l1.m3.1.1.3.4.cmml" xref="alg1.l1.m3.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l1.m3.1.1.3.4.1.cmml" xref="alg1.l1.m3.1.1.3.4">subscript</csymbol><ci id="alg1.l1.m3.1.1.3.4.2.cmml" xref="alg1.l1.m3.1.1.3.4.2">𝐵</ci><ci id="alg1.l1.m3.1.1.3.4.3.cmml" xref="alg1.l1.m3.1.1.3.4.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m3.1c">D_{gNB_{y}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m3.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>: BPL dataset for <math alttext="\text{gNB}_{y}" class="ltx_Math" display="inline" id="alg1.l1.m4.1"><semantics id="alg1.l1.m4.1a"><msub id="alg1.l1.m4.1.1" xref="alg1.l1.m4.1.1.cmml"><mtext id="alg1.l1.m4.1.1.2" xref="alg1.l1.m4.1.1.2a.cmml">gNB</mtext><mi id="alg1.l1.m4.1.1.3" xref="alg1.l1.m4.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l1.m4.1b"><apply id="alg1.l1.m4.1.1.cmml" xref="alg1.l1.m4.1.1"><csymbol cd="ambiguous" id="alg1.l1.m4.1.1.1.cmml" xref="alg1.l1.m4.1.1">subscript</csymbol><ci id="alg1.l1.m4.1.1.2a.cmml" xref="alg1.l1.m4.1.1.2"><mtext id="alg1.l1.m4.1.1.2.cmml" xref="alg1.l1.m4.1.1.2">gNB</mtext></ci><ci id="alg1.l1.m4.1.1.3.cmml" xref="alg1.l1.m4.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m4.1c">\text{gNB}_{y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m4.1d">gNB start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span><math alttext="w_{gNB_{y}}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><msub id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi id="alg1.l2.m1.1.1.2" xref="alg1.l2.m1.1.1.2.cmml">w</mi><mrow id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml"><mi id="alg1.l2.m1.1.1.3.2" xref="alg1.l2.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l2.m1.1.1.3.1" xref="alg1.l2.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l2.m1.1.1.3.3" xref="alg1.l2.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l2.m1.1.1.3.1a" xref="alg1.l2.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l2.m1.1.1.3.4" xref="alg1.l2.m1.1.1.3.4.cmml"><mi id="alg1.l2.m1.1.1.3.4.2" xref="alg1.l2.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l2.m1.1.1.3.4.3" xref="alg1.l2.m1.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1">subscript</csymbol><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">𝑤</ci><apply id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3"><times id="alg1.l2.m1.1.1.3.1.cmml" xref="alg1.l2.m1.1.1.3.1"></times><ci id="alg1.l2.m1.1.1.3.2.cmml" xref="alg1.l2.m1.1.1.3.2">𝑔</ci><ci id="alg1.l2.m1.1.1.3.3.cmml" xref="alg1.l2.m1.1.1.3.3">𝑁</ci><apply id="alg1.l2.m1.1.1.3.4.cmml" xref="alg1.l2.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.3.4.1.cmml" xref="alg1.l2.m1.1.1.3.4">subscript</csymbol><ci id="alg1.l2.m1.1.1.3.4.2.cmml" xref="alg1.l2.m1.1.1.3.4.2">𝐵</ci><ci id="alg1.l2.m1.1.1.3.4.3.cmml" xref="alg1.l2.m1.1.1.3.4.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">w_{gNB_{y}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">italic_w start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>: Model parameters for <math alttext="\text{gNB}_{y}" class="ltx_Math" display="inline" id="alg1.l2.m2.1"><semantics id="alg1.l2.m2.1a"><msub id="alg1.l2.m2.1.1" xref="alg1.l2.m2.1.1.cmml"><mtext id="alg1.l2.m2.1.1.2" xref="alg1.l2.m2.1.1.2a.cmml">gNB</mtext><mi id="alg1.l2.m2.1.1.3" xref="alg1.l2.m2.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m2.1b"><apply id="alg1.l2.m2.1.1.cmml" xref="alg1.l2.m2.1.1"><csymbol cd="ambiguous" id="alg1.l2.m2.1.1.1.cmml" xref="alg1.l2.m2.1.1">subscript</csymbol><ci id="alg1.l2.m2.1.1.2a.cmml" xref="alg1.l2.m2.1.1.2"><mtext id="alg1.l2.m2.1.1.2.cmml" xref="alg1.l2.m2.1.1.2">gNB</mtext></ci><ci id="alg1.l2.m2.1.1.3.cmml" xref="alg1.l2.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m2.1c">\text{gNB}_{y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m2.1d">gNB start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.2.1.1" style="font-size:80%;">3:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l3.1">Step1: Model Training for <math alttext="\text{gNB}_{x}" class="ltx_Math" display="inline" id="alg1.l3.1.m1.1"><semantics id="alg1.l3.1.m1.1a"><msub id="alg1.l3.1.m1.1.1" xref="alg1.l3.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="alg1.l3.1.m1.1.1.2" xref="alg1.l3.1.m1.1.1.2a.cmml">gNB</mtext><mi id="alg1.l3.1.m1.1.1.3" xref="alg1.l3.1.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l3.1.m1.1b"><apply id="alg1.l3.1.m1.1.1.cmml" xref="alg1.l3.1.m1.1.1"><csymbol cd="ambiguous" id="alg1.l3.1.m1.1.1.1.cmml" xref="alg1.l3.1.m1.1.1">subscript</csymbol><ci id="alg1.l3.1.m1.1.1.2a.cmml" xref="alg1.l3.1.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="alg1.l3.1.m1.1.1.2.cmml" xref="alg1.l3.1.m1.1.1.2">gNB</mtext></ci><ci id="alg1.l3.1.m1.1.1.3.cmml" xref="alg1.l3.1.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.1.m1.1c">\text{gNB}_{x}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.1.m1.1d">gNB start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math></span> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span>Training, validation and testing split of <math alttext="D_{gNB_{x}}" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><msub id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><mi id="alg1.l4.m1.1.1.2" xref="alg1.l4.m1.1.1.2.cmml">D</mi><mrow id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3.cmml"><mi id="alg1.l4.m1.1.1.3.2" xref="alg1.l4.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l4.m1.1.1.3.1" xref="alg1.l4.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.3.3" xref="alg1.l4.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l4.m1.1.1.3.1a" xref="alg1.l4.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l4.m1.1.1.3.4" xref="alg1.l4.m1.1.1.3.4.cmml"><mi id="alg1.l4.m1.1.1.3.4.2" xref="alg1.l4.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l4.m1.1.1.3.4.3" xref="alg1.l4.m1.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><apply id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.cmml" xref="alg1.l4.m1.1.1">subscript</csymbol><ci id="alg1.l4.m1.1.1.2.cmml" xref="alg1.l4.m1.1.1.2">𝐷</ci><apply id="alg1.l4.m1.1.1.3.cmml" xref="alg1.l4.m1.1.1.3"><times id="alg1.l4.m1.1.1.3.1.cmml" xref="alg1.l4.m1.1.1.3.1"></times><ci id="alg1.l4.m1.1.1.3.2.cmml" xref="alg1.l4.m1.1.1.3.2">𝑔</ci><ci id="alg1.l4.m1.1.1.3.3.cmml" xref="alg1.l4.m1.1.1.3.3">𝑁</ci><apply id="alg1.l4.m1.1.1.3.4.cmml" xref="alg1.l4.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.3.4.1.cmml" xref="alg1.l4.m1.1.1.3.4">subscript</csymbol><ci id="alg1.l4.m1.1.1.3.4.2.cmml" xref="alg1.l4.m1.1.1.3.4.2">𝐵</ci><ci id="alg1.l4.m1.1.1.3.4.3.cmml" xref="alg1.l4.m1.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">D_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span>Model training using training set of <math alttext="D_{gNB_{x}}" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><msub id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml"><mi id="alg1.l5.m1.1.1.2" xref="alg1.l5.m1.1.1.2.cmml">D</mi><mrow id="alg1.l5.m1.1.1.3" xref="alg1.l5.m1.1.1.3.cmml"><mi id="alg1.l5.m1.1.1.3.2" xref="alg1.l5.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l5.m1.1.1.3.1" xref="alg1.l5.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l5.m1.1.1.3.3" xref="alg1.l5.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l5.m1.1.1.3.1a" xref="alg1.l5.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l5.m1.1.1.3.4" xref="alg1.l5.m1.1.1.3.4.cmml"><mi id="alg1.l5.m1.1.1.3.4.2" xref="alg1.l5.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l5.m1.1.1.3.4.3" xref="alg1.l5.m1.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><apply id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.1.cmml" xref="alg1.l5.m1.1.1">subscript</csymbol><ci id="alg1.l5.m1.1.1.2.cmml" xref="alg1.l5.m1.1.1.2">𝐷</ci><apply id="alg1.l5.m1.1.1.3.cmml" xref="alg1.l5.m1.1.1.3"><times id="alg1.l5.m1.1.1.3.1.cmml" xref="alg1.l5.m1.1.1.3.1"></times><ci id="alg1.l5.m1.1.1.3.2.cmml" xref="alg1.l5.m1.1.1.3.2">𝑔</ci><ci id="alg1.l5.m1.1.1.3.3.cmml" xref="alg1.l5.m1.1.1.3.3">𝑁</ci><apply id="alg1.l5.m1.1.1.3.4.cmml" xref="alg1.l5.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.3.4.1.cmml" xref="alg1.l5.m1.1.1.3.4">subscript</csymbol><ci id="alg1.l5.m1.1.1.3.4.2.cmml" xref="alg1.l5.m1.1.1.3.4.2">𝐵</ci><ci id="alg1.l5.m1.1.1.3.4.3.cmml" xref="alg1.l5.m1.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">D_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span><math alttext="w_{gNB_{x}}\leftarrow" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><msub id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml"><mi id="alg1.l6.m1.1.1.2.2" xref="alg1.l6.m1.1.1.2.2.cmml">w</mi><mrow id="alg1.l6.m1.1.1.2.3" xref="alg1.l6.m1.1.1.2.3.cmml"><mi id="alg1.l6.m1.1.1.2.3.2" xref="alg1.l6.m1.1.1.2.3.2.cmml">g</mi><mo id="alg1.l6.m1.1.1.2.3.1" xref="alg1.l6.m1.1.1.2.3.1.cmml">⁢</mo><mi id="alg1.l6.m1.1.1.2.3.3" xref="alg1.l6.m1.1.1.2.3.3.cmml">N</mi><mo id="alg1.l6.m1.1.1.2.3.1a" xref="alg1.l6.m1.1.1.2.3.1.cmml">⁢</mo><msub id="alg1.l6.m1.1.1.2.3.4" xref="alg1.l6.m1.1.1.2.3.4.cmml"><mi id="alg1.l6.m1.1.1.2.3.4.2" xref="alg1.l6.m1.1.1.2.3.4.2.cmml">B</mi><mi id="alg1.l6.m1.1.1.2.3.4.3" xref="alg1.l6.m1.1.1.2.3.4.3.cmml">x</mi></msub></mrow></msub><mo id="alg1.l6.m1.1.1.1" stretchy="false" xref="alg1.l6.m1.1.1.1.cmml">←</mo><mi id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><ci id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1">←</ci><apply id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.2.1.cmml" xref="alg1.l6.m1.1.1.2">subscript</csymbol><ci id="alg1.l6.m1.1.1.2.2.cmml" xref="alg1.l6.m1.1.1.2.2">𝑤</ci><apply id="alg1.l6.m1.1.1.2.3.cmml" xref="alg1.l6.m1.1.1.2.3"><times id="alg1.l6.m1.1.1.2.3.1.cmml" xref="alg1.l6.m1.1.1.2.3.1"></times><ci id="alg1.l6.m1.1.1.2.3.2.cmml" xref="alg1.l6.m1.1.1.2.3.2">𝑔</ci><ci id="alg1.l6.m1.1.1.2.3.3.cmml" xref="alg1.l6.m1.1.1.2.3.3">𝑁</ci><apply id="alg1.l6.m1.1.1.2.3.4.cmml" xref="alg1.l6.m1.1.1.2.3.4"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.2.3.4.1.cmml" xref="alg1.l6.m1.1.1.2.3.4">subscript</csymbol><ci id="alg1.l6.m1.1.1.2.3.4.2.cmml" xref="alg1.l6.m1.1.1.2.3.4.2">𝐵</ci><ci id="alg1.l6.m1.1.1.2.3.4.3.cmml" xref="alg1.l6.m1.1.1.2.3.4.3">𝑥</ci></apply></apply></apply><csymbol cd="latexml" id="alg1.l6.m1.1.1.3.cmml" xref="alg1.l6.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">w_{gNB_{x}}\leftarrow</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">italic_w start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT ←</annotation></semantics></math> Save model parameters </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.2.1.1" style="font-size:80%;">7:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l7.1">Step 2: Model Transfer to <math alttext="\text{gNB}_{y}" class="ltx_Math" display="inline" id="alg1.l7.1.m1.1"><semantics id="alg1.l7.1.m1.1a"><msub id="alg1.l7.1.m1.1.1" xref="alg1.l7.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="alg1.l7.1.m1.1.1.2" xref="alg1.l7.1.m1.1.1.2a.cmml">gNB</mtext><mi id="alg1.l7.1.m1.1.1.3" xref="alg1.l7.1.m1.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l7.1.m1.1b"><apply id="alg1.l7.1.m1.1.1.cmml" xref="alg1.l7.1.m1.1.1"><csymbol cd="ambiguous" id="alg1.l7.1.m1.1.1.1.cmml" xref="alg1.l7.1.m1.1.1">subscript</csymbol><ci id="alg1.l7.1.m1.1.1.2a.cmml" xref="alg1.l7.1.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="alg1.l7.1.m1.1.1.2.cmml" xref="alg1.l7.1.m1.1.1.2">gNB</mtext></ci><ci id="alg1.l7.1.m1.1.1.3.cmml" xref="alg1.l7.1.m1.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.1.m1.1c">\text{gNB}_{y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.1.m1.1d">gNB start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math></span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span>Initialize the model with parameters <math alttext="w_{gNB_{x}}" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><msub id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml"><mi id="alg1.l8.m1.1.1.2" xref="alg1.l8.m1.1.1.2.cmml">w</mi><mrow id="alg1.l8.m1.1.1.3" xref="alg1.l8.m1.1.1.3.cmml"><mi id="alg1.l8.m1.1.1.3.2" xref="alg1.l8.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l8.m1.1.1.3.1" xref="alg1.l8.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l8.m1.1.1.3.3" xref="alg1.l8.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l8.m1.1.1.3.1a" xref="alg1.l8.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l8.m1.1.1.3.4" xref="alg1.l8.m1.1.1.3.4.cmml"><mi id="alg1.l8.m1.1.1.3.4.2" xref="alg1.l8.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l8.m1.1.1.3.4.3" xref="alg1.l8.m1.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><apply id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.1.cmml" xref="alg1.l8.m1.1.1">subscript</csymbol><ci id="alg1.l8.m1.1.1.2.cmml" xref="alg1.l8.m1.1.1.2">𝑤</ci><apply id="alg1.l8.m1.1.1.3.cmml" xref="alg1.l8.m1.1.1.3"><times id="alg1.l8.m1.1.1.3.1.cmml" xref="alg1.l8.m1.1.1.3.1"></times><ci id="alg1.l8.m1.1.1.3.2.cmml" xref="alg1.l8.m1.1.1.3.2">𝑔</ci><ci id="alg1.l8.m1.1.1.3.3.cmml" xref="alg1.l8.m1.1.1.3.3">𝑁</ci><apply id="alg1.l8.m1.1.1.3.4.cmml" xref="alg1.l8.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.4.1.cmml" xref="alg1.l8.m1.1.1.3.4">subscript</csymbol><ci id="alg1.l8.m1.1.1.3.4.2.cmml" xref="alg1.l8.m1.1.1.3.4.2">𝐵</ci><ci id="alg1.l8.m1.1.1.3.4.3.cmml" xref="alg1.l8.m1.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.1c">w_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.1d">italic_w start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span><math alttext="\tilde{D}_{gNB_{y}}\leftarrow" class="ltx_Math" display="inline" id="alg1.l9.m1.1"><semantics id="alg1.l9.m1.1a"><mrow id="alg1.l9.m1.1.1" xref="alg1.l9.m1.1.1.cmml"><msub id="alg1.l9.m1.1.1.2" xref="alg1.l9.m1.1.1.2.cmml"><mover accent="true" id="alg1.l9.m1.1.1.2.2" xref="alg1.l9.m1.1.1.2.2.cmml"><mi id="alg1.l9.m1.1.1.2.2.2" xref="alg1.l9.m1.1.1.2.2.2.cmml">D</mi><mo id="alg1.l9.m1.1.1.2.2.1" xref="alg1.l9.m1.1.1.2.2.1.cmml">~</mo></mover><mrow id="alg1.l9.m1.1.1.2.3" xref="alg1.l9.m1.1.1.2.3.cmml"><mi id="alg1.l9.m1.1.1.2.3.2" xref="alg1.l9.m1.1.1.2.3.2.cmml">g</mi><mo id="alg1.l9.m1.1.1.2.3.1" xref="alg1.l9.m1.1.1.2.3.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.2.3.3" xref="alg1.l9.m1.1.1.2.3.3.cmml">N</mi><mo id="alg1.l9.m1.1.1.2.3.1a" xref="alg1.l9.m1.1.1.2.3.1.cmml">⁢</mo><msub id="alg1.l9.m1.1.1.2.3.4" xref="alg1.l9.m1.1.1.2.3.4.cmml"><mi id="alg1.l9.m1.1.1.2.3.4.2" xref="alg1.l9.m1.1.1.2.3.4.2.cmml">B</mi><mi id="alg1.l9.m1.1.1.2.3.4.3" xref="alg1.l9.m1.1.1.2.3.4.3.cmml">y</mi></msub></mrow></msub><mo id="alg1.l9.m1.1.1.1" stretchy="false" xref="alg1.l9.m1.1.1.1.cmml">←</mo><mi id="alg1.l9.m1.1.1.3" xref="alg1.l9.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.1b"><apply id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1"><ci id="alg1.l9.m1.1.1.1.cmml" xref="alg1.l9.m1.1.1.1">←</ci><apply id="alg1.l9.m1.1.1.2.cmml" xref="alg1.l9.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.2.1.cmml" xref="alg1.l9.m1.1.1.2">subscript</csymbol><apply id="alg1.l9.m1.1.1.2.2.cmml" xref="alg1.l9.m1.1.1.2.2"><ci id="alg1.l9.m1.1.1.2.2.1.cmml" xref="alg1.l9.m1.1.1.2.2.1">~</ci><ci id="alg1.l9.m1.1.1.2.2.2.cmml" xref="alg1.l9.m1.1.1.2.2.2">𝐷</ci></apply><apply id="alg1.l9.m1.1.1.2.3.cmml" xref="alg1.l9.m1.1.1.2.3"><times id="alg1.l9.m1.1.1.2.3.1.cmml" xref="alg1.l9.m1.1.1.2.3.1"></times><ci id="alg1.l9.m1.1.1.2.3.2.cmml" xref="alg1.l9.m1.1.1.2.3.2">𝑔</ci><ci id="alg1.l9.m1.1.1.2.3.3.cmml" xref="alg1.l9.m1.1.1.2.3.3">𝑁</ci><apply id="alg1.l9.m1.1.1.2.3.4.cmml" xref="alg1.l9.m1.1.1.2.3.4"><csymbol cd="ambiguous" id="alg1.l9.m1.1.1.2.3.4.1.cmml" xref="alg1.l9.m1.1.1.2.3.4">subscript</csymbol><ci id="alg1.l9.m1.1.1.2.3.4.2.cmml" xref="alg1.l9.m1.1.1.2.3.4.2">𝐵</ci><ci id="alg1.l9.m1.1.1.2.3.4.3.cmml" xref="alg1.l9.m1.1.1.2.3.4.3">𝑦</ci></apply></apply></apply><csymbol cd="latexml" id="alg1.l9.m1.1.1.3.cmml" xref="alg1.l9.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.1c">\tilde{D}_{gNB_{y}}\leftarrow</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.1d">over~ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ←</annotation></semantics></math> subsampling <math alttext="D_{gNB_{y}}" class="ltx_Math" display="inline" id="alg1.l9.m2.1"><semantics id="alg1.l9.m2.1a"><msub id="alg1.l9.m2.1.1" xref="alg1.l9.m2.1.1.cmml"><mi id="alg1.l9.m2.1.1.2" xref="alg1.l9.m2.1.1.2.cmml">D</mi><mrow id="alg1.l9.m2.1.1.3" xref="alg1.l9.m2.1.1.3.cmml"><mi id="alg1.l9.m2.1.1.3.2" xref="alg1.l9.m2.1.1.3.2.cmml">g</mi><mo id="alg1.l9.m2.1.1.3.1" xref="alg1.l9.m2.1.1.3.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.3.3" xref="alg1.l9.m2.1.1.3.3.cmml">N</mi><mo id="alg1.l9.m2.1.1.3.1a" xref="alg1.l9.m2.1.1.3.1.cmml">⁢</mo><msub id="alg1.l9.m2.1.1.3.4" xref="alg1.l9.m2.1.1.3.4.cmml"><mi id="alg1.l9.m2.1.1.3.4.2" xref="alg1.l9.m2.1.1.3.4.2.cmml">B</mi><mi id="alg1.l9.m2.1.1.3.4.3" xref="alg1.l9.m2.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l9.m2.1b"><apply id="alg1.l9.m2.1.1.cmml" xref="alg1.l9.m2.1.1"><csymbol cd="ambiguous" 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end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span>Training, validation and testing split of <math alttext="\tilde{D}_{gNB_{y}}" class="ltx_Math" display="inline" id="alg1.l10.m1.1"><semantics id="alg1.l10.m1.1a"><msub id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml"><mover accent="true" id="alg1.l10.m1.1.1.2" xref="alg1.l10.m1.1.1.2.cmml"><mi id="alg1.l10.m1.1.1.2.2" xref="alg1.l10.m1.1.1.2.2.cmml">D</mi><mo id="alg1.l10.m1.1.1.2.1" xref="alg1.l10.m1.1.1.2.1.cmml">~</mo></mover><mrow id="alg1.l10.m1.1.1.3" xref="alg1.l10.m1.1.1.3.cmml"><mi id="alg1.l10.m1.1.1.3.2" xref="alg1.l10.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l10.m1.1.1.3.1" xref="alg1.l10.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.3.3" xref="alg1.l10.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l10.m1.1.1.3.1a" xref="alg1.l10.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l10.m1.1.1.3.4" xref="alg1.l10.m1.1.1.3.4.cmml"><mi id="alg1.l10.m1.1.1.3.4.2" xref="alg1.l10.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l10.m1.1.1.3.4.3" xref="alg1.l10.m1.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.1b"><apply id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"><csymbol cd="ambiguous" id="alg1.l10.m1.1.1.1.cmml" xref="alg1.l10.m1.1.1">subscript</csymbol><apply id="alg1.l10.m1.1.1.2.cmml" xref="alg1.l10.m1.1.1.2"><ci id="alg1.l10.m1.1.1.2.1.cmml" xref="alg1.l10.m1.1.1.2.1">~</ci><ci id="alg1.l10.m1.1.1.2.2.cmml" xref="alg1.l10.m1.1.1.2.2">𝐷</ci></apply><apply id="alg1.l10.m1.1.1.3.cmml" xref="alg1.l10.m1.1.1.3"><times id="alg1.l10.m1.1.1.3.1.cmml" xref="alg1.l10.m1.1.1.3.1"></times><ci id="alg1.l10.m1.1.1.3.2.cmml" xref="alg1.l10.m1.1.1.3.2">𝑔</ci><ci id="alg1.l10.m1.1.1.3.3.cmml" xref="alg1.l10.m1.1.1.3.3">𝑁</ci><apply id="alg1.l10.m1.1.1.3.4.cmml" xref="alg1.l10.m1.1.1.3.4"><csymbol cd="ambiguous" id="alg1.l10.m1.1.1.3.4.1.cmml" xref="alg1.l10.m1.1.1.3.4">subscript</csymbol><ci id="alg1.l10.m1.1.1.3.4.2.cmml" xref="alg1.l10.m1.1.1.3.4.2">𝐵</ci><ci id="alg1.l10.m1.1.1.3.4.3.cmml" xref="alg1.l10.m1.1.1.3.4.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.1c">\tilde{D}_{gNB_{y}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.1d">over~ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span>Fine-tuning using training set of <math alttext="\tilde{D}_{gNB_{y}}" class="ltx_Math" display="inline" id="alg1.l11.m1.1"><semantics id="alg1.l11.m1.1a"><msub id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml"><mover accent="true" id="alg1.l11.m1.1.1.2" xref="alg1.l11.m1.1.1.2.cmml"><mi id="alg1.l11.m1.1.1.2.2" xref="alg1.l11.m1.1.1.2.2.cmml">D</mi><mo id="alg1.l11.m1.1.1.2.1" xref="alg1.l11.m1.1.1.2.1.cmml">~</mo></mover><mrow id="alg1.l11.m1.1.1.3" xref="alg1.l11.m1.1.1.3.cmml"><mi id="alg1.l11.m1.1.1.3.2" xref="alg1.l11.m1.1.1.3.2.cmml">g</mi><mo id="alg1.l11.m1.1.1.3.1" xref="alg1.l11.m1.1.1.3.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.3.3" xref="alg1.l11.m1.1.1.3.3.cmml">N</mi><mo id="alg1.l11.m1.1.1.3.1a" xref="alg1.l11.m1.1.1.3.1.cmml">⁢</mo><msub id="alg1.l11.m1.1.1.3.4" xref="alg1.l11.m1.1.1.3.4.cmml"><mi id="alg1.l11.m1.1.1.3.4.2" xref="alg1.l11.m1.1.1.3.4.2.cmml">B</mi><mi id="alg1.l11.m1.1.1.3.4.3" xref="alg1.l11.m1.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><apply id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1"><csymbol cd="ambiguous" id="alg1.l11.m1.1.1.1.cmml" xref="alg1.l11.m1.1.1">subscript</csymbol><apply 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end_ARG start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l12.1.1.1" style="font-size:80%;">12:</span></span><math alttext="w_{gNB_{y}}\leftarrow" class="ltx_Math" display="inline" id="alg1.l12.m1.1"><semantics id="alg1.l12.m1.1a"><mrow id="alg1.l12.m1.1.1" xref="alg1.l12.m1.1.1.cmml"><msub id="alg1.l12.m1.1.1.2" xref="alg1.l12.m1.1.1.2.cmml"><mi id="alg1.l12.m1.1.1.2.2" xref="alg1.l12.m1.1.1.2.2.cmml">w</mi><mrow id="alg1.l12.m1.1.1.2.3" xref="alg1.l12.m1.1.1.2.3.cmml"><mi id="alg1.l12.m1.1.1.2.3.2" xref="alg1.l12.m1.1.1.2.3.2.cmml">g</mi><mo id="alg1.l12.m1.1.1.2.3.1" xref="alg1.l12.m1.1.1.2.3.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.2.3.3" xref="alg1.l12.m1.1.1.2.3.3.cmml">N</mi><mo id="alg1.l12.m1.1.1.2.3.1a" xref="alg1.l12.m1.1.1.2.3.1.cmml">⁢</mo><msub id="alg1.l12.m1.1.1.2.3.4" xref="alg1.l12.m1.1.1.2.3.4.cmml"><mi id="alg1.l12.m1.1.1.2.3.4.2" xref="alg1.l12.m1.1.1.2.3.4.2.cmml">B</mi><mi id="alg1.l12.m1.1.1.2.3.4.3" xref="alg1.l12.m1.1.1.2.3.4.3.cmml">y</mi></msub></mrow></msub><mo id="alg1.l12.m1.1.1.1" stretchy="false" xref="alg1.l12.m1.1.1.1.cmml">←</mo><mi id="alg1.l12.m1.1.1.3" xref="alg1.l12.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l12.m1.1b"><apply id="alg1.l12.m1.1.1.cmml" xref="alg1.l12.m1.1.1"><ci id="alg1.l12.m1.1.1.1.cmml" xref="alg1.l12.m1.1.1.1">←</ci><apply id="alg1.l12.m1.1.1.2.cmml" xref="alg1.l12.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l12.m1.1.1.2.1.cmml" xref="alg1.l12.m1.1.1.2">subscript</csymbol><ci id="alg1.l12.m1.1.1.2.2.cmml" xref="alg1.l12.m1.1.1.2.2">𝑤</ci><apply id="alg1.l12.m1.1.1.2.3.cmml" xref="alg1.l12.m1.1.1.2.3"><times id="alg1.l12.m1.1.1.2.3.1.cmml" xref="alg1.l12.m1.1.1.2.3.1"></times><ci id="alg1.l12.m1.1.1.2.3.2.cmml" xref="alg1.l12.m1.1.1.2.3.2">𝑔</ci><ci id="alg1.l12.m1.1.1.2.3.3.cmml" xref="alg1.l12.m1.1.1.2.3.3">𝑁</ci><apply id="alg1.l12.m1.1.1.2.3.4.cmml" xref="alg1.l12.m1.1.1.2.3.4"><csymbol cd="ambiguous" id="alg1.l12.m1.1.1.2.3.4.1.cmml" xref="alg1.l12.m1.1.1.2.3.4">subscript</csymbol><ci id="alg1.l12.m1.1.1.2.3.4.2.cmml" xref="alg1.l12.m1.1.1.2.3.4.2">𝐵</ci><ci id="alg1.l12.m1.1.1.2.3.4.3.cmml" xref="alg1.l12.m1.1.1.2.3.4.3">𝑦</ci></apply></apply></apply><csymbol cd="latexml" id="alg1.l12.m1.1.1.3.cmml" xref="alg1.l12.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l12.m1.1c">w_{gNB_{y}}\leftarrow</annotation><annotation encoding="application/x-llamapun" id="alg1.l12.m1.1d">italic_w start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ←</annotation></semantics></math> Save fine-tuned model parameters for <math alttext="\text{gNB}_{y}" class="ltx_Math" display="inline" id="alg1.l12.m2.1"><semantics id="alg1.l12.m2.1a"><msub id="alg1.l12.m2.1.1" xref="alg1.l12.m2.1.1.cmml"><mtext id="alg1.l12.m2.1.1.2" xref="alg1.l12.m2.1.1.2a.cmml">gNB</mtext><mi id="alg1.l12.m2.1.1.3" xref="alg1.l12.m2.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l12.m2.1b"><apply id="alg1.l12.m2.1.1.cmml" xref="alg1.l12.m2.1.1"><csymbol cd="ambiguous" id="alg1.l12.m2.1.1.1.cmml" xref="alg1.l12.m2.1.1">subscript</csymbol><ci id="alg1.l12.m2.1.1.2a.cmml" xref="alg1.l12.m2.1.1.2"><mtext id="alg1.l12.m2.1.1.2.cmml" xref="alg1.l12.m2.1.1.2">gNB</mtext></ci><ci id="alg1.l12.m2.1.1.3.cmml" xref="alg1.l12.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l12.m2.1c">\text{gNB}_{y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l12.m2.1d">gNB start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> </div> </div> </figure> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.12">We propose to learn the mapping function between the UE location and the best BPL serving that location for a reference <math alttext="\text{gNB}_{x}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><msub id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mtext id="S3.SS2.p1.1.m1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.2a.cmml">gNB</mtext><mi id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p1.1.m1.1.1.2a.cmml" xref="S3.SS2.p1.1.m1.1.1.2"><mtext id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">gNB</mtext></ci><ci id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">\text{gNB}_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">gNB start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> using a standard fully-connected NN and transfer the knowledge acquired by the model to another target <math alttext="\text{gNB}_{y}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><msub id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><mtext id="S3.SS2.p1.2.m2.1.1.2" xref="S3.SS2.p1.2.m2.1.1.2a.cmml">gNB</mtext><mi id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><apply id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.p1.2.m2.1.1.2a.cmml" xref="S3.SS2.p1.2.m2.1.1.2"><mtext id="S3.SS2.p1.2.m2.1.1.2.cmml" xref="S3.SS2.p1.2.m2.1.1.2">gNB</mtext></ci><ci id="S3.SS2.p1.2.m2.1.1.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\text{gNB}_{y}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">gNB start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>, as described in Alg. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#alg1" title="Algorithm 1 ‣ III-B Proposed Location-Based Transfer Learning Solution ‣ III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">1</span></a>. In the first step, we consider the complete BPL dataset <math alttext="D_{gNB_{x}}" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><msub id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mi id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml">D</mi><mrow id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml"><mi id="S3.SS2.p1.3.m3.1.1.3.2" xref="S3.SS2.p1.3.m3.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p1.3.m3.1.1.3.1" xref="S3.SS2.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.3.m3.1.1.3.3" xref="S3.SS2.p1.3.m3.1.1.3.3.cmml">N</mi><mo id="S3.SS2.p1.3.m3.1.1.3.1a" xref="S3.SS2.p1.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S3.SS2.p1.3.m3.1.1.3.4" xref="S3.SS2.p1.3.m3.1.1.3.4.cmml"><mi id="S3.SS2.p1.3.m3.1.1.3.4.2" xref="S3.SS2.p1.3.m3.1.1.3.4.2.cmml">B</mi><mi id="S3.SS2.p1.3.m3.1.1.3.4.3" xref="S3.SS2.p1.3.m3.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply 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encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> specific to the reference <math alttext="\text{gNB}_{x}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.1"><semantics id="S3.SS2.p1.4.m4.1a"><msub id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml"><mtext id="S3.SS2.p1.4.m4.1.1.2" xref="S3.SS2.p1.4.m4.1.1.2a.cmml">gNB</mtext><mi id="S3.SS2.p1.4.m4.1.1.3" xref="S3.SS2.p1.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.1b"><apply id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m4.1.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.p1.4.m4.1.1.2a.cmml" xref="S3.SS2.p1.4.m4.1.1.2"><mtext id="S3.SS2.p1.4.m4.1.1.2.cmml" xref="S3.SS2.p1.4.m4.1.1.2">gNB</mtext></ci><ci id="S3.SS2.p1.4.m4.1.1.3.cmml" xref="S3.SS2.p1.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.1c">\text{gNB}_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.1d">gNB start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and we split it into training, validation, and testing sets with a ratio of <math alttext="[60\%,20\%,20\%]" class="ltx_Math" display="inline" id="S3.SS2.p1.5.m5.3"><semantics id="S3.SS2.p1.5.m5.3a"><mrow id="S3.SS2.p1.5.m5.3.3.3" xref="S3.SS2.p1.5.m5.3.3.4.cmml"><mo id="S3.SS2.p1.5.m5.3.3.3.4" stretchy="false" xref="S3.SS2.p1.5.m5.3.3.4.cmml">[</mo><mrow id="S3.SS2.p1.5.m5.1.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.1.cmml"><mn id="S3.SS2.p1.5.m5.1.1.1.1.2" xref="S3.SS2.p1.5.m5.1.1.1.1.2.cmml">60</mn><mo id="S3.SS2.p1.5.m5.1.1.1.1.1" xref="S3.SS2.p1.5.m5.1.1.1.1.1.cmml">%</mo></mrow><mo id="S3.SS2.p1.5.m5.3.3.3.5" xref="S3.SS2.p1.5.m5.3.3.4.cmml">,</mo><mrow id="S3.SS2.p1.5.m5.2.2.2.2" 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We train the NN using the training set and save the resulting weights <math alttext="w_{gNB_{x}}" class="ltx_Math" display="inline" id="S3.SS2.p1.6.m6.1"><semantics id="S3.SS2.p1.6.m6.1a"><msub id="S3.SS2.p1.6.m6.1.1" xref="S3.SS2.p1.6.m6.1.1.cmml"><mi id="S3.SS2.p1.6.m6.1.1.2" xref="S3.SS2.p1.6.m6.1.1.2.cmml">w</mi><mrow id="S3.SS2.p1.6.m6.1.1.3" xref="S3.SS2.p1.6.m6.1.1.3.cmml"><mi id="S3.SS2.p1.6.m6.1.1.3.2" xref="S3.SS2.p1.6.m6.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p1.6.m6.1.1.3.1" xref="S3.SS2.p1.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.6.m6.1.1.3.3" xref="S3.SS2.p1.6.m6.1.1.3.3.cmml">N</mi><mo id="S3.SS2.p1.6.m6.1.1.3.1a" xref="S3.SS2.p1.6.m6.1.1.3.1.cmml">⁢</mo><msub id="S3.SS2.p1.6.m6.1.1.3.4" xref="S3.SS2.p1.6.m6.1.1.3.4.cmml"><mi id="S3.SS2.p1.6.m6.1.1.3.4.2" xref="S3.SS2.p1.6.m6.1.1.3.4.2.cmml">B</mi><mi id="S3.SS2.p1.6.m6.1.1.3.4.3" xref="S3.SS2.p1.6.m6.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m6.1b"><apply id="S3.SS2.p1.6.m6.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.1.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS2.p1.6.m6.1.1.2.cmml" xref="S3.SS2.p1.6.m6.1.1.2">𝑤</ci><apply id="S3.SS2.p1.6.m6.1.1.3.cmml" xref="S3.SS2.p1.6.m6.1.1.3"><times id="S3.SS2.p1.6.m6.1.1.3.1.cmml" xref="S3.SS2.p1.6.m6.1.1.3.1"></times><ci id="S3.SS2.p1.6.m6.1.1.3.2.cmml" xref="S3.SS2.p1.6.m6.1.1.3.2">𝑔</ci><ci id="S3.SS2.p1.6.m6.1.1.3.3.cmml" xref="S3.SS2.p1.6.m6.1.1.3.3">𝑁</ci><apply id="S3.SS2.p1.6.m6.1.1.3.4.cmml" xref="S3.SS2.p1.6.m6.1.1.3.4"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.1.1.3.4.1.cmml" xref="S3.SS2.p1.6.m6.1.1.3.4">subscript</csymbol><ci id="S3.SS2.p1.6.m6.1.1.3.4.2.cmml" xref="S3.SS2.p1.6.m6.1.1.3.4.2">𝐵</ci><ci id="S3.SS2.p1.6.m6.1.1.3.4.3.cmml" xref="S3.SS2.p1.6.m6.1.1.3.4.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m6.1c">w_{gNB_{x}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.6.m6.1d">italic_w start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. In the second step, the NN initialized with <math alttext="w_{gNB_{x}}" class="ltx_Math" display="inline" id="S3.SS2.p1.7.m7.1"><semantics id="S3.SS2.p1.7.m7.1a"><msub id="S3.SS2.p1.7.m7.1.1" xref="S3.SS2.p1.7.m7.1.1.cmml"><mi id="S3.SS2.p1.7.m7.1.1.2" xref="S3.SS2.p1.7.m7.1.1.2.cmml">w</mi><mrow id="S3.SS2.p1.7.m7.1.1.3" xref="S3.SS2.p1.7.m7.1.1.3.cmml"><mi id="S3.SS2.p1.7.m7.1.1.3.2" xref="S3.SS2.p1.7.m7.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p1.7.m7.1.1.3.1" xref="S3.SS2.p1.7.m7.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.7.m7.1.1.3.3" xref="S3.SS2.p1.7.m7.1.1.3.3.cmml">N</mi><mo id="S3.SS2.p1.7.m7.1.1.3.1a" xref="S3.SS2.p1.7.m7.1.1.3.1.cmml">⁢</mo><msub id="S3.SS2.p1.7.m7.1.1.3.4" xref="S3.SS2.p1.7.m7.1.1.3.4.cmml"><mi id="S3.SS2.p1.7.m7.1.1.3.4.2" xref="S3.SS2.p1.7.m7.1.1.3.4.2.cmml">B</mi><mi id="S3.SS2.p1.7.m7.1.1.3.4.3" xref="S3.SS2.p1.7.m7.1.1.3.4.3.cmml">x</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.7.m7.1b"><apply id="S3.SS2.p1.7.m7.1.1.cmml" 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id="S3.SS2.p1.9.m9.1.1.3.2.cmml" xref="S3.SS2.p1.9.m9.1.1.3.2">𝑔</ci><ci id="S3.SS2.p1.9.m9.1.1.3.3.cmml" xref="S3.SS2.p1.9.m9.1.1.3.3">𝑁</ci><apply id="S3.SS2.p1.9.m9.1.1.3.4.cmml" xref="S3.SS2.p1.9.m9.1.1.3.4"><csymbol cd="ambiguous" id="S3.SS2.p1.9.m9.1.1.3.4.1.cmml" xref="S3.SS2.p1.9.m9.1.1.3.4">subscript</csymbol><ci id="S3.SS2.p1.9.m9.1.1.3.4.2.cmml" xref="S3.SS2.p1.9.m9.1.1.3.4.2">𝐵</ci><ci id="S3.SS2.p1.9.m9.1.1.3.4.3.cmml" xref="S3.SS2.p1.9.m9.1.1.3.4.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.9.m9.1c">\tilde{D}_{gNB_{y}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.9.m9.1d">over~ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> uniformly sampled from the corresponding BPL dataset <math alttext="D_{gNB_{y}}" class="ltx_Math" display="inline" id="S3.SS2.p1.10.m10.1"><semantics id="S3.SS2.p1.10.m10.1a"><msub id="S3.SS2.p1.10.m10.1.1" xref="S3.SS2.p1.10.m10.1.1.cmml"><mi id="S3.SS2.p1.10.m10.1.1.2" xref="S3.SS2.p1.10.m10.1.1.2.cmml">D</mi><mrow id="S3.SS2.p1.10.m10.1.1.3" xref="S3.SS2.p1.10.m10.1.1.3.cmml"><mi id="S3.SS2.p1.10.m10.1.1.3.2" xref="S3.SS2.p1.10.m10.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p1.10.m10.1.1.3.1" xref="S3.SS2.p1.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p1.10.m10.1.1.3.3" xref="S3.SS2.p1.10.m10.1.1.3.3.cmml">N</mi><mo id="S3.SS2.p1.10.m10.1.1.3.1a" xref="S3.SS2.p1.10.m10.1.1.3.1.cmml">⁢</mo><msub id="S3.SS2.p1.10.m10.1.1.3.4" xref="S3.SS2.p1.10.m10.1.1.3.4.cmml"><mi id="S3.SS2.p1.10.m10.1.1.3.4.2" xref="S3.SS2.p1.10.m10.1.1.3.4.2.cmml">B</mi><mi id="S3.SS2.p1.10.m10.1.1.3.4.3" xref="S3.SS2.p1.10.m10.1.1.3.4.3.cmml">y</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.10.m10.1b"><apply id="S3.SS2.p1.10.m10.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.10.m10.1.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1">subscript</csymbol><ci id="S3.SS2.p1.10.m10.1.1.2.cmml" xref="S3.SS2.p1.10.m10.1.1.2">𝐷</ci><apply id="S3.SS2.p1.10.m10.1.1.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3"><times id="S3.SS2.p1.10.m10.1.1.3.1.cmml" xref="S3.SS2.p1.10.m10.1.1.3.1"></times><ci id="S3.SS2.p1.10.m10.1.1.3.2.cmml" xref="S3.SS2.p1.10.m10.1.1.3.2">𝑔</ci><ci id="S3.SS2.p1.10.m10.1.1.3.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3.3">𝑁</ci><apply id="S3.SS2.p1.10.m10.1.1.3.4.cmml" xref="S3.SS2.p1.10.m10.1.1.3.4"><csymbol cd="ambiguous" id="S3.SS2.p1.10.m10.1.1.3.4.1.cmml" xref="S3.SS2.p1.10.m10.1.1.3.4">subscript</csymbol><ci id="S3.SS2.p1.10.m10.1.1.3.4.2.cmml" xref="S3.SS2.p1.10.m10.1.1.3.4.2">𝐵</ci><ci id="S3.SS2.p1.10.m10.1.1.3.4.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3.4.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.10.m10.1c">D_{gNB_{y}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.10.m10.1d">italic_D start_POSTSUBSCRIPT italic_g italic_N italic_B start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. We split this reduced dataset into training, validation, and testing with the same ratio, and we use the training set for fine-tuning the model. 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In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#bib.bib4" title="">4</a>]</cite>, the UE location is used as a training feature to select the best gNB codebook entry in a search space of size <math alttext="N_{gNB}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><msub id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">N</mi><mrow id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml"><mi id="S3.SS2.p2.1.m1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.3.2.cmml">g</mi><mo id="S3.SS2.p2.1.m1.1.1.3.1" xref="S3.SS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml">N</mi><mo id="S3.SS2.p2.1.m1.1.1.3.1a" xref="S3.SS2.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.3.4" xref="S3.SS2.p2.1.m1.1.1.3.4.cmml">B</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝑁</ci><apply id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3"><times id="S3.SS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.1"></times><ci id="S3.SS2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2">𝑔</ci><ci id="S3.SS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3">𝑁</ci><ci id="S3.SS2.p2.1.m1.1.1.3.4.cmml" xref="S3.SS2.p2.1.m1.1.1.3.4">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">N_{gNB}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_N start_POSTSUBSCRIPT italic_g italic_N italic_B end_POSTSUBSCRIPT</annotation></semantics></math>, while assuming an omni-directional UE antenna. 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Given the more challenging beam selection task, we evaluate the accuracy of correctly predicting the best BPL, i.e., top-1 accuracy, as well as the accuracy of the best BPL being within the top-5 BPLs predicted by the model, i.e. best-in-top-5 accuracy. We emphasize that the selected best-in-top-5 metric allows us to evaluate the possibility of narrowing the search space to only five BPLs instead of performing a full sweep over the complete search space. Therefore, our training labels comprise the top five BPLs for the given UE location.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.1">To address the limitation of the cross-entropy loss function, commonly used for classification tasks and which only favors the best prediction, we employ the weighted cross-entropy loss function given by</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="L=\sum_{k=1}^{5}-w_{k}\log\hat{p}_{k}" class="ltx_Math" display="block" id="S3.E3.m1.1"><semantics id="S3.E3.m1.1a"><mrow id="S3.E3.m1.1.1" xref="S3.E3.m1.1.1.cmml"><mi id="S3.E3.m1.1.1.2" xref="S3.E3.m1.1.1.2.cmml">L</mi><mo id="S3.E3.m1.1.1.1" rspace="0.111em" xref="S3.E3.m1.1.1.1.cmml">=</mo><mrow id="S3.E3.m1.1.1.3" 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id="S3.E3.m1.1c">L=\sum_{k=1}^{5}-w_{k}\log\hat{p}_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.1d">italic_L = ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_log over^ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.4">where <math alttext="\hat{p}_{k}" class="ltx_Math" display="inline" id="S3.SS2.p5.1.m1.1"><semantics id="S3.SS2.p5.1.m1.1a"><msub id="S3.SS2.p5.1.m1.1.1" xref="S3.SS2.p5.1.m1.1.1.cmml"><mover accent="true" id="S3.SS2.p5.1.m1.1.1.2" 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start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> denotes the predicted probability that the <math alttext="\text{k}^{th}" class="ltx_Math" display="inline" id="S3.SS2.p5.2.m2.1"><semantics id="S3.SS2.p5.2.m2.1a"><msup id="S3.SS2.p5.2.m2.1.1" xref="S3.SS2.p5.2.m2.1.1.cmml"><mtext id="S3.SS2.p5.2.m2.1.1.2" xref="S3.SS2.p5.2.m2.1.1.2a.cmml">k</mtext><mrow id="S3.SS2.p5.2.m2.1.1.3" xref="S3.SS2.p5.2.m2.1.1.3.cmml"><mi id="S3.SS2.p5.2.m2.1.1.3.2" xref="S3.SS2.p5.2.m2.1.1.3.2.cmml">t</mi><mo id="S3.SS2.p5.2.m2.1.1.3.1" xref="S3.SS2.p5.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.p5.2.m2.1.1.3.3" xref="S3.SS2.p5.2.m2.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p5.2.m2.1b"><apply id="S3.SS2.p5.2.m2.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p5.2.m2.1.1.1.cmml" xref="S3.SS2.p5.2.m2.1.1">superscript</csymbol><ci id="S3.SS2.p5.2.m2.1.1.2a.cmml" xref="S3.SS2.p5.2.m2.1.1.2"><mtext id="S3.SS2.p5.2.m2.1.1.2.cmml" xref="S3.SS2.p5.2.m2.1.1.2">k</mtext></ci><apply id="S3.SS2.p5.2.m2.1.1.3.cmml" xref="S3.SS2.p5.2.m2.1.1.3"><times id="S3.SS2.p5.2.m2.1.1.3.1.cmml" xref="S3.SS2.p5.2.m2.1.1.3.1"></times><ci id="S3.SS2.p5.2.m2.1.1.3.2.cmml" xref="S3.SS2.p5.2.m2.1.1.3.2">𝑡</ci><ci id="S3.SS2.p5.2.m2.1.1.3.3.cmml" xref="S3.SS2.p5.2.m2.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p5.2.m2.1c">\text{k}^{th}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.2.m2.1d">k start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> best BPL is selected, and <math alttext="w_{k}" class="ltx_Math" display="inline" id="S3.SS2.p5.3.m3.1"><semantics id="S3.SS2.p5.3.m3.1a"><msub id="S3.SS2.p5.3.m3.1.1" xref="S3.SS2.p5.3.m3.1.1.cmml"><mi id="S3.SS2.p5.3.m3.1.1.2" xref="S3.SS2.p5.3.m3.1.1.2.cmml">w</mi><mi id="S3.SS2.p5.3.m3.1.1.3" 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id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.1">We evaluate the effectiveness of our cross-environment transfer learning by examining the top-1 and best-in-top-5 accuracies on the testing set relative to the target gNB.</p> </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S3.T1.6.1.1" style="font-size:90%;">TABLE I</span>: </span><span class="ltx_text" id="S3.T1.7.2" style="font-size:90%;">NN architecture and hyperparameters</span></figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S3.T1.4"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S3.T1.4.5.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.4.5.1.1"><span class="ltx_text ltx_font_bold" id="S3.T1.4.5.1.1.1">Parameters</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S3.T1.4.5.1.2"><span class="ltx_text ltx_font_bold" id="S3.T1.4.5.1.2.1">Values</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S3.T1.2.2.3">Input size</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S3.T1.2.2.2"> <math alttext="2" class="ltx_Math" display="inline" id="S3.T1.1.1.1.m1.1"><semantics id="S3.T1.1.1.1.m1.1a"><mn id="S3.T1.1.1.1.m1.1.1" xref="S3.T1.1.1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.T1.1.1.1.m1.1b"><cn id="S3.T1.1.1.1.m1.1.1.cmml" type="integer" xref="S3.T1.1.1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.1.1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.T1.1.1.1.m1.1d">2</annotation></semantics></math> <math alttext="\{x,y\}" class="ltx_Math" display="inline" id="S3.T1.2.2.2.m2.2"><semantics id="S3.T1.2.2.2.m2.2a"><mrow id="S3.T1.2.2.2.m2.2.3.2" xref="S3.T1.2.2.2.m2.2.3.1.cmml"><mo id="S3.T1.2.2.2.m2.2.3.2.1" stretchy="false" xref="S3.T1.2.2.2.m2.2.3.1.cmml">{</mo><mi id="S3.T1.2.2.2.m2.1.1" xref="S3.T1.2.2.2.m2.1.1.cmml">x</mi><mo id="S3.T1.2.2.2.m2.2.3.2.2" xref="S3.T1.2.2.2.m2.2.3.1.cmml">,</mo><mi id="S3.T1.2.2.2.m2.2.2" xref="S3.T1.2.2.2.m2.2.2.cmml">y</mi><mo id="S3.T1.2.2.2.m2.2.3.2.3" stretchy="false" xref="S3.T1.2.2.2.m2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.2.2.2.m2.2b"><set id="S3.T1.2.2.2.m2.2.3.1.cmml" xref="S3.T1.2.2.2.m2.2.3.2"><ci id="S3.T1.2.2.2.m2.1.1.cmml" xref="S3.T1.2.2.2.m2.1.1">𝑥</ci><ci id="S3.T1.2.2.2.m2.2.2.cmml" xref="S3.T1.2.2.2.m2.2.2">𝑦</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.2.2.2.m2.2c">\{x,y\}</annotation><annotation encoding="application/x-llamapun" id="S3.T1.2.2.2.m2.2d">{ italic_x , italic_y }</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S3.T1.4.6.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.6.1.1">Label size</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.6.1.2">5</td> </tr> <tr class="ltx_tr" id="S3.T1.4.7.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.7.2.1">Hidden Layers</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.7.2.2">3 layers, 128 nodes each</td> </tr> <tr class="ltx_tr" id="S3.T1.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.3.3.2">Output size</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.3.3.1"> <math alttext="64\times 16" class="ltx_Math" display="inline" id="S3.T1.3.3.1.m1.1"><semantics id="S3.T1.3.3.1.m1.1a"><mrow id="S3.T1.3.3.1.m1.1.1" xref="S3.T1.3.3.1.m1.1.1.cmml"><mn id="S3.T1.3.3.1.m1.1.1.2" xref="S3.T1.3.3.1.m1.1.1.2.cmml">64</mn><mo id="S3.T1.3.3.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.T1.3.3.1.m1.1.1.1.cmml">×</mo><mn id="S3.T1.3.3.1.m1.1.1.3" xref="S3.T1.3.3.1.m1.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.3.3.1.m1.1b"><apply id="S3.T1.3.3.1.m1.1.1.cmml" xref="S3.T1.3.3.1.m1.1.1"><times id="S3.T1.3.3.1.m1.1.1.1.cmml" xref="S3.T1.3.3.1.m1.1.1.1"></times><cn id="S3.T1.3.3.1.m1.1.1.2.cmml" type="integer" xref="S3.T1.3.3.1.m1.1.1.2">64</cn><cn id="S3.T1.3.3.1.m1.1.1.3.cmml" type="integer" xref="S3.T1.3.3.1.m1.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.3.3.1.m1.1c">64\times 16</annotation><annotation encoding="application/x-llamapun" id="S3.T1.3.3.1.m1.1d">64 × 16</annotation></semantics></math> (number of available BPLs)</td> </tr> <tr class="ltx_tr" id="S3.T1.4.8.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.8.3.1">Activation</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.8.3.2">ReLu on hidden layers</td> </tr> <tr class="ltx_tr" id="S3.T1.4.9.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.9.4.1">Loss function</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.9.4.2">Customized weighted cross-entropy loss</td> </tr> <tr class="ltx_tr" id="S3.T1.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.4.2">Training/Validation/Test</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.4.1"><math alttext="[60\%,20\%,20\%]" class="ltx_Math" display="inline" id="S3.T1.4.4.1.m1.3"><semantics id="S3.T1.4.4.1.m1.3a"><mrow id="S3.T1.4.4.1.m1.3.3.3" xref="S3.T1.4.4.1.m1.3.3.4.cmml"><mo id="S3.T1.4.4.1.m1.3.3.3.4" stretchy="false" xref="S3.T1.4.4.1.m1.3.3.4.cmml">[</mo><mrow id="S3.T1.4.4.1.m1.1.1.1.1" xref="S3.T1.4.4.1.m1.1.1.1.1.cmml"><mn id="S3.T1.4.4.1.m1.1.1.1.1.2" xref="S3.T1.4.4.1.m1.1.1.1.1.2.cmml">60</mn><mo id="S3.T1.4.4.1.m1.1.1.1.1.1" xref="S3.T1.4.4.1.m1.1.1.1.1.1.cmml">%</mo></mrow><mo id="S3.T1.4.4.1.m1.3.3.3.5" xref="S3.T1.4.4.1.m1.3.3.4.cmml">,</mo><mrow id="S3.T1.4.4.1.m1.2.2.2.2" xref="S3.T1.4.4.1.m1.2.2.2.2.cmml"><mn id="S3.T1.4.4.1.m1.2.2.2.2.2" xref="S3.T1.4.4.1.m1.2.2.2.2.2.cmml">20</mn><mo id="S3.T1.4.4.1.m1.2.2.2.2.1" xref="S3.T1.4.4.1.m1.2.2.2.2.1.cmml">%</mo></mrow><mo id="S3.T1.4.4.1.m1.3.3.3.6" xref="S3.T1.4.4.1.m1.3.3.4.cmml">,</mo><mrow id="S3.T1.4.4.1.m1.3.3.3.3" xref="S3.T1.4.4.1.m1.3.3.3.3.cmml"><mn id="S3.T1.4.4.1.m1.3.3.3.3.2" xref="S3.T1.4.4.1.m1.3.3.3.3.2.cmml">20</mn><mo id="S3.T1.4.4.1.m1.3.3.3.3.1" xref="S3.T1.4.4.1.m1.3.3.3.3.1.cmml">%</mo></mrow><mo id="S3.T1.4.4.1.m1.3.3.3.7" stretchy="false" xref="S3.T1.4.4.1.m1.3.3.4.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.T1.4.4.1.m1.3b"><list id="S3.T1.4.4.1.m1.3.3.4.cmml" xref="S3.T1.4.4.1.m1.3.3.3"><apply id="S3.T1.4.4.1.m1.1.1.1.1.cmml" xref="S3.T1.4.4.1.m1.1.1.1.1"><csymbol cd="latexml" id="S3.T1.4.4.1.m1.1.1.1.1.1.cmml" xref="S3.T1.4.4.1.m1.1.1.1.1.1">percent</csymbol><cn id="S3.T1.4.4.1.m1.1.1.1.1.2.cmml" type="integer" xref="S3.T1.4.4.1.m1.1.1.1.1.2">60</cn></apply><apply id="S3.T1.4.4.1.m1.2.2.2.2.cmml" xref="S3.T1.4.4.1.m1.2.2.2.2"><csymbol cd="latexml" id="S3.T1.4.4.1.m1.2.2.2.2.1.cmml" xref="S3.T1.4.4.1.m1.2.2.2.2.1">percent</csymbol><cn id="S3.T1.4.4.1.m1.2.2.2.2.2.cmml" type="integer" xref="S3.T1.4.4.1.m1.2.2.2.2.2">20</cn></apply><apply id="S3.T1.4.4.1.m1.3.3.3.3.cmml" xref="S3.T1.4.4.1.m1.3.3.3.3"><csymbol cd="latexml" id="S3.T1.4.4.1.m1.3.3.3.3.1.cmml" xref="S3.T1.4.4.1.m1.3.3.3.3.1">percent</csymbol><cn id="S3.T1.4.4.1.m1.3.3.3.3.2.cmml" type="integer" xref="S3.T1.4.4.1.m1.3.3.3.3.2">20</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.4.4.1.m1.3c">[60\%,20\%,20\%]</annotation><annotation encoding="application/x-llamapun" id="S3.T1.4.4.1.m1.3d">[ 60 % , 20 % , 20 % ]</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S3.T1.4.10.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.10.5.1">Training batch size</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.10.5.2">128</td> </tr> <tr class="ltx_tr" id="S3.T1.4.11.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.11.6.1">Learning epochs</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.11.6.2">60</td> </tr> <tr class="ltx_tr" id="S3.T1.4.12.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.12.7.1">Optimizer</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.12.7.2">Adam</td> </tr> <tr class="ltx_tr" id="S3.T1.4.13.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r" id="S3.T1.4.13.8.1">Initial learning rate</th> <td class="ltx_td ltx_align_center ltx_border_r" id="S3.T1.4.13.8.2">0.2</td> </tr> <tr class="ltx_tr" id="S3.T1.4.14.9"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r" id="S3.T1.4.14.9.1">Learning rate scheduler</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S3.T1.4.14.9.2">MultiStepLR, steps at 20 and 40</td> </tr> </tbody> </table> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Results</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section we present a performance evaluation of our transfer learning solution proposed in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3" title="III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">III</span></a>. Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS1" title="IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-A</span></span></a> presents the performance of transfer learning from a reference gNB to a target gNB in the same city, i.e., intra-city transfer. Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS2" title="IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-B</span></span></a> extends the analysis by evaluating the performance of transferring between gNBs in different cities, i.e., inter-city transfer. In Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS3" title="IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-C</span></span></a> we study how the amount of fine-tuning data impacts the performance.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2">Intra-City Transfer Learning Performance</span> </h3> <figure class="ltx_figure" id="S4.F2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F2.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F2.sf1.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F2.sf1.3.2" style="font-size:90%;">Frankfurt generalization.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F2.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F2.sf2.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F2.sf2.3.2" style="font-size:90%;">Frankfurt transfer.</span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F2.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F2.sf3.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf3.2.1.1" style="font-size:90%;">(c)</span> </span><span class="ltx_text" id="S4.F2.sf3.3.2" style="font-size:90%;">Seoul generalization.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F2.sf4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F2.sf4.g1" src="x6.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.sf4.2.1.1" style="font-size:90%;">(d)</span> </span><span class="ltx_text" id="S4.F2.sf4.3.2" style="font-size:90%;">Seoul transfer.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F2.2.1.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S4.F2.3.2" style="font-size:90%;">Top-1 BPL prediction: generalization and transfer learning accuracy using 5% fine-tuning dataset size across all gNB combinations of reference and target gNBs in (a)-(b) Frankfurt and (c)-(d) Seoul. The diagonal entries correspond to the accuracy of training and testing on the same gNB. </span></figcaption> </figure> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Let us first evaluate the generalization accuracy of our model by training it for a reference gNB and testing it on a target gNB <span class="ltx_text ltx_font_italic" id="S4.SS1.p1.1.1">without</span> fine-tuning. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf1" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(a)</span></a> shows this result in terms of top-1 BPL prediction accuracy for all the gNB combinations in Frankfurt. We note that the diagonal entries correspond to the baseline accuracy of the model, i.e., training and testing on the same gNB. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf1" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(a)</span></a> shows that the model generalizes very poorly, with a mean top-1 accuracy of 5% for gNB combinations where the reference and target gNBs do not coincide. This clearly demonstrates the site-specific nature of the learning task and highlights the necessity of fine-tuning data specific to the target gNB for effective transfer learning. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a> shows the top-1 accuracy achieved by transferring the model between gNB combinations in Frankfurt and fine-tuning it with only 5% of the target gNB dataset. Again, the diagonal entries of the matrix represent the baseline accuracy achieved by training and testing on the same gNB. Comparing Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf1" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(a)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a> shows that transfer learning enables the model to adapt better to the new environment, significantly improving the top-1 beam prediction accuracy. Specifically, the transfer learning accuracy is up to 75%, corresponding in an improvement of 70 percentage points from the generalization accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf1" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(a)</span></a>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a> also reveals a distinct vertical line pattern for most target gNBs, indicating that the model adapts better for some target gNBs than others. To investigate the underlying reasons, let us consider Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>, which shows the coverage characteristics for the gNBs in Frankfurt. We observe that the gNBs covering smaller areas in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a> generally correspond to the target gNBs with low transfer learning accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>, i.e., gNBs 1, 6, 11, 20, 21, and especially 25. This is because less extensive coverage corresponds to a reduced dataset size and likely insufficient data for fine-tuning the model. This is also consistent with the lower baseline accuracy (diagonal in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf1" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(a)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>) for these gNBs. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a> also shows that this group of gNBs with small coverage generally perform worse as the reference gNB, suggesting that the size of the dataset plays a key role also in selecting which reference gNB the model should be transferred from.</p> </div> <figure class="ltx_figure" id="S4.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="296" id="S4.F3.sf1.g1" src="x7.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F3.sf1.3.2" style="font-size:90%;">Frankfurt.</span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="296" id="S4.F3.sf2.g1" src="x8.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F3.sf2.3.2" style="font-size:90%;">Seoul.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.14.6.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.10.5" style="font-size:90%;">Percentage coverage with respect to the total <math alttext="500" class="ltx_Math" display="inline" id="S4.F3.6.1.m1.1"><semantics id="S4.F3.6.1.m1.1b"><mn id="S4.F3.6.1.m1.1.1" xref="S4.F3.6.1.m1.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.F3.6.1.m1.1c"><cn id="S4.F3.6.1.m1.1.1.cmml" type="integer" xref="S4.F3.6.1.m1.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.6.1.m1.1d">500</annotation><annotation encoding="application/x-llamapun" id="S4.F3.6.1.m1.1e">500</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S4.F3.10.5.1">m</span> <math alttext="\times" class="ltx_Math" display="inline" id="S4.F3.8.3.m3.1"><semantics id="S4.F3.8.3.m3.1b"><mo id="S4.F3.8.3.m3.1.1" xref="S4.F3.8.3.m3.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S4.F3.8.3.m3.1c"><times id="S4.F3.8.3.m3.1.1.cmml" xref="S4.F3.8.3.m3.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.8.3.m3.1d">\times</annotation><annotation encoding="application/x-llamapun" id="S4.F3.8.3.m3.1e">×</annotation></semantics></math> <math alttext="500" class="ltx_Math" display="inline" id="S4.F3.9.4.m4.1"><semantics id="S4.F3.9.4.m4.1b"><mn id="S4.F3.9.4.m4.1.1" xref="S4.F3.9.4.m4.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.F3.9.4.m4.1c"><cn id="S4.F3.9.4.m4.1.1.cmml" type="integer" xref="S4.F3.9.4.m4.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.9.4.m4.1d">500</annotation><annotation encoding="application/x-llamapun" id="S4.F3.9.4.m4.1e">500</annotation></semantics></math> <span class="ltx_text ltx_markedasmath" id="S4.F3.10.5.2">m</span> study area and corresponding dataset size of the gNBs in (a) Frankfurt and (b) Seoul, differentiating between LoS and NLoS coverage.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1">We conduct the same analysis in Seoul to validate the aforementioned insights, with Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf3" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(c)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf4" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(d)</span></a> presenting the generalization and transfer accuracies, respectively. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf3" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(c)</span></a> confirms that the model exhibits very poor generalization also in Seoul, as evidenced by the low top-1 accuracies outside the diagonal. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf4" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(d)</span></a> shows that transfer learning with 5% fine-tuning leads to substantial accuracy improvements in Seoul, resulting in beam prediction accuracy ranging between 60% and 80%, corresponding to an improvement between 55 and 75 percentage points respect to the mean generalization accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf3" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(c)</span></a>. The transfer learning accuracies in Seoul in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf4" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(d)</span></a> often surpass those observed in Frankfurt in Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>, likely because Seoul gNBs generally cover more extensive areas (<span class="ltx_text ltx_font_italic" id="S4.SS1.p2.1.1">cf</span>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a> vs Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf2" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(b)</span></a>), resulting in larger datasets used for training and fine-tuning the model. Furthermore, the accuracies in Seoul in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf4" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(d)</span></a> are more homogeneous among the gNB combinations with respect to Frankfurt in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>. However, we still observe that the model achieves lower accuracies when adapting to target gNBs 5, 6 and 26. This is consistent with the small dataset size of gNBs 5 and 6 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf2" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(b)</span></a>, while gNB 26’s coverage is comparable to e.g. gNBs 16 and 20 which exhibits better performance in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf4" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(d)</span></a>. This suggests that other factors, such as e.g., the spatial distribution of the type of coverage beam orientation, likely play a role in the transfer learning accuracy, although their effect is not straightforward to explain, as evidenced by the lack of clear correlation between the e.g. LoS/NLoS per gNB coverage reported in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3" title="Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3</span></a> and the accuracies in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2" title="Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2</span></a>. Instead, in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.SS3" title="IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-C</span></span></a> we further investigate the impact of the dominant observable factor of the training dataset size.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2">Inter-City Transfer Learning Performance</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">Let us now examine how transfer learning performs between different cities to determine to what extent our approach be applied between network environments with varying urban densities and layouts. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4.sf1" title="In Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4(a)</span></a> shows the top-1 accuracies achieved when transferring from reference gNBs in Seoul to target gNBs in Frankfurt with a fine-tuning percentage of 5%, while Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4.sf2" title="In Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4(b)</span></a> shows the reverse scenario, i.e. transferring from reference gNBs in Frankfurt to target gNBs in Seoul with the same fine-tuning percentage. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4.sf1" title="In Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4(a)</span></a> shows that the per-target gNB transfer learning accuracy when transferring models trained on Seoul gNBs is comparable to the inter-city accuracy in Frankfurt in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>. The dominant trend in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4.sf1" title="In Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4(a)</span></a> is again the difficulty of adapting to target gNBs 1, 6, 11, 21, 22, and 25, for which the fine-tuning dataset might be insufficient. This suggests that despite the reference gNBs being in another city, the model can adapt well as long as it is provided with sufficient fine-tuning data from the target gNB. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4.sf2" title="In Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4(b)</span></a> shows that gNBs providing small area coverage in Frankfurt, e.g. gNBs 1, 2, 21, 22, and 25 in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>, perform relatively poorly as reference gNBs for transfer learning to Seoul. In contrast, effective transfer learning is enabled by choosing reference gNBs with larger area coverage in Frankfurt, e.g., 10 and 13 in Fig.<a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>. Overall, the high inter-city transfer learning accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F4" title="Figure 4 ‣ IV-B Inter-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">4</span></a> of up to 80% is highly encouraging, suggesting that our approach is practically scalable also across different cities.</p> </div> <figure class="ltx_figure" id="S4.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F4.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F4.sf1.g1" src="x9.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F4.sf1.3.2" style="font-size:90%;">Seoul-to-Frankfurt.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F4.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F4.sf2.g1" src="x10.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F4.sf2.3.2" style="font-size:90%;">Frankfurt-to-Seoul.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.2.1.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S4.F4.3.2" style="font-size:90%;">Top-1 transfer learning accuracy across the cities using 5% fine-tuning for gNB combinations of (a) reference gNBs in Seoul and target gNBs in Frankfurt and (b) vice-versa. </span></figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS3.4.1.1">IV-C</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS3.5.2">Fine-Tuning Effect</span> </h3> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="233" id="S4.F5.sf1.g1" src="x11.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf1.2.1.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F5.sf1.3.2" style="font-size:90%;">From reference gNB 20 to target gNB 14.</span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="233" id="S4.F5.sf2.g1" src="x12.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf2.2.1.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F5.sf2.3.2" style="font-size:90%;">From reference gNB 17 to target gNB 11.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.2.1.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S4.F5.3.2" style="font-size:90%;">Test accuracy vs. percentage of target gNB dataset used for fine-tuning in transferring between selected gNB combinations in Frankfurt. The plots also include the baseline accuracy of training and testing on target gNB.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">Let us now explicitly explore the impact of varying the fine-tuning dataset percentage on the transfer learning performance. We focus on the Frankfurt environment given the greater variance of the results among the target gNBs observed in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F5.sf1" title="In Figure 5 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">5(a)</span></a> presents the top-1 transfer accuracy versus the size of fine-tuning dataset as in percentage to the target gNB dataset, for transferring from gNB 20 to gNB 14 in Frankfurt. Similarly, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F5.sf2" title="In Figure 5 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">5(b)</span></a> presents these results for transferring from gNB 17 to gNB 11 in the same city. Moreover, to fully evaluate the performance of transfer learning in practice, in this section we additionally consider the best-in-top-5 transfer accuracy, as introduced in Sec. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S3.SS2" title="III-B Proposed Location-Based Transfer Learning Solution ‣ III Beam Prediction Problem Formulation & Proposed Transfer Learning Solution ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag"><span class="ltx_text">III-B</span></span></a>. This corresponds to the accuracy of having the best BPL within the top-5 predicted BPLs, allowing us to evaluate the possibility of narrowing the search space to the top-5 BPLs predicted by the fine-tuned model. The corresponding top-1 and best-in-top-5 accuracies for training and testing on the target gNB are reported as a baseline. We emphasize that, as shown in Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>, gNB 11 has a smaller coverage area than gNB 14 and consequently a smaller overall dataset size (6150 and 59778 respectively). Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F5.sf1" title="In Figure 5 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">5(a)</span></a> shows that for target gNB 14 the model reaches top-1 transfer accuracy of 73% at 5% fine-tuning and only slightly improves to 80% with larger fine-tuning percentages, suggesting there is no significant benefit from additional fine-tuning data. The corresponding best-in-top-5 curve follows a similar trend, achieving an excellent accuracy of 88% at 5% fine-tuning (vs. 94% baseline accuracy). By contrast, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F5.sf2" title="In Figure 5 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">5(b)</span></a> shows that for target gNB 11 the model significantly improves its top-1 transfer accuracy by increasing the fine-tuning percentages beyond 5%, reaching the corresponding baseline of 71% at 50% fine-tuning dataset size and stabilizing around 75% for higher percentages<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>We note that the transfer model can even outperform the baseline as it leverages the complete reference gNB training data plus target gNB fine-tuning, whereas the baseline model only uses data from the target gNB.</span></span></span>. These results confirm that the low transfer learning accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a> for this gNB is due to insufficient fine-tuning data rather than model saturation, as gNB 11’s dataset is significantly smaller than those of other gNBs (<span class="ltx_text ltx_font_italic" id="S4.SS3.p1.1.1">cf</span>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>). Thus, increasing the fine-tuning percentage for gNBs with smaller coverage areas and datasets should be a means to achieve good transfer learning performance in practice.</p> </div> <figure class="ltx_figure" id="S4.F6"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F6.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="570" id="S4.F6.sf1.g1" src="x13.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf1.2.1.1" style="font-size:90%;">(a)</span> </span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F6.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="582" id="S4.F6.sf2.g1" src="x14.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf2.2.1.1" style="font-size:90%;">(b)</span> </span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F6.sf3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F6.sf3.g1" src="x15.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf3.2.1.1" style="font-size:90%;">(c)</span> </span><span class="ltx_text" id="S4.F6.sf3.3.2" style="font-size:90%;">Top-1.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F6.sf4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="723" id="S4.F6.sf4.g1" src="x16.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.sf4.2.1.1" style="font-size:90%;">(d)</span> </span><span class="ltx_text" id="S4.F6.sf4.3.2" style="font-size:90%;">Best-in-top-5.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F6.2.1.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S4.F6.3.2" style="font-size:90%;">(c) Top-1 and (d) best-in-top-5 transfer learning accuracy in Frankfurt with gNB-specific fine-tuning dataset size, given in (a) as absolute dataset size and in (b) as a percentage of to the total dataset size of the target gNB. </span></figcaption> </figure> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.1">To study this, we now consider the performance of our transfer learning solution with a minimum fine-tuning dataset size of 1000. Namely, we set the fine-tuning dataset size to 1000 for all the datasets where 5% of the total is lower than 1000 and to 5% otherwise (gNB 25 being the exception, as its dataset consists of only 389 data points). Fig <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf1" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(a)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf2" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(b)</span></a> show the resulting size of fine-tuning datasets, as absolute size and as a percentage relative to the total dataset size for each gNB (<span class="ltx_text ltx_font_italic" id="S4.SS3.p2.1.1">cf</span>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>). Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf3" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(c)</span></a> and Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf4" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(d)</span></a> present the resulting top-1 and best-in-top-5 beam prediction transfer learning accuracies for all gNB combinations in Frankfurt. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf3" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(c)</span></a> confirms that the target gNBs suffering from lower accuracy in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F2.sf2" title="In Figure 2 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">2(b)</span></a>, i.e., gNBs 1, 6, 11, 21, 22, and 25, can now better adapt from other models thanks to the additional fine-tuning data. Notably, the very low top-1 baseline accuracy of gNB 25 of 3% is significantly improved by first transferring the model from any other gNB and then fine-tuning it: e.g. transferring from gNB 7 achieves a top-1 accuracy of 73%. This suggests that transferring previously trained models on other gNBs can be extremely helpful for gNBs with limited coverage area, provided a reasonable minimum amount of site-specific fine-tuning data. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf3" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(c)</span></a> also shows that increasing the fine-tuning for target gNBs 1 and 2 results in very high top-1 accuracies of around 90%. This may be due to their area coverage being completely LoS (<span class="ltx_text ltx_font_italic" id="S4.SS3.p2.1.2">cf</span>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F3.sf1" title="In Figure 3 ‣ IV-A Intra-City Transfer Learning Performance ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">3(a)</span></a>), making their optimal BPL patterns geometrically uniform and thus easier to learn and adapt to. Finally, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6.sf4" title="In Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6(d)</span></a> shows that the best BPL is generally included in the top five BPLs predicted with very high reliability. In particular, for every target gNB, there exists a reference gNB ensuring best-in-top-5 transfer accuracy above 75%. Overall, Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.14287v1#S4.F6" title="Figure 6 ‣ IV-C Fine-Tuning Effect ‣ IV Results ‣ Cross-Environment Transfer Learning for Location-Aided Beam Prediction in 5G and Beyond Millimeter-Wave Networks This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038."><span class="ltx_text ltx_ref_tag">6</span></a> demonstrates that our transfer learning approach performs well even in highly site-specific scenarios in densely urbanized areas like Frankfurt, provided there is a sufficient fine-tuning data size which is typically between 5% and 40% of the available dataset.</p> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Conclusions</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We investigated cross-environment transfer learning to enhance BPL prediction efficiency for 5G and beyond mm-wave networks. We proposed training the model on a reference gNB and then transferring it to a target gNB in the same or a different city by fine-tuning the model with a much more limited dataset from the target gNB. Our results show that minimal fine-tuning (5% of the target gNB dataset) can be sufficient to adapt the model to a new environment and achieve an accuracy of up to 80% in predicting the best BPL. Moreover, the high accuracy obtained in inter-city transfer between Frankfurt and Seoul suggests that transfer learning is practically scalable also across different cities with varying urban densities and layouts. Finally, we showed that our method remains effective even in highly site-specific scenarios in densely urbanized areas like Frankfurt, provided a reasonable fine-tuning data size is used. Our ongoing work aims to further optimize transfer learning by exploring different learning algorithms and fine-tuning strategies.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Z. Zhang <em class="ltx_emph ltx_font_italic" id="bib.bib1.1.1">et al.</em>, “6G wireless networks: Vision, requirements, architecture, and key technologies,” <em class="ltx_emph ltx_font_italic" id="bib.bib1.2.2">IEEE Vehicular Technology Magazine</em>, vol. 14, no. 3, pp. 28–41, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> A. M. Nor, S. Halunga, and O. 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