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<h1 class="ltx_title ltx_title_document">The Platonic Representation Hypothesis</h1>
<div class="ltx_authors">
<span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Minyoung Huh
</span></span>
<span class="ltx_author_before">  </span><span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Brian Cheung
</span></span>
<span class="ltx_author_before">  </span><span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Tongzhou Wang
</span></span>
<span class="ltx_author_before">  </span><span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Phillip Isola
</span></span>
</div>

<div class="ltx_abstract">
<h6 class="ltx_title ltx_title_abstract">Abstract</h6>
<p id="id1.id1" class="ltx_p">We argue that representations in AI models, particularly deep networks, are converging. First, we survey many examples of convergence in the literature: over time and across multiple domains, the ways by which different neural networks represent data are becoming more aligned.
Next, we demonstrate convergence across data modalities: as vision models and language models get larger, they measure distance between datapoints in a more and more alike way.
We hypothesize that this convergence is driving toward a shared statistical model of reality, akin to Plato’s concept of an ideal reality. We term such a representation the <span id="id1.id1.1" class="ltx_text ltx_font_italic">platonic representation</span> and discuss several possible selective pressures toward it. Finally, we discuss the implications of these trends, their limitations, and counterexamples to our analysis.</p>
<table id="id2.id2" class="ltx_tabular ltx_align_middle">
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<td id="id2.id2.1.1" class="ltx_td ltx_align_left" style="padding-bottom:3.44444pt;"><span id="id2.id2.1.1.1" class="ltx_text ltx_font_bold">Project Page:</span></td>
<td id="id2.id2.1.2" class="ltx_td ltx_nopad_r ltx_align_right" style="padding-bottom:3.44444pt;"><a target="_blank" href="https://phillipi.github.io/prh/" title="" class="ltx_ref ltx_href ltx_font_typewriter">phillipi.github.io/prh</a></td>
</tr>
<tr id="id2.id2.2" class="ltx_tr">
<td id="id2.id2.2.1" class="ltx_td ltx_align_left"><span id="id2.id2.2.1.1" class="ltx_text ltx_font_bold">Code:</span></td>
<td id="id2.id2.2.2" class="ltx_td ltx_nopad_r ltx_align_right">
<a target="_blank" href="https://github.com/minyoungg/platonic-rep/" title="" class="ltx_ref ltx_href ltx_font_typewriter">github.com/minyoungg/platonic-rep</a>
</td>
</tr>
</table>
</div>
<div class="ltx_keywords">Machine Learning, Representation, Artificial Intelligence, Multimodality
</div>
<div id="p2" class="ltx_para ltx_noindent">
<br class="ltx_break">
</div>
<section id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2>

<div id="S1.p1" class="ltx_para ltx_noindent">
<p id="S1.p1.1" class="ltx_p">AI systems are rapidly evolving into highly multifunctional entities. For example, whereas in the past we had special-purpose solutions for different language processing tasks (<span id="S1.p1.1.1" class="ltx_text ltx_font_italic">e.g.</span>, sentiment analysis, parsing, dialogue), modern large language models (LLMs) are competent at all these tasks using a single set of weights <cite class="ltx_cite ltx_citemacro_citep">(Srivastava et al., <a href="#bib.bib112" title="" class="ltx_ref">2022</a>)</cite>. Unified systems are also being built across data modalities: instead of using a different architecture for processing images versus text, recent models, such as GPT4-V <cite class="ltx_cite ltx_citemacro_citep">(Achiam et al., <a href="#bib.bib2" title="" class="ltx_ref">2023</a>)</cite>, Gemini <cite class="ltx_cite ltx_citemacro_citep">(Anil et al., <a href="#bib.bib4" title="" class="ltx_ref">2023</a>)</cite>, and LLaVA <cite class="ltx_cite ltx_citemacro_citep">(Liu et al., <a href="#bib.bib70" title="" class="ltx_ref">2023</a>)</cite>, handle both modalities with a combined architecture.
More and more systems are built off of general-purpose pretrained backbones, sometimes called foundation models <cite class="ltx_cite ltx_citemacro_citep">(Bommasani et al., <a href="#bib.bib15" title="" class="ltx_ref">2021</a>)</cite>, that support a large range of tasks, including robotics <cite class="ltx_cite ltx_citemacro_citep">(Driess et al., <a href="#bib.bib30" title="" class="ltx_ref">2023</a>; Brohan et al., <a href="#bib.bib16" title="" class="ltx_ref">2023</a>)</cite>, bioinformatics <cite class="ltx_cite ltx_citemacro_citep">(Ma et al., <a href="#bib.bib77" title="" class="ltx_ref">2024</a>)</cite>, and healthcare <cite class="ltx_cite ltx_citemacro_citep">(Steinberg et al., <a href="#bib.bib113" title="" class="ltx_ref">2021</a>)</cite>.
In short, AI systems are becoming increasingly homogeneous in both their architectures and their capabilities.</p>
</div>
<figure id="S1.F1" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><svg id="S1.F1.pic1" class="ltx_picture ltx_centering ltx_figure_panel" height="102.91" overflow="visible" version="1.1" width="600"><g transform="translate(0,102.91) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#4D4D4D" fill-opacity="1.0"><path d="M 0 5.91 L 0 97.01 C 0 100.27 2.64 102.91 5.91 102.91 L 594.09 102.91 C 597.36 102.91 600 100.27 600 97.01 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#FAFAFA" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 97.01 C 1.97 99.18 3.73 100.94 5.91 100.94 L 594.09 100.94 C 596.27 100.94 598.03 99.18 598.03 97.01 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 19.24 11.36)"><foreignObject width="561.52" height="78.72" transform="matrix(1 0 0 -1 0 16.6)" overflow="visible" color="#000000">
<span id="S1.F1.pic1.1.1.1.1.1" class="ltx_inline-block ltx_minipage ltx_align_bottom" style="width:405.8pt;">
<span id="S1.F1.pic1.1.1.1.1.1.1" class="ltx_p"><span id="S1.F1.pic1.1.1.1.1.1.1.1" class="ltx_text ltx_font_bold">The Platonic Representation Hypothesis
<br class="ltx_break"></span>
<span id="S1.F1.pic1.1.1.1.1.1.1.2" class="ltx_text ltx_inline-block" style="width:433.6pt;"><span class="ltx_rule" style="width:433.6pt;height:0.4pt;background:black;display:inline-block;"> </span></span></span>
<span id="S1.F1.pic1.1.1.1.1.1.2" class="ltx_p">Neural networks, trained with different objectives on different data and modalities, are converging to a shared statistical model of reality in their representation spaces.</span>
</span></foreignObject></g></g></svg></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2405.07987/assets/x1.png" id="S1.F1.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" width="392" height="353" alt="Refer to caption"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span><span id="S1.F1.8.4" class="ltx_text ltx_font_bold" style="font-size:90%;">The Platonic Representation Hypothesis:<span id="S1.F1.8.4.4" class="ltx_text ltx_font_medium"> Images (<math id="S1.F1.5.1.1.m1.1" class="ltx_Math" alttext="X" display="inline"><semantics id="S1.F1.5.1.1.m1.1b"><mi id="S1.F1.5.1.1.m1.1.1" xref="S1.F1.5.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.F1.5.1.1.m1.1c"><ci id="S1.F1.5.1.1.m1.1.1.cmml" xref="S1.F1.5.1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.5.1.1.m1.1d">X</annotation></semantics></math>) and text (<math id="S1.F1.6.2.2.m2.1" class="ltx_Math" alttext="Y" display="inline"><semantics id="S1.F1.6.2.2.m2.1b"><mi id="S1.F1.6.2.2.m2.1.1" xref="S1.F1.6.2.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S1.F1.6.2.2.m2.1c"><ci id="S1.F1.6.2.2.m2.1.1.cmml" xref="S1.F1.6.2.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.6.2.2.m2.1d">Y</annotation></semantics></math>) are projections of a common underlying reality (<math id="S1.F1.7.3.3.m3.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.F1.7.3.3.m3.1b"><mi id="S1.F1.7.3.3.m3.1.1" xref="S1.F1.7.3.3.m3.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.F1.7.3.3.m3.1c"><ci id="S1.F1.7.3.3.m3.1.1.cmml" xref="S1.F1.7.3.3.m3.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.7.3.3.m3.1d">Z</annotation></semantics></math>). We conjecture that representation learning algorithms will converge on a shared representation of <math id="S1.F1.8.4.4.m4.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.F1.8.4.4.m4.1b"><mi id="S1.F1.8.4.4.m4.1.1" xref="S1.F1.8.4.4.m4.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.F1.8.4.4.m4.1c"><ci id="S1.F1.8.4.4.m4.1.1.cmml" xref="S1.F1.8.4.4.m4.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.F1.8.4.4.m4.1d">Z</annotation></semantics></math>, and scaling model size, as well as data and task diversity, drives this convergence. </span></span></figcaption>
</figure>
<div id="S1.p2" class="ltx_para ltx_noindent">
<p id="S1.p2.1" class="ltx_p">This paper explores one aspect of this trend: representational convergence. We argue that there is a growing similarity in how datapoints are represented in different neural network models. This similarity spans across different model architectures, training objectives, and even data modalities.</p>
</div>
<div id="S1.p3" class="ltx_para ltx_noindent">
<p id="S1.p3.1" class="ltx_p">What has led to this convergence? Will it continue? And ultimately, where does it end?</p>
</div>
<div id="S1.p4" class="ltx_para ltx_noindent">
<p id="S1.p4.7" class="ltx_p">Our central hypothesis, stated above in <a href="#S1.F1" title="In 1 Introduction ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>, is that there is indeed an endpoint to this convergence and a principle that drives it: different models are all trying to arrive at a <span id="S1.p4.7.1" class="ltx_text ltx_font_italic">representation of reality</span>, meaning a representation of the joint distribution over events in the world that generate the data we observe. Figure <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> conveys this hypothesis: there exists a real world (labeled <math id="S1.p4.1.m1.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">Z</annotation></semantics></math>), which we measure with various sensors, such as the camera shown to the left (<math id="S1.p4.2.m2.1" class="ltx_Math" alttext="X" display="inline"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">X</annotation></semantics></math>). Other <span id="S1.p4.7.2" class="ltx_text ltx_font_italic">projections</span> of these measurements, such as the textual description shown, can be produced from the first set of measurements or mediated by some other set of measurements, <span id="S1.p4.7.3" class="ltx_text ltx_font_italic">e.g.</span>, touch or other camera views (dotted arrow from <math id="S1.p4.3.m3.1" class="ltx_Math" alttext="X" display="inline"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">X</annotation></semantics></math> to <math id="S1.p4.4.m4.1" class="ltx_Math" alttext="Y" display="inline"><semantics id="S1.p4.4.m4.1a"><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">Y</annotation></semantics></math>)<span id="footnote1" class="ltx_note ltx_role_footnote"><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>Touch could convey the shapes in this example but not the colors. This is an important limitation to our hypothesis that we discuss at several points in the paper: different sensors and views might capture different information, which may limit their potential to converge to identical representations.
</span></span></span>. Representation learning algorithms find vector embeddings that statistically model the various measurements and projections. The resulting vector embeddings are all derived from the underlying reality in <math id="S1.p4.5.m5.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.p4.5.m5.1a"><mi id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><ci id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">Z</annotation></semantics></math> and thereby become aligned. As models are trained on more data and for more tasks, they require representations that capture more and more information about <math id="S1.p4.6.m6.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.p4.6.m6.1a"><mi id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">Z</annotation></semantics></math>, and hence alignment toward <math id="S1.p4.7.m7.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S1.p4.7.m7.1a"><mi id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S1.p4.7.m7.1b"><ci id="S1.p4.7.m7.1.1.cmml" xref="S1.p4.7.m7.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m7.1c">Z</annotation></semantics></math> increases toward a convergent point as a function of scale.</p>
</div>
<div id="S1.p5" class="ltx_para ltx_noindent">
<p id="S1.p5.1" class="ltx_p">We call this converged hypothetical representation the “platonic representation” in reference to Plato’s Allegory of the Cave <cite class="ltx_cite ltx_citemacro_citep">(Plato, <a href="#bib.bib95" title="" class="ltx_ref">c. 375 BC</a>)</cite>, and his idea of an ideal reality that underlies our sensations. The training data for our algorithms are shadows on the cave wall, yet, we hypothesize, models are recovering ever better representations of the actual world outside the cave. This idea is not unique to Plato; our hypothesis is also related to the notion of “convergent realism” <cite class="ltx_cite ltx_citemacro_citep">(Newton-Smith, <a href="#bib.bib86" title="" class="ltx_ref">1981</a>; Putnam, <a href="#bib.bib96" title="" class="ltx_ref">1982</a>; Doppelt, <a href="#bib.bib27" title="" class="ltx_ref">2007</a>; Hardin &amp; Rosenberg, <a href="#bib.bib43" title="" class="ltx_ref">1982</a>)</cite> in the philosophy of science (<span id="S1.p5.1.1" class="ltx_text ltx_font_italic">i.e.</span>, that science is converging on truth), and to many arguments that have been put forth in the representation learning literature (<span id="S1.p5.1.2" class="ltx_text ltx_font_italic">e.g.</span>, <cite class="ltx_cite ltx_citemacro_citet">Tian et al. (<a href="#bib.bib117" title="" class="ltx_ref">2020a</a>); Zimmermann et al. (<a href="#bib.bib139" title="" class="ltx_ref">2021</a>); Richens &amp; Everitt (<a href="#bib.bib101" title="" class="ltx_ref">2024</a>); Cao &amp; Yamins (<a href="#bib.bib17" title="" class="ltx_ref">2024</a>)</cite>).</p>
</div>
<div id="S1.p6" class="ltx_para ltx_noindent">
<p id="S1.p6.1" class="ltx_p">Also closely related to our hypothesis is the “Anna Karenina scenario” described by <cite class="ltx_cite ltx_citemacro_citet">Bansal et al. (<a href="#bib.bib9" title="" class="ltx_ref">2021</a>)</cite>, referring to the possibility that all well-performing neural nets represent the world in the same way. We discuss the evidence they give for this possibility in Section <a href="#S2" title="2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>.<span id="footnote2" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>Borrowed from <cite class="ltx_cite ltx_citemacro_citet">Tolstoy (<a href="#bib.bib119" title="" class="ltx_ref">1877</a>)</cite>, similar analogies have been made in other domains, such as the “Anna Karenina principle” popularized by <cite class="ltx_cite ltx_citemacro_citet">Diamond (<a href="#bib.bib25" title="" class="ltx_ref">1998</a>)</cite> to explain animal domestication.</span></span></span> The platonic representation hypothesis refers to the situation where we are in an Anna Karenina scenario <span id="S1.p6.1.1" class="ltx_text ltx_font_italic">and</span> the “happy representation” that is converged upon is one that reflects a statistical model of the underlying reality. We discuss the potential nature of this statistical model in more detail in Section <a href="#S4" title="4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a>.</p>
</div>
</section>
<section id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">2 </span>Representations are converging</h2>

<section id="S2.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Preliminaries</h4>

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<p id="S2.SS0.SSS0.Px1.p1.1" class="ltx_p">We restrict our attention to representations that are <span id="S2.SS0.SSS0.Px1.p1.1.1" class="ltx_text ltx_font_italic">vector embeddings</span>. We characterize such a representation by the similarity structure it induces, referred to as its kernel. Kernels are commonly used to assess representations <cite class="ltx_cite ltx_citemacro_citep">(Kornblith et al., <a href="#bib.bib61" title="" class="ltx_ref">2019</a>; Klabunde et al., <a href="#bib.bib59" title="" class="ltx_ref">2023</a>)</cite>; this can be justified by the fact that they capture the relative structures among data samples, which are also the learning signal for many machine learning algorithms  <cite class="ltx_cite ltx_citemacro_citep">(Aronszajn, <a href="#bib.bib5" title="" class="ltx_ref">1950</a>; Smola &amp; Schölkopf, <a href="#bib.bib108" title="" class="ltx_ref">1998</a>)</cite>. Following prior literature, we define <span id="S2.SS0.SSS0.Px1.p1.1.2" class="ltx_text ltx_font_italic">representational alignment</span> as a measure of the similarity of the similarity structures induced by two representations, <span id="S2.SS0.SSS0.Px1.p1.1.3" class="ltx_text ltx_font_italic">i.e.</span>, a similarity metric over kernels.
We give the mathematical definition of these concepts below:</p>
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<p id="S2.I1.i1.p1.2" class="ltx_p">A <span id="S2.I1.i1.p1.2.1" class="ltx_text ltx_font_bold">representation</span> is a function <math id="S2.I1.i1.p1.1.m1.1" class="ltx_Math" alttext="f\colon\mathcal{X}\rightarrow\mathbb{R}^{n}" display="inline"><semantics id="S2.I1.i1.p1.1.m1.1a"><mrow id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml"><mi id="S2.I1.i1.p1.1.m1.1.1.2" xref="S2.I1.i1.p1.1.m1.1.1.2.cmml">f</mi><mo lspace="0.278em" rspace="0.278em" id="S2.I1.i1.p1.1.m1.1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.I1.i1.p1.1.m1.1.1.3" xref="S2.I1.i1.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i1.p1.1.m1.1.1.3.2" xref="S2.I1.i1.p1.1.m1.1.1.3.2.cmml">𝒳</mi><mo stretchy="false" id="S2.I1.i1.p1.1.m1.1.1.3.1" xref="S2.I1.i1.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.I1.i1.p1.1.m1.1.1.3.3" xref="S2.I1.i1.p1.1.m1.1.1.3.3.cmml"><mi mathvariant="normal" id="S2.I1.i1.p1.1.m1.1.1.3.3.2" xref="S2.I1.i1.p1.1.m1.1.1.3.3.2.cmml">ℝ</mi><mi id="S2.I1.i1.p1.1.m1.1.1.3.3.3" xref="S2.I1.i1.p1.1.m1.1.1.3.3.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><apply id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1"><ci id="S2.I1.i1.p1.1.m1.1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1.1">:</ci><ci id="S2.I1.i1.p1.1.m1.1.1.2.cmml" xref="S2.I1.i1.p1.1.m1.1.1.2">𝑓</ci><apply id="S2.I1.i1.p1.1.m1.1.1.3.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3"><ci id="S2.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.1">→</ci><ci id="S2.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.2">𝒳</ci><apply id="S2.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.I1.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.I1.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.3.2">ℝ</ci><ci id="S2.I1.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">f\colon\mathcal{X}\rightarrow\mathbb{R}^{n}</annotation></semantics></math> that assigns a feature vector to each input in some data domain <math id="S2.I1.i1.p1.2.m2.1" class="ltx_Math" alttext="\mathcal{X}" display="inline"><semantics id="S2.I1.i1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><ci id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">\mathcal{X}</annotation></semantics></math>.</p>
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<p id="S2.I1.i2.p1.5" class="ltx_p">A <span id="S2.I1.i2.p1.5.1" class="ltx_text ltx_font_bold">kernel</span>, <math id="S2.I1.i2.p1.1.m1.1" class="ltx_Math" alttext="K\colon\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}" display="inline"><semantics id="S2.I1.i2.p1.1.m1.1a"><mrow id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml"><mi id="S2.I1.i2.p1.1.m1.1.1.2" xref="S2.I1.i2.p1.1.m1.1.1.2.cmml">K</mi><mo lspace="0.278em" rspace="0.278em" id="S2.I1.i2.p1.1.m1.1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.I1.i2.p1.1.m1.1.1.3" xref="S2.I1.i2.p1.1.m1.1.1.3.cmml"><mrow id="S2.I1.i2.p1.1.m1.1.1.3.2" xref="S2.I1.i2.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.1.m1.1.1.3.2.2" xref="S2.I1.i2.p1.1.m1.1.1.3.2.2.cmml">𝒳</mi><mo lspace="0.222em" rspace="0.222em" id="S2.I1.i2.p1.1.m1.1.1.3.2.1" xref="S2.I1.i2.p1.1.m1.1.1.3.2.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.1.m1.1.1.3.2.3" xref="S2.I1.i2.p1.1.m1.1.1.3.2.3.cmml">𝒳</mi></mrow><mo stretchy="false" id="S2.I1.i2.p1.1.m1.1.1.3.1" xref="S2.I1.i2.p1.1.m1.1.1.3.1.cmml">→</mo><mi mathvariant="normal" id="S2.I1.i2.p1.1.m1.1.1.3.3" xref="S2.I1.i2.p1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><apply id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1"><ci id="S2.I1.i2.p1.1.m1.1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1.1">:</ci><ci id="S2.I1.i2.p1.1.m1.1.1.2.cmml" xref="S2.I1.i2.p1.1.m1.1.1.2">𝐾</ci><apply id="S2.I1.i2.p1.1.m1.1.1.3.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3"><ci id="S2.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.1">→</ci><apply id="S2.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.2"><times id="S2.I1.i2.p1.1.m1.1.1.3.2.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.2.1"></times><ci id="S2.I1.i2.p1.1.m1.1.1.3.2.2.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.2.2">𝒳</ci><ci id="S2.I1.i2.p1.1.m1.1.1.3.2.3.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.2.3">𝒳</ci></apply><ci id="S2.I1.i2.p1.1.m1.1.1.3.3.cmml" xref="S2.I1.i2.p1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">K\colon\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}</annotation></semantics></math>, characterizes how a representation measures distance/similarity between datapoints. <math id="S2.I1.i2.p1.2.m2.4" class="ltx_Math" alttext="K(x_{i},x_{j})=\langle f(x_{i}),f(x_{j})\rangle" display="inline"><semantics id="S2.I1.i2.p1.2.m2.4a"><mrow id="S2.I1.i2.p1.2.m2.4.4" xref="S2.I1.i2.p1.2.m2.4.4.cmml"><mrow id="S2.I1.i2.p1.2.m2.2.2.2" xref="S2.I1.i2.p1.2.m2.2.2.2.cmml"><mi id="S2.I1.i2.p1.2.m2.2.2.2.4" xref="S2.I1.i2.p1.2.m2.2.2.2.4.cmml">K</mi><mo lspace="0em" rspace="0em" id="S2.I1.i2.p1.2.m2.2.2.2.3" xref="S2.I1.i2.p1.2.m2.2.2.2.3.cmml">​</mo><mrow id="S2.I1.i2.p1.2.m2.2.2.2.2.2" xref="S2.I1.i2.p1.2.m2.2.2.2.2.3.cmml"><mo stretchy="false" 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id="S2.I1.i2.p1.2.m2.4.4.4.2.3" xref="S2.I1.i2.p1.2.m2.4.4.4.3.cmml">⟨</mo><mrow id="S2.I1.i2.p1.2.m2.3.3.3.1.1" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.cmml"><mi id="S2.I1.i2.p1.2.m2.3.3.3.1.1.3" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.3.cmml">f</mi><mo lspace="0em" rspace="0em" id="S2.I1.i2.p1.2.m2.3.3.3.1.1.2" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.2.cmml">​</mo><mrow id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.2" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.cmml">(</mo><msub id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.cmml"><mi id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.2" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.2.cmml">x</mi><mi id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.3" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.3" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I1.i2.p1.2.m2.4.4.4.2.4" xref="S2.I1.i2.p1.2.m2.4.4.4.3.cmml">,</mo><mrow id="S2.I1.i2.p1.2.m2.4.4.4.2.2" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.cmml"><mi id="S2.I1.i2.p1.2.m2.4.4.4.2.2.3" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.3.cmml">f</mi><mo lspace="0em" rspace="0em" id="S2.I1.i2.p1.2.m2.4.4.4.2.2.2" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.2.cmml">​</mo><mrow id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.cmml"><mo stretchy="false" id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.2" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.cmml">(</mo><msub id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.cmml"><mi id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.2" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.2.cmml">x</mi><mi id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.3" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.3" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S2.I1.i2.p1.2.m2.4.4.4.2.5" xref="S2.I1.i2.p1.2.m2.4.4.4.3.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.4b"><apply id="S2.I1.i2.p1.2.m2.4.4.cmml" xref="S2.I1.i2.p1.2.m2.4.4"><eq id="S2.I1.i2.p1.2.m2.4.4.5.cmml" xref="S2.I1.i2.p1.2.m2.4.4.5"></eq><apply id="S2.I1.i2.p1.2.m2.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2"><times id="S2.I1.i2.p1.2.m2.2.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.3"></times><ci id="S2.I1.i2.p1.2.m2.2.2.2.4.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.4">𝐾</ci><interval closure="open" id="S2.I1.i2.p1.2.m2.2.2.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.2.2"><apply id="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.2">𝑥</ci><ci id="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.I1.i2.p1.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.2">𝑥</ci><ci id="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><list id="S2.I1.i2.p1.2.m2.4.4.4.3.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2"><apply id="S2.I1.i2.p1.2.m2.3.3.3.1.1.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1"><times id="S2.I1.i2.p1.2.m2.3.3.3.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.2"></times><ci id="S2.I1.i2.p1.2.m2.3.3.3.1.1.3.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.3">𝑓</ci><apply id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.2">𝑥</ci><ci id="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.3.cmml" xref="S2.I1.i2.p1.2.m2.3.3.3.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S2.I1.i2.p1.2.m2.4.4.4.2.2.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2"><times id="S2.I1.i2.p1.2.m2.4.4.4.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.2"></times><ci id="S2.I1.i2.p1.2.m2.4.4.4.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.3">𝑓</ci><apply id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.2">𝑥</ci><ci id="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.3.cmml" xref="S2.I1.i2.p1.2.m2.4.4.4.2.2.1.1.1.3">𝑗</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.4c">K(x_{i},x_{j})=\langle f(x_{i}),f(x_{j})\rangle</annotation></semantics></math>, where <math id="S2.I1.i2.p1.3.m3.2" class="ltx_Math" alttext="\langle{{}\cdot{}},{{}\cdot{}}\rangle" display="inline"><semantics id="S2.I1.i2.p1.3.m3.2a"><mrow id="S2.I1.i2.p1.3.m3.2.3.2" xref="S2.I1.i2.p1.3.m3.2.3.1.cmml"><mo stretchy="false" id="S2.I1.i2.p1.3.m3.2.3.2.1" xref="S2.I1.i2.p1.3.m3.2.3.1.cmml">⟨</mo><mo lspace="0em" rspace="0em" id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml">⋅</mo><mo rspace="0em" id="S2.I1.i2.p1.3.m3.2.3.2.2" xref="S2.I1.i2.p1.3.m3.2.3.1.cmml">,</mo><mo lspace="0em" rspace="0em" id="S2.I1.i2.p1.3.m3.2.2" xref="S2.I1.i2.p1.3.m3.2.2.cmml">⋅</mo><mo stretchy="false" id="S2.I1.i2.p1.3.m3.2.3.2.3" xref="S2.I1.i2.p1.3.m3.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.2b"><list id="S2.I1.i2.p1.3.m3.2.3.1.cmml" xref="S2.I1.i2.p1.3.m3.2.3.2"><ci id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1">⋅</ci><ci id="S2.I1.i2.p1.3.m3.2.2.cmml" xref="S2.I1.i2.p1.3.m3.2.2">⋅</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.2c">\langle{{}\cdot{}},{{}\cdot{}}\rangle</annotation></semantics></math> denotes inner product, <math id="S2.I1.i2.p1.4.m4.2" class="ltx_Math" alttext="x_{i},x_{j}\in\mathcal{X}" display="inline"><semantics id="S2.I1.i2.p1.4.m4.2a"><mrow id="S2.I1.i2.p1.4.m4.2.2" xref="S2.I1.i2.p1.4.m4.2.2.cmml"><mrow id="S2.I1.i2.p1.4.m4.2.2.2.2" xref="S2.I1.i2.p1.4.m4.2.2.2.3.cmml"><msub id="S2.I1.i2.p1.4.m4.1.1.1.1.1" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1.cmml"><mi id="S2.I1.i2.p1.4.m4.1.1.1.1.1.2" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1.2.cmml">x</mi><mi id="S2.I1.i2.p1.4.m4.1.1.1.1.1.3" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.I1.i2.p1.4.m4.2.2.2.2.3" xref="S2.I1.i2.p1.4.m4.2.2.2.3.cmml">,</mo><msub id="S2.I1.i2.p1.4.m4.2.2.2.2.2" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2.cmml"><mi id="S2.I1.i2.p1.4.m4.2.2.2.2.2.2" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2.2.cmml">x</mi><mi id="S2.I1.i2.p1.4.m4.2.2.2.2.2.3" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2.3.cmml">j</mi></msub></mrow><mo id="S2.I1.i2.p1.4.m4.2.2.3" xref="S2.I1.i2.p1.4.m4.2.2.3.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.4.m4.2.2.4" xref="S2.I1.i2.p1.4.m4.2.2.4.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.4.m4.2b"><apply id="S2.I1.i2.p1.4.m4.2.2.cmml" xref="S2.I1.i2.p1.4.m4.2.2"><in id="S2.I1.i2.p1.4.m4.2.2.3.cmml" xref="S2.I1.i2.p1.4.m4.2.2.3"></in><list id="S2.I1.i2.p1.4.m4.2.2.2.3.cmml" xref="S2.I1.i2.p1.4.m4.2.2.2.2"><apply id="S2.I1.i2.p1.4.m4.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1.2">𝑥</ci><ci id="S2.I1.i2.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.I1.i2.p1.4.m4.1.1.1.1.1.3">𝑖</ci></apply><apply id="S2.I1.i2.p1.4.m4.2.2.2.2.2.cmml" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.I1.i2.p1.4.m4.2.2.2.2.2.1.cmml" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2">subscript</csymbol><ci id="S2.I1.i2.p1.4.m4.2.2.2.2.2.2.cmml" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2.2">𝑥</ci><ci id="S2.I1.i2.p1.4.m4.2.2.2.2.2.3.cmml" xref="S2.I1.i2.p1.4.m4.2.2.2.2.2.3">𝑗</ci></apply></list><ci id="S2.I1.i2.p1.4.m4.2.2.4.cmml" xref="S2.I1.i2.p1.4.m4.2.2.4">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.4.m4.2c">x_{i},x_{j}\in\mathcal{X}</annotation></semantics></math> and <math id="S2.I1.i2.p1.5.m5.1" class="ltx_Math" alttext="K\in\mathcal{K}" display="inline"><semantics id="S2.I1.i2.p1.5.m5.1a"><mrow id="S2.I1.i2.p1.5.m5.1.1" xref="S2.I1.i2.p1.5.m5.1.1.cmml"><mi id="S2.I1.i2.p1.5.m5.1.1.2" xref="S2.I1.i2.p1.5.m5.1.1.2.cmml">K</mi><mo id="S2.I1.i2.p1.5.m5.1.1.1" xref="S2.I1.i2.p1.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.5.m5.1.1.3" xref="S2.I1.i2.p1.5.m5.1.1.3.cmml">𝒦</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.5.m5.1b"><apply id="S2.I1.i2.p1.5.m5.1.1.cmml" xref="S2.I1.i2.p1.5.m5.1.1"><in id="S2.I1.i2.p1.5.m5.1.1.1.cmml" xref="S2.I1.i2.p1.5.m5.1.1.1"></in><ci id="S2.I1.i2.p1.5.m5.1.1.2.cmml" xref="S2.I1.i2.p1.5.m5.1.1.2">𝐾</ci><ci id="S2.I1.i2.p1.5.m5.1.1.3.cmml" xref="S2.I1.i2.p1.5.m5.1.1.3">𝒦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.5.m5.1c">K\in\mathcal{K}</annotation></semantics></math>.</p>
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<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S2.I1.i3.p1" class="ltx_para ltx_noindent">
<p id="S2.I1.i3.p1.1" class="ltx_p">A <span id="S2.I1.i3.p1.1.1" class="ltx_text ltx_font_bold">kernel-alignment metric</span>, <math id="S2.I1.i3.p1.1.m1.1" class="ltx_Math" alttext="m\colon\mathcal{K}\times\mathcal{K}\rightarrow\mathbb{R}" display="inline"><semantics id="S2.I1.i3.p1.1.m1.1a"><mrow id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml"><mi id="S2.I1.i3.p1.1.m1.1.1.2" xref="S2.I1.i3.p1.1.m1.1.1.2.cmml">m</mi><mo lspace="0.278em" rspace="0.278em" id="S2.I1.i3.p1.1.m1.1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.I1.i3.p1.1.m1.1.1.3" xref="S2.I1.i3.p1.1.m1.1.1.3.cmml"><mrow id="S2.I1.i3.p1.1.m1.1.1.3.2" xref="S2.I1.i3.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.1.m1.1.1.3.2.2" xref="S2.I1.i3.p1.1.m1.1.1.3.2.2.cmml">𝒦</mi><mo lspace="0.222em" rspace="0.222em" id="S2.I1.i3.p1.1.m1.1.1.3.2.1" xref="S2.I1.i3.p1.1.m1.1.1.3.2.1.cmml">×</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.1.m1.1.1.3.2.3" xref="S2.I1.i3.p1.1.m1.1.1.3.2.3.cmml">𝒦</mi></mrow><mo stretchy="false" id="S2.I1.i3.p1.1.m1.1.1.3.1" xref="S2.I1.i3.p1.1.m1.1.1.3.1.cmml">→</mo><mi mathvariant="normal" id="S2.I1.i3.p1.1.m1.1.1.3.3" xref="S2.I1.i3.p1.1.m1.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><apply id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1"><ci id="S2.I1.i3.p1.1.m1.1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1.1">:</ci><ci id="S2.I1.i3.p1.1.m1.1.1.2.cmml" xref="S2.I1.i3.p1.1.m1.1.1.2">𝑚</ci><apply id="S2.I1.i3.p1.1.m1.1.1.3.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3"><ci id="S2.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.1">→</ci><apply id="S2.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.2"><times id="S2.I1.i3.p1.1.m1.1.1.3.2.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.2.1"></times><ci id="S2.I1.i3.p1.1.m1.1.1.3.2.2.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.2.2">𝒦</ci><ci id="S2.I1.i3.p1.1.m1.1.1.3.2.3.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.2.3">𝒦</ci></apply><ci id="S2.I1.i3.p1.1.m1.1.1.3.3.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">m\colon\mathcal{K}\times\mathcal{K}\rightarrow\mathbb{R}</annotation></semantics></math>, measures the similarity between two kernels, <span id="S2.I1.i3.p1.1.2" class="ltx_text ltx_font_italic">i.e.</span>, how similar is the distance measure induced by one representation to the distance measure induced by another. Examples include Centered Kernel Distance (CKA) <cite class="ltx_cite ltx_citemacro_citep">(Kornblith et al., <a href="#bib.bib61" title="" class="ltx_ref">2019</a>)</cite>, SVCCA <cite class="ltx_cite ltx_citemacro_citep">(Raghu et al., <a href="#bib.bib100" title="" class="ltx_ref">2017</a>)</cite>, and nearest-neighbor metrics <cite class="ltx_cite ltx_citemacro_citep">(Klabunde et al., <a href="#bib.bib59" title="" class="ltx_ref">2023</a>)</cite>.</p>
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</div>
<div id="S2.SS0.SSS0.Px1.p2" class="ltx_para ltx_noindent">
<p id="S2.SS0.SSS0.Px1.p2.4" class="ltx_p">In our experiments, we use a <em id="S2.SS0.SSS0.Px1.p2.4.1" class="ltx_emph ltx_font_italic">mutual nearest-neighbor metric</em> that measures the mean intersection of the <math id="S2.SS0.SSS0.Px1.p2.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.SS0.SSS0.Px1.p2.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.1.m1.1c">k</annotation></semantics></math>-nearest neighbor sets induced by two kernels, <math id="S2.SS0.SSS0.Px1.p2.2.m2.1" class="ltx_Math" alttext="K_{1}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p2.2.m2.1a"><msub id="S2.SS0.SSS0.Px1.p2.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2.cmml">K</mi><mn id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.3" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2">𝐾</ci><cn type="integer" id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.2.m2.1c">K_{1}</annotation></semantics></math> and <math id="S2.SS0.SSS0.Px1.p2.3.m3.1" class="ltx_Math" alttext="K_{2}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p2.3.m3.1a"><msub id="S2.SS0.SSS0.Px1.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml">K</mi><mn id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.3" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2">𝐾</ci><cn type="integer" id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.3.m3.1c">K_{2}</annotation></semantics></math>, normalized by <math id="S2.SS0.SSS0.Px1.p2.4.m4.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.SS0.SSS0.Px1.p2.4.m4.1a"><mi id="S2.SS0.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.4.m4.1c">k</annotation></semantics></math>.
This metric is a variant of those proposed in <cite class="ltx_cite ltx_citemacro_citet">Klabunde et al. (<a href="#bib.bib59" title="" class="ltx_ref">2023</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Oron et al. (<a href="#bib.bib93" title="" class="ltx_ref">2017</a>)</cite>.
See Appendix <a href="#A1" title="Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">A</span></a> for the exact definition and Appendix <a href="#A2" title="Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B</span></a> for comparisons with alternative alignment metrics.</p>
</div>
<div id="S2.SS0.SSS0.Px1.p3" class="ltx_para ltx_noindent">
<p id="S2.SS0.SSS0.Px1.p3.1" class="ltx_p">Next, we explore several ways in which representations are converging. First, we argue that different neural networks are converging to aligned representations. Then, we show that this continues to hold across modalities, where image embeddings in vision models align with text embeddings in language models.</p>
</div>
<figure id="S2.F2.6" class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_top" style="width:151.8pt;">
<p id="S2.F2.2.2" class="ltx_p ltx_figure_panel ltx_minipage ltx_align_top" style="width:281.9pt;"><span id="S2.F2.1.1.1" class="ltx_text" style="position:relative; bottom:0.0pt;"><img src="/html/2405.07987/assets/x2.png" id="S2.F2.1.1.1.g1" class="ltx_graphics ltx_img_square" width="155" height="171" alt="Refer to caption"></span>

<span id="S2.F2.2.2.2" class="ltx_text" style="position:relative; bottom:0.0pt;"><img src="/html/2405.07987/assets/x3.png" id="S2.F2.2.2.2.g1" class="ltx_graphics ltx_img_square" width="171" height="156" alt="Refer to caption"></span></p>
<br class="ltx_break ltx_break">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span><span id="S2.F2.6.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">VISION models converge as COMPETENCE increases:<span id="S2.F2.5.3.1.1" class="ltx_text ltx_font_medium"> We measure alignment among <math id="S2.F2.5.3.1.1.m1.1" class="ltx_Math" alttext="78" display="inline"><semantics id="S2.F2.5.3.1.1.m1.1b"><mn id="S2.F2.5.3.1.1.m1.1.1" xref="S2.F2.5.3.1.1.m1.1.1.cmml">78</mn><annotation-xml encoding="MathML-Content" id="S2.F2.5.3.1.1.m1.1c"><cn type="integer" id="S2.F2.5.3.1.1.m1.1.1.cmml" xref="S2.F2.5.3.1.1.m1.1.1">78</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.5.3.1.1.m1.1d">78</annotation></semantics></math> models using mutual nearest-neighbors on Places-365 <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a href="#bib.bib137" title="" class="ltx_ref">2017</a>)</cite>, and evaluate their performance on downstream tasks from the Visual Task Adaptation Benchmark (VTAB; <cite class="ltx_cite ltx_citemacro_citet">Zhai et al. (<a href="#bib.bib135" title="" class="ltx_ref">2019</a>)</cite>). </span>LEFT:<span id="S2.F2.6.4.2.3" class="ltx_text ltx_font_medium"> Models that solve more VTAB tasks tend to be more aligned with each other. Error bars show standard error. </span>RIGHT:<span id="S2.F2.6.4.2.2" class="ltx_text ltx_font_medium"> We use UMAP to embed <em id="S2.F2.6.4.2.2.1" class="ltx_emph ltx_font_italic">models</em> into a 2D space, based on <math id="S2.F2.6.4.2.2.m1.2" class="ltx_Math" alttext="\mathsf{distance}\triangleq-\log(\mathsf{alignment})" display="inline"><semantics id="S2.F2.6.4.2.2.m1.2b"><mrow id="S2.F2.6.4.2.2.m1.2.3" xref="S2.F2.6.4.2.2.m1.2.3.cmml"><mi id="S2.F2.6.4.2.2.m1.2.3.2" xref="S2.F2.6.4.2.2.m1.2.3.2.cmml">𝖽𝗂𝗌𝗍𝖺𝗇𝖼𝖾</mi><mo id="S2.F2.6.4.2.2.m1.2.3.1" xref="S2.F2.6.4.2.2.m1.2.3.1.cmml">≜</mo><mrow id="S2.F2.6.4.2.2.m1.2.3.3" xref="S2.F2.6.4.2.2.m1.2.3.3.cmml"><mo rspace="0.167em" id="S2.F2.6.4.2.2.m1.2.3.3b" xref="S2.F2.6.4.2.2.m1.2.3.3.cmml">−</mo><mrow id="S2.F2.6.4.2.2.m1.2.3.3.2.2" xref="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml"><mi id="S2.F2.6.4.2.2.m1.1.1" xref="S2.F2.6.4.2.2.m1.1.1.cmml">log</mi><mo id="S2.F2.6.4.2.2.m1.2.3.3.2.2b" xref="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml">⁡</mo><mrow id="S2.F2.6.4.2.2.m1.2.3.3.2.2.1" xref="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml"><mo stretchy="false" id="S2.F2.6.4.2.2.m1.2.3.3.2.2.1.1" xref="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml">(</mo><mi id="S2.F2.6.4.2.2.m1.2.2" xref="S2.F2.6.4.2.2.m1.2.2.cmml">𝖺𝗅𝗂𝗀𝗇𝗆𝖾𝗇𝗍</mi><mo stretchy="false" id="S2.F2.6.4.2.2.m1.2.3.3.2.2.1.2" xref="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.6.4.2.2.m1.2c"><apply id="S2.F2.6.4.2.2.m1.2.3.cmml" xref="S2.F2.6.4.2.2.m1.2.3"><ci id="S2.F2.6.4.2.2.m1.2.3.1.cmml" xref="S2.F2.6.4.2.2.m1.2.3.1">≜</ci><ci id="S2.F2.6.4.2.2.m1.2.3.2.cmml" xref="S2.F2.6.4.2.2.m1.2.3.2">𝖽𝗂𝗌𝗍𝖺𝗇𝖼𝖾</ci><apply id="S2.F2.6.4.2.2.m1.2.3.3.cmml" xref="S2.F2.6.4.2.2.m1.2.3.3"><minus id="S2.F2.6.4.2.2.m1.2.3.3.1.cmml" xref="S2.F2.6.4.2.2.m1.2.3.3"></minus><apply id="S2.F2.6.4.2.2.m1.2.3.3.2.1.cmml" xref="S2.F2.6.4.2.2.m1.2.3.3.2.2"><log id="S2.F2.6.4.2.2.m1.1.1.cmml" xref="S2.F2.6.4.2.2.m1.1.1"></log><ci id="S2.F2.6.4.2.2.m1.2.2.cmml" xref="S2.F2.6.4.2.2.m1.2.2">𝖺𝗅𝗂𝗀𝗇𝗆𝖾𝗇𝗍</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.6.4.2.2.m1.2d">\mathsf{distance}\triangleq-\log(\mathsf{alignment})</annotation></semantics></math>. More competent and general models (blue) have more similar representations.</span></span></figcaption>
</figure>
</section>
<section id="S2.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.1 </span>Different models, with different architectures and objectives, can have aligned representations</h3>

<div id="S2.SS1.p1" class="ltx_para ltx_noindent">
<p id="S2.SS1.p1.1" class="ltx_p">One indication of representational convergence is the rising number of systems built on top of pre-trained foundation models. These models are becoming standard backbones across a growing spectrum of tasks. Their versatility across numerous applications implies a level of universality in the way they represent data.</p>
</div>
<div id="S2.SS1.p2" class="ltx_para ltx_noindent">
<p id="S2.SS1.p2.1" class="ltx_p">While this trend implies convergence toward a relatively small set of foundation models, it does not imply that <span id="S2.SS1.p2.1.1" class="ltx_text ltx_font_italic">different</span> foundation models will arrive at the same representation. Yet that is what has been observed by several recent papers.</p>
</div>
<div id="S2.SS1.p3" class="ltx_para ltx_noindent">
<p id="S2.SS1.p3.13" class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Lenc &amp; Vedaldi (<a href="#bib.bib65" title="" class="ltx_ref">2015</a>)</cite> conducted one such study, in which they measured representational similarity through a technique called <span id="S2.SS1.p3.13.1" class="ltx_text ltx_font_italic">model stitching</span>. Given two models, <math id="S2.SS1.p3.1.m1.1" class="ltx_Math" alttext="f" display="inline"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">f</annotation></semantics></math> and <math id="S2.SS1.p3.2.m2.1" class="ltx_Math" alttext="g" display="inline"><semantics id="S2.SS1.p3.2.m2.1a"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">g</annotation></semantics></math>, each composed of multiple layers (<math id="S2.SS1.p3.3.m3.1" class="ltx_Math" alttext="f=f_{1}\circ\cdots\circ f_{n}" display="inline"><semantics id="S2.SS1.p3.3.m3.1a"><mrow id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml"><mi id="S2.SS1.p3.3.m3.1.1.2" xref="S2.SS1.p3.3.m3.1.1.2.cmml">f</mi><mo id="S2.SS1.p3.3.m3.1.1.1" xref="S2.SS1.p3.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.3.m3.1.1.3" xref="S2.SS1.p3.3.m3.1.1.3.cmml"><msub id="S2.SS1.p3.3.m3.1.1.3.2" xref="S2.SS1.p3.3.m3.1.1.3.2.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.2.2" xref="S2.SS1.p3.3.m3.1.1.3.2.2.cmml">f</mi><mn id="S2.SS1.p3.3.m3.1.1.3.2.3" xref="S2.SS1.p3.3.m3.1.1.3.2.3.cmml">1</mn></msub><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.3.m3.1.1.3.1" xref="S2.SS1.p3.3.m3.1.1.3.1.cmml">∘</mo><mi mathvariant="normal" id="S2.SS1.p3.3.m3.1.1.3.3" xref="S2.SS1.p3.3.m3.1.1.3.3.cmml">⋯</mi><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.3.m3.1.1.3.1a" xref="S2.SS1.p3.3.m3.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p3.3.m3.1.1.3.4" xref="S2.SS1.p3.3.m3.1.1.3.4.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.4.2" xref="S2.SS1.p3.3.m3.1.1.3.4.2.cmml">f</mi><mi id="S2.SS1.p3.3.m3.1.1.3.4.3" xref="S2.SS1.p3.3.m3.1.1.3.4.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1"><eq id="S2.SS1.p3.3.m3.1.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1.1"></eq><ci id="S2.SS1.p3.3.m3.1.1.2.cmml" xref="S2.SS1.p3.3.m3.1.1.2">𝑓</ci><apply id="S2.SS1.p3.3.m3.1.1.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3"><compose id="S2.SS1.p3.3.m3.1.1.3.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.1"></compose><apply id="S2.SS1.p3.3.m3.1.1.3.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2.2">𝑓</ci><cn type="integer" id="S2.SS1.p3.3.m3.1.1.3.2.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.2.3">1</cn></apply><ci id="S2.SS1.p3.3.m3.1.1.3.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.3">⋯</ci><apply id="S2.SS1.p3.3.m3.1.1.3.4.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.1.1.3.4.1.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p3.3.m3.1.1.3.4.2.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.2">𝑓</ci><ci id="S2.SS1.p3.3.m3.1.1.3.4.3.cmml" xref="S2.SS1.p3.3.m3.1.1.3.4.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">f=f_{1}\circ\cdots\circ f_{n}</annotation></semantics></math>, <math id="S2.SS1.p3.4.m4.1" class="ltx_Math" alttext="g=g_{1}\circ\cdots\circ g_{m}" display="inline"><semantics id="S2.SS1.p3.4.m4.1a"><mrow id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><mi id="S2.SS1.p3.4.m4.1.1.2" xref="S2.SS1.p3.4.m4.1.1.2.cmml">g</mi><mo id="S2.SS1.p3.4.m4.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml"><msub id="S2.SS1.p3.4.m4.1.1.3.2" xref="S2.SS1.p3.4.m4.1.1.3.2.cmml"><mi id="S2.SS1.p3.4.m4.1.1.3.2.2" xref="S2.SS1.p3.4.m4.1.1.3.2.2.cmml">g</mi><mn id="S2.SS1.p3.4.m4.1.1.3.2.3" xref="S2.SS1.p3.4.m4.1.1.3.2.3.cmml">1</mn></msub><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.4.m4.1.1.3.1" xref="S2.SS1.p3.4.m4.1.1.3.1.cmml">∘</mo><mi mathvariant="normal" id="S2.SS1.p3.4.m4.1.1.3.3" xref="S2.SS1.p3.4.m4.1.1.3.3.cmml">⋯</mi><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.4.m4.1.1.3.1a" xref="S2.SS1.p3.4.m4.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p3.4.m4.1.1.3.4" xref="S2.SS1.p3.4.m4.1.1.3.4.cmml"><mi id="S2.SS1.p3.4.m4.1.1.3.4.2" xref="S2.SS1.p3.4.m4.1.1.3.4.2.cmml">g</mi><mi id="S2.SS1.p3.4.m4.1.1.3.4.3" xref="S2.SS1.p3.4.m4.1.1.3.4.3.cmml">m</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><apply id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1"><eq id="S2.SS1.p3.4.m4.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1"></eq><ci id="S2.SS1.p3.4.m4.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.2">𝑔</ci><apply id="S2.SS1.p3.4.m4.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3"><compose id="S2.SS1.p3.4.m4.1.1.3.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3.1"></compose><apply id="S2.SS1.p3.4.m4.1.1.3.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.3.2.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.3.2.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2.2">𝑔</ci><cn type="integer" id="S2.SS1.p3.4.m4.1.1.3.2.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2.3">1</cn></apply><ci id="S2.SS1.p3.4.m4.1.1.3.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.3">⋯</ci><apply id="S2.SS1.p3.4.m4.1.1.3.4.cmml" xref="S2.SS1.p3.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.3.4.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.3.4.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.4.2">𝑔</ci><ci id="S2.SS1.p3.4.m4.1.1.3.4.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.4.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">g=g_{1}\circ\cdots\circ g_{m}</annotation></semantics></math>), an intermediate representation from <math id="S2.SS1.p3.5.m5.1" class="ltx_Math" alttext="f" display="inline"><semantics id="S2.SS1.p3.5.m5.1a"><mi id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><ci id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">f</annotation></semantics></math> is integrated into <math id="S2.SS1.p3.6.m6.1" class="ltx_Math" alttext="g" display="inline"><semantics id="S2.SS1.p3.6.m6.1a"><mi id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><ci id="S2.SS1.p3.6.m6.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">g</annotation></semantics></math> via a learned affine stitching layer <math id="S2.SS1.p3.7.m7.1" class="ltx_Math" alttext="h" display="inline"><semantics id="S2.SS1.p3.7.m7.1a"><mi id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.1b"><ci id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.1c">h</annotation></semantics></math>, resulting in a new stitched model <math id="S2.SS1.p3.8.m8.1" class="ltx_Math" alttext="F=f_{1}\circ\cdots\circ f_{k}\circ h\circ g_{k+1}\circ\cdots\circ g_{m}" display="inline"><semantics id="S2.SS1.p3.8.m8.1a"><mrow id="S2.SS1.p3.8.m8.1.1" xref="S2.SS1.p3.8.m8.1.1.cmml"><mi id="S2.SS1.p3.8.m8.1.1.2" xref="S2.SS1.p3.8.m8.1.1.2.cmml">F</mi><mo id="S2.SS1.p3.8.m8.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.8.m8.1.1.3" xref="S2.SS1.p3.8.m8.1.1.3.cmml"><msub id="S2.SS1.p3.8.m8.1.1.3.2" xref="S2.SS1.p3.8.m8.1.1.3.2.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.2.2" xref="S2.SS1.p3.8.m8.1.1.3.2.2.cmml">f</mi><mn id="S2.SS1.p3.8.m8.1.1.3.2.3" xref="S2.SS1.p3.8.m8.1.1.3.2.3.cmml">1</mn></msub><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><mi mathvariant="normal" id="S2.SS1.p3.8.m8.1.1.3.3" xref="S2.SS1.p3.8.m8.1.1.3.3.cmml">⋯</mi><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1a" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p3.8.m8.1.1.3.4" xref="S2.SS1.p3.8.m8.1.1.3.4.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.4.2" xref="S2.SS1.p3.8.m8.1.1.3.4.2.cmml">f</mi><mi id="S2.SS1.p3.8.m8.1.1.3.4.3" xref="S2.SS1.p3.8.m8.1.1.3.4.3.cmml">k</mi></msub><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1b" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><mi id="S2.SS1.p3.8.m8.1.1.3.5" xref="S2.SS1.p3.8.m8.1.1.3.5.cmml">h</mi><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1c" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p3.8.m8.1.1.3.6" xref="S2.SS1.p3.8.m8.1.1.3.6.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.6.2" xref="S2.SS1.p3.8.m8.1.1.3.6.2.cmml">g</mi><mrow id="S2.SS1.p3.8.m8.1.1.3.6.3" xref="S2.SS1.p3.8.m8.1.1.3.6.3.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.6.3.2" xref="S2.SS1.p3.8.m8.1.1.3.6.3.2.cmml">k</mi><mo id="S2.SS1.p3.8.m8.1.1.3.6.3.1" xref="S2.SS1.p3.8.m8.1.1.3.6.3.1.cmml">+</mo><mn id="S2.SS1.p3.8.m8.1.1.3.6.3.3" xref="S2.SS1.p3.8.m8.1.1.3.6.3.3.cmml">1</mn></mrow></msub><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1d" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><mi mathvariant="normal" id="S2.SS1.p3.8.m8.1.1.3.7" xref="S2.SS1.p3.8.m8.1.1.3.7.cmml">⋯</mi><mo lspace="0.222em" rspace="0.222em" id="S2.SS1.p3.8.m8.1.1.3.1e" xref="S2.SS1.p3.8.m8.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p3.8.m8.1.1.3.8" xref="S2.SS1.p3.8.m8.1.1.3.8.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.8.2" xref="S2.SS1.p3.8.m8.1.1.3.8.2.cmml">g</mi><mi id="S2.SS1.p3.8.m8.1.1.3.8.3" xref="S2.SS1.p3.8.m8.1.1.3.8.3.cmml">m</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.1b"><apply id="S2.SS1.p3.8.m8.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1"><eq id="S2.SS1.p3.8.m8.1.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1.1"></eq><ci id="S2.SS1.p3.8.m8.1.1.2.cmml" xref="S2.SS1.p3.8.m8.1.1.2">𝐹</ci><apply id="S2.SS1.p3.8.m8.1.1.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3"><compose id="S2.SS1.p3.8.m8.1.1.3.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.1"></compose><apply id="S2.SS1.p3.8.m8.1.1.3.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.3.2.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.3.2.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.2.2">𝑓</ci><cn type="integer" id="S2.SS1.p3.8.m8.1.1.3.2.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.2.3">1</cn></apply><ci id="S2.SS1.p3.8.m8.1.1.3.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.3">⋯</ci><apply id="S2.SS1.p3.8.m8.1.1.3.4.cmml" xref="S2.SS1.p3.8.m8.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.3.4.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.3.4.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.4.2">𝑓</ci><ci id="S2.SS1.p3.8.m8.1.1.3.4.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.4.3">𝑘</ci></apply><ci id="S2.SS1.p3.8.m8.1.1.3.5.cmml" xref="S2.SS1.p3.8.m8.1.1.3.5">ℎ</ci><apply id="S2.SS1.p3.8.m8.1.1.3.6.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.3.6.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6">subscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.3.6.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6.2">𝑔</ci><apply id="S2.SS1.p3.8.m8.1.1.3.6.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6.3"><plus id="S2.SS1.p3.8.m8.1.1.3.6.3.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6.3.1"></plus><ci id="S2.SS1.p3.8.m8.1.1.3.6.3.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6.3.2">𝑘</ci><cn type="integer" id="S2.SS1.p3.8.m8.1.1.3.6.3.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.6.3.3">1</cn></apply></apply><ci id="S2.SS1.p3.8.m8.1.1.3.7.cmml" xref="S2.SS1.p3.8.m8.1.1.3.7">⋯</ci><apply id="S2.SS1.p3.8.m8.1.1.3.8.cmml" xref="S2.SS1.p3.8.m8.1.1.3.8"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.3.8.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3.8">subscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.3.8.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.8.2">𝑔</ci><ci id="S2.SS1.p3.8.m8.1.1.3.8.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3.8.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.1c">F=f_{1}\circ\cdots\circ f_{k}\circ h\circ g_{k+1}\circ\cdots\circ g_{m}</annotation></semantics></math>. If <math id="S2.SS1.p3.9.m9.1" class="ltx_Math" alttext="F" display="inline"><semantics id="S2.SS1.p3.9.m9.1a"><mi id="S2.SS1.p3.9.m9.1.1" xref="S2.SS1.p3.9.m9.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m9.1b"><ci id="S2.SS1.p3.9.m9.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m9.1c">F</annotation></semantics></math> has good performance, it indicates that <math id="S2.SS1.p3.10.m10.1" class="ltx_Math" alttext="f" display="inline"><semantics id="S2.SS1.p3.10.m10.1a"><mi id="S2.SS1.p3.10.m10.1.1" xref="S2.SS1.p3.10.m10.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.10.m10.1b"><ci id="S2.SS1.p3.10.m10.1.1.cmml" xref="S2.SS1.p3.10.m10.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.10.m10.1c">f</annotation></semantics></math> and <math id="S2.SS1.p3.11.m11.1" class="ltx_Math" alttext="g" display="inline"><semantics id="S2.SS1.p3.11.m11.1a"><mi id="S2.SS1.p3.11.m11.1.1" xref="S2.SS1.p3.11.m11.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.11.m11.1b"><ci id="S2.SS1.p3.11.m11.1.1.cmml" xref="S2.SS1.p3.11.m11.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.11.m11.1c">g</annotation></semantics></math> have compatible representations at layer <math id="S2.SS1.p3.12.m12.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.SS1.p3.12.m12.1a"><mi id="S2.SS1.p3.12.m12.1.1" xref="S2.SS1.p3.12.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.12.m12.1b"><ci id="S2.SS1.p3.12.m12.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.12.m12.1c">k</annotation></semantics></math>, up to the transform <math id="S2.SS1.p3.13.m13.1" class="ltx_Math" alttext="h" display="inline"><semantics id="S2.SS1.p3.13.m13.1a"><mi id="S2.SS1.p3.13.m13.1.1" xref="S2.SS1.p3.13.m13.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.13.m13.1b"><ci id="S2.SS1.p3.13.m13.1.1.cmml" xref="S2.SS1.p3.13.m13.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.13.m13.1c">h</annotation></semantics></math>.</p>
</div>
<div id="S2.SS1.p4" class="ltx_para ltx_noindent">
<p id="S2.SS1.p4.1" class="ltx_p">In their study, <cite class="ltx_cite ltx_citemacro_citet">Lenc &amp; Vedaldi (<a href="#bib.bib65" title="" class="ltx_ref">2015</a>)</cite> made two notable findings: (1) A vision model trained on ImageNet <cite class="ltx_cite ltx_citemacro_citep">(Russakovsky et al., <a href="#bib.bib102" title="" class="ltx_ref">2015</a>)</cite> can be aligned with a model trained on the Places-365 dataset <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a href="#bib.bib137" title="" class="ltx_ref">2017</a>)</cite> while maintaining good performance; (2) The early layers of these convolutional networks are more interchangeable than later layers. The first finding illustrates a level of data independence where distinct image datasets lead to similar representations. The second finding agrees with extensive research that oriented Gabor-like filters are common in both artificial and biological vision systems. This suggests a convergence to a similar initial layer of representation across various neural network architectures <cite class="ltx_cite ltx_citemacro_citep">(Olshausen &amp; Field, <a href="#bib.bib89" title="" class="ltx_ref">1996</a>; Krizhevsky et al., <a href="#bib.bib63" title="" class="ltx_ref">2017</a>)</cite>.
<cite class="ltx_cite ltx_citemacro_citet">Bansal et al. (<a href="#bib.bib9" title="" class="ltx_ref">2021</a>)</cite> expanded on the idea of model stitching, uncovering that models trained using self-supervised objectives align closely with their supervised counterparts.</p>
</div>
<div id="S2.SS1.p5" class="ltx_para ltx_noindent">
<p id="S2.SS1.p5.2" class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Moschella et al. (<a href="#bib.bib83" title="" class="ltx_ref">2022</a>)</cite> further demonstrated the feasibility of “zero-shot” model stitching without learning a stitching layer.
Despite the fact that different text models were trained on different modalities, they found that the models often
embed data in remarkably similar ways. In particular, they considered the kernel <math id="S2.SS1.p5.1.m1.1" class="ltx_Math" alttext="K" display="inline"><semantics id="S2.SS1.p5.1.m1.1a"><mi id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><ci id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">K</annotation></semantics></math> defined by learned representations and showed that <math id="S2.SS1.p5.2.m2.1" class="ltx_Math" alttext="K" display="inline"><semantics id="S2.SS1.p5.2.m2.1a"><mi id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><ci id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">K</annotation></semantics></math> serves as a bridge between models, allowing an encoder trained in one language, like English, to work effectively with a decoder in another, like French.</p>
</div>
<div id="S2.SS1.p6" class="ltx_para ltx_noindent">
<p id="S2.SS1.p6.1" class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Dravid et al. (<a href="#bib.bib29" title="" class="ltx_ref">2023</a>)</cite> extended this idea to individual neurons, and found “Rosetta Neurons” that are activated by the same pattern across a range of vision models. Such neurons form a common dictionary independently discovered by all models.</p>
</div>
<figure id="S2.F3" class="ltx_figure"><img src="/html/2405.07987/assets/x4.png" id="S2.F3.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="332" height="130" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span><span id="S2.F3.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">LANGUAGE and VISION models align:<span id="S2.F3.4.2.2" class="ltx_text ltx_font_medium">
We measure alignment using mutual nearest-neighbor on the Wikipedia caption dataset (WIT) <cite class="ltx_cite ltx_citemacro_citep">(Srinivasan et al., <a href="#bib.bib111" title="" class="ltx_ref">2021</a>)</cite>. The x-axis is the language model performance measured over 4M tokens from the OpenWebText dataset <cite class="ltx_cite ltx_citemacro_citep">(Gokaslan &amp; Cohen, <a href="#bib.bib35" title="" class="ltx_ref">2019</a>)</cite> (see Appendix <a href="#A2" title="Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B</span></a> for plots with model names).
We measure performance using <math id="S2.F3.3.1.1.m1.1" class="ltx_Math" alttext="1-\texttt{bits-per-byte}" display="inline"><semantics id="S2.F3.3.1.1.m1.1b"><mrow id="S2.F3.3.1.1.m1.1.1" xref="S2.F3.3.1.1.m1.1.1.cmml"><mn id="S2.F3.3.1.1.m1.1.1.2" xref="S2.F3.3.1.1.m1.1.1.2.cmml">1</mn><mo id="S2.F3.3.1.1.m1.1.1.1" xref="S2.F3.3.1.1.m1.1.1.1.cmml">−</mo><mtext class="ltx_mathvariant_monospace" id="S2.F3.3.1.1.m1.1.1.3" xref="S2.F3.3.1.1.m1.1.1.3a.cmml">bits-per-byte</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.3.1.1.m1.1c"><apply id="S2.F3.3.1.1.m1.1.1.cmml" xref="S2.F3.3.1.1.m1.1.1"><minus id="S2.F3.3.1.1.m1.1.1.1.cmml" xref="S2.F3.3.1.1.m1.1.1.1"></minus><cn type="integer" id="S2.F3.3.1.1.m1.1.1.2.cmml" xref="S2.F3.3.1.1.m1.1.1.2">1</cn><ci id="S2.F3.3.1.1.m1.1.1.3a.cmml" xref="S2.F3.3.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S2.F3.3.1.1.m1.1.1.3.cmml" xref="S2.F3.3.1.1.m1.1.1.3">bits-per-byte</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.3.1.1.m1.1d">1-\texttt{bits-per-byte}</annotation></semantics></math>, where <span id="S2.F3.4.2.2.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">bits-per-byte</span> normalizes the cross-entropy by the total bytes in the input text string.
The results show a linear relationship between language-vision alignment and language modeling score, where a general trend is that more capable language models align better with more capable vision models.
We find that CLIP models, which are trained with explicit language supervision, exhibit a higher level of alignment. However, this alignment decreases after being fine-tuned on ImageNet classification (labeled CLIP (I12K ft)).
</span></span></figcaption>
</figure>
</section>
<section id="S2.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.2 </span>Alignment increases with scale and performance</h3>

<div id="S2.SS2.p1" class="ltx_para ltx_noindent">
<p id="S2.SS2.p1.1" class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Kornblith et al. (<a href="#bib.bib61" title="" class="ltx_ref">2019</a>)</cite> observed model alignment not only exists but also increases with model scale. On CIFAR-10 classification <cite class="ltx_cite ltx_citemacro_citep">(Krizhevsky et al., <a href="#bib.bib62" title="" class="ltx_ref">2009</a>)</cite>, they found that larger models exhibit greater alignment with each other compared to smaller ones.
Theoretically, <cite class="ltx_cite ltx_citemacro_citet">Balestriero &amp; Baraniuk (<a href="#bib.bib8" title="" class="ltx_ref">2018</a>)</cite> showed that models with similar outputs (<span id="S2.SS2.p1.1.1" class="ltx_text ltx_font_italic">e.g.</span>, as a result of having high performance) also have similar internal activations.
With the continuing trend of models scaling up, this suggests model alignment will increase over time – we might expect that the next generation of bigger, better models will be even more aligned with each other.</p>
</div>
<div id="S2.SS2.p2" class="ltx_para ltx_noindent">
<p id="S2.SS2.p2.1" class="ltx_p">We expand upon this observation by evaluating the transfer performance of <math id="S2.SS2.p2.1.m1.1" class="ltx_Math" alttext="78" display="inline"><semantics id="S2.SS2.p2.1.m1.1a"><mn id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">78</mn><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><cn type="integer" id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">78</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">78</annotation></semantics></math> vision models. These models were trained with varying architectures, training objectives, and datasets (detailed in <a href="#A3.SS1" title="C.1 Vision-Vision Alignment and Representation Quality ‣ Appendix C Experiments on Evaluating Alignment and Convergence ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">C.1</span></a>). In Figure <a href="#S2.F2.6" title="Figure 2 ‣ Preliminaries ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (left), we bin these models based on their average transfer performance on the VTAB dataset <cite class="ltx_cite ltx_citemacro_citep">(Zhai et al., <a href="#bib.bib135" title="" class="ltx_ref">2019</a>)</cite>, and then measure the average kernel alignment of the models within each bin. The results indicate that models with high transfer performance form a tightly clustered set of representations, while models with weak performance have more variable representations. We further visualize this structure with UMAP <cite class="ltx_cite ltx_citemacro_citep">(McInnes et al., <a href="#bib.bib78" title="" class="ltx_ref">2018</a>)</cite> over models representation in Figure <a href="#S2.F2.6" title="Figure 2 ‣ Preliminaries ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> (right). This suggests that models that are competent all represent data in a similar way. Echoing <cite class="ltx_cite ltx_citemacro_citet">Bansal et al. (<a href="#bib.bib9" title="" class="ltx_ref">2021</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Tolstoy (<a href="#bib.bib119" title="" class="ltx_ref">1877</a>)</cite>, we might say: all strong models are alike, each weak model is weak in its own way.</p>
</div>
<div id="S2.SS2.p3" class="ltx_para ltx_noindent">
<p id="S2.SS2.p3.1" class="ltx_p">The discussion so far indicates that various models are aligning toward a unified representation. But does the convergence extend to model weights? While models with different architectures might not have compatible weight spaces, there exists ample evidence that models with the same architecture will often converge to the same basin of weights <cite class="ltx_cite ltx_citemacro_citep">(Nagarajan &amp; Kolter, <a href="#bib.bib84" title="" class="ltx_ref">2019</a>; Garipov et al., <a href="#bib.bib32" title="" class="ltx_ref">2018</a>; Lubana et al., <a href="#bib.bib76" title="" class="ltx_ref">2023</a>)</cite>. This holds even for models with different initializations, up to permutations over weight space <cite class="ltx_cite ltx_citemacro_citep">(Ainsworth et al., <a href="#bib.bib3" title="" class="ltx_ref">2022</a>)</cite>. Because of this, it is possible to merge separately trained models of the same architecture, and achieve some of the capabilities of all models in the mixture <cite class="ltx_cite ltx_citemacro_citep">(Stoica et al., <a href="#bib.bib114" title="" class="ltx_ref">2023</a>; Jordan et al., <a href="#bib.bib55" title="" class="ltx_ref">2022</a>; Wortsman et al., <a href="#bib.bib130" title="" class="ltx_ref">2022</a>)</cite>.</p>
</div>
<figure id="S2.F4" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_2"><img src="/html/2405.07987/assets/x5.png" id="S2.F4.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="226" height="128" alt="Refer to caption"></div>
<div class="ltx_flex_cell ltx_flex_size_2"><img src="/html/2405.07987/assets/x6.png" id="S2.F4.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="226" height="128" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span><span id="S2.F4.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Alignment predicts downstream performance:<span id="S2.F4.2.1.1" class="ltx_text ltx_font_medium"> We visualize correlation between LLM alignment score to DINOv2 <cite class="ltx_cite ltx_citemacro_citep">(Oquab et al., <a href="#bib.bib92" title="" class="ltx_ref">2023</a>)</cite> and downstream task performance on Hellaswag (common-sense) <cite class="ltx_cite ltx_citemacro_citep">(Zellers et al., <a href="#bib.bib134" title="" class="ltx_ref">2019</a>)</cite> and GSM8K (math) <cite class="ltx_cite ltx_citemacro_citep">(Cobbe et al., <a href="#bib.bib21" title="" class="ltx_ref">2021</a>)</cite>. LLMs are plotted with radii proportional to the size of the model, and color-coded by their rank order in language modeling scores (<math id="S2.F4.2.1.1.m1.1" class="ltx_Math" alttext="1-\texttt{bits-per-byte}" display="inline"><semantics id="S2.F4.2.1.1.m1.1b"><mrow id="S2.F4.2.1.1.m1.1.1" xref="S2.F4.2.1.1.m1.1.1.cmml"><mn id="S2.F4.2.1.1.m1.1.1.2" xref="S2.F4.2.1.1.m1.1.1.2.cmml">1</mn><mo id="S2.F4.2.1.1.m1.1.1.1" xref="S2.F4.2.1.1.m1.1.1.1.cmml">−</mo><mtext class="ltx_mathvariant_monospace" id="S2.F4.2.1.1.m1.1.1.3" xref="S2.F4.2.1.1.m1.1.1.3a.cmml">bits-per-byte</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.F4.2.1.1.m1.1c"><apply id="S2.F4.2.1.1.m1.1.1.cmml" xref="S2.F4.2.1.1.m1.1.1"><minus id="S2.F4.2.1.1.m1.1.1.1.cmml" xref="S2.F4.2.1.1.m1.1.1.1"></minus><cn type="integer" id="S2.F4.2.1.1.m1.1.1.2.cmml" xref="S2.F4.2.1.1.m1.1.1.2">1</cn><ci id="S2.F4.2.1.1.m1.1.1.3a.cmml" xref="S2.F4.2.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S2.F4.2.1.1.m1.1.1.3.cmml" xref="S2.F4.2.1.1.m1.1.1.3">bits-per-byte</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F4.2.1.1.m1.1d">1-\texttt{bits-per-byte}</annotation></semantics></math>). We observe that models aligned more closely with vision also show better performance on downstream language tasks. For Hellaswag, there is a linear relationship with alignment score, while GSM8K exhibits an “emergence”-esque trend.
</span></span></figcaption>
</figure>
</section>
<section id="S2.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.3 </span>Representations are converging across modalities</h3>

<div id="S2.SS3.p1" class="ltx_para ltx_noindent">
<p id="S2.SS3.p1.1" class="ltx_p">Do models trained on different data modalities also converge?
Several works indicate that the answer is <em id="S2.SS3.p1.1.1" class="ltx_emph ltx_font_italic">yes</em>.</p>
</div>
<div id="S2.SS3.p2" class="ltx_para ltx_noindent">
<p id="S2.SS3.p2.1" class="ltx_p"><cite class="ltx_cite ltx_citemacro_citet">Merullo et al. (<a href="#bib.bib79" title="" class="ltx_ref">2022</a>)</cite> extended model stitching to the cross-modal setting, finding that a single linear projection is sufficient to stitch a vision model to an LLM and achieve good performance on visual question answering and image captioning. <cite class="ltx_cite ltx_citemacro_citet">Koh et al. (<a href="#bib.bib60" title="" class="ltx_ref">2023</a>)</cite> showed that linear stitching can also work in the opposite direction, aligning text inputs to visual outputs.
In fact, many recent language-vision models stitch pre-trained language and vision models together. For example, LLaVA <cite class="ltx_cite ltx_citemacro_citep">(Liu et al., <a href="#bib.bib70" title="" class="ltx_ref">2023</a>)</cite> demonstrated state-of-the-art results by projecting visual features into a language model with a 2-layer MLP.</p>
</div>
<div id="S2.SS3.p3" class="ltx_para ltx_noindent">
<p id="S2.SS3.p3.1" class="ltx_p">Other works show further kinds of evidence of cross-modal synergy. <cite class="ltx_cite ltx_citemacro_citet">Achiam et al. (<a href="#bib.bib2" title="" class="ltx_ref">2023</a>)</cite> found that jointly training a language model with a vision model improves performance on language tasks, compared to training the language model on its own. <cite class="ltx_cite ltx_citemacro_citet">Sharma et al. (<a href="#bib.bib105" title="" class="ltx_ref">2024</a>)</cite> probed the visual knowledge of LLMs trained <span id="S2.SS3.p3.1.1" class="ltx_text ltx_font_italic">only</span> on language data, by converting images into code that an LLM can process. They found that LLMs have rich knowledge of visual structures, to the extent that decent visual representations can be trained on images generated solely by querying an LLM to produce code and rendering the response. In visual generation, LLMs show abilities to augment captions with visual structures (<span id="S2.SS3.p3.1.2" class="ltx_text ltx_font_italic">e.g.</span>, bounding boxes and locations) and improve generation quality <cite class="ltx_cite ltx_citemacro_citep">(Betker et al., <a href="#bib.bib13" title="" class="ltx_ref">2023</a>; Lian et al., <a href="#bib.bib67" title="" class="ltx_ref">2023a</a>, <a href="#bib.bib68" title="" class="ltx_ref">b</a>; Wu et al., <a href="#bib.bib131" title="" class="ltx_ref">2023</a>)</cite>. In other modalities, <cite class="ltx_cite ltx_citemacro_citet">Ngo &amp; Kim (<a href="#bib.bib88" title="" class="ltx_ref">2024</a>)</cite> showed auditory models are also roughly aligned with language models up to a linear transformation, and <cite class="ltx_cite ltx_citemacro_citet">Ng et al. (<a href="#bib.bib87" title="" class="ltx_ref">2023</a>)</cite> demonstrated the effectiveness of using pre-trained language models for facial motion prediction.</p>
</div>
<div id="S2.SS3.p4" class="ltx_para ltx_noindent">
<p id="S2.SS3.p4.1" class="ltx_p">We set out to address these claims in a broader scope to determine whether models are indeed learning an increasingly modality-agnostic representation of the world. We sampled a variety of models trained either solely on vision or solely on language, and compared their representations as they became larger and more competent over many tasks.</p>
</div>
<div id="S2.SS3.p5" class="ltx_para ltx_noindent">
<p id="S2.SS3.p5.5" class="ltx_p">In <a href="#S2.F3" title="In 2.1 Different models, with different architectures and objectives, can have aligned representations ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a>, we assess alignment between a suite of language models and vision models. So far we have only defined alignment for two kernels defined over the same input space. To measure cross-modal alignment, we use paired datasets to bridge the two modalities. For vision and text, we use the Wikipedia captions dataset <math id="S2.SS3.p5.1.m1.1" class="ltx_Math" alttext="\{(x_{i},y_{i})\}_{i}" display="inline"><semantics id="S2.SS3.p5.1.m1.1a"><msub id="S2.SS3.p5.1.m1.1.1" xref="S2.SS3.p5.1.m1.1.1.cmml"><mrow id="S2.SS3.p5.1.m1.1.1.1.1" xref="S2.SS3.p5.1.m1.1.1.1.2.cmml"><mo stretchy="false" id="S2.SS3.p5.1.m1.1.1.1.1.2" xref="S2.SS3.p5.1.m1.1.1.1.2.cmml">{</mo><mrow id="S2.SS3.p5.1.m1.1.1.1.1.1.2" xref="S2.SS3.p5.1.m1.1.1.1.1.1.3.cmml"><mo stretchy="false" id="S2.SS3.p5.1.m1.1.1.1.1.1.2.3" xref="S2.SS3.p5.1.m1.1.1.1.1.1.3.cmml">(</mo><msub id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.2" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.3" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS3.p5.1.m1.1.1.1.1.1.2.4" xref="S2.SS3.p5.1.m1.1.1.1.1.1.3.cmml">,</mo><msub id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.cmml"><mi id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.2" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.2.cmml">y</mi><mi id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.3" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="S2.SS3.p5.1.m1.1.1.1.1.1.2.5" xref="S2.SS3.p5.1.m1.1.1.1.1.1.3.cmml">)</mo></mrow><mo stretchy="false" id="S2.SS3.p5.1.m1.1.1.1.1.3" xref="S2.SS3.p5.1.m1.1.1.1.2.cmml">}</mo></mrow><mi id="S2.SS3.p5.1.m1.1.1.3" xref="S2.SS3.p5.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.1.m1.1b"><apply id="S2.SS3.p5.1.m1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.1.m1.1.1.2.cmml" xref="S2.SS3.p5.1.m1.1.1">subscript</csymbol><set id="S2.SS3.p5.1.m1.1.1.1.2.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1"><interval closure="open" id="S2.SS3.p5.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2"><apply id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.1.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.2.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.2">𝑦</ci><ci id="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.3.cmml" xref="S2.SS3.p5.1.m1.1.1.1.1.1.2.2.3">𝑖</ci></apply></interval></set><ci id="S2.SS3.p5.1.m1.1.1.3.cmml" xref="S2.SS3.p5.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.1.m1.1c">\{(x_{i},y_{i})\}_{i}</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Srinivasan et al., <a href="#bib.bib111" title="" class="ltx_ref">2021</a>)</cite>, composed of images from Wikipedia (<math id="S2.SS3.p5.2.m2.1" class="ltx_Math" alttext="x_{i}" display="inline"><semantics id="S2.SS3.p5.2.m2.1a"><msub id="S2.SS3.p5.2.m2.1.1" xref="S2.SS3.p5.2.m2.1.1.cmml"><mi id="S2.SS3.p5.2.m2.1.1.2" xref="S2.SS3.p5.2.m2.1.1.2.cmml">x</mi><mi id="S2.SS3.p5.2.m2.1.1.3" xref="S2.SS3.p5.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.2.m2.1b"><apply id="S2.SS3.p5.2.m2.1.1.cmml" xref="S2.SS3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.2.m2.1.1.1.cmml" xref="S2.SS3.p5.2.m2.1.1">subscript</csymbol><ci id="S2.SS3.p5.2.m2.1.1.2.cmml" xref="S2.SS3.p5.2.m2.1.1.2">𝑥</ci><ci id="S2.SS3.p5.2.m2.1.1.3.cmml" xref="S2.SS3.p5.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.2.m2.1c">x_{i}</annotation></semantics></math>) and their corresponding captions (<math id="S2.SS3.p5.3.m3.1" class="ltx_Math" alttext="y_{i}" display="inline"><semantics id="S2.SS3.p5.3.m3.1a"><msub id="S2.SS3.p5.3.m3.1.1" xref="S2.SS3.p5.3.m3.1.1.cmml"><mi id="S2.SS3.p5.3.m3.1.1.2" xref="S2.SS3.p5.3.m3.1.1.2.cmml">y</mi><mi id="S2.SS3.p5.3.m3.1.1.3" xref="S2.SS3.p5.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.3.m3.1b"><apply id="S2.SS3.p5.3.m3.1.1.cmml" xref="S2.SS3.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.3.m3.1.1.1.cmml" xref="S2.SS3.p5.3.m3.1.1">subscript</csymbol><ci id="S2.SS3.p5.3.m3.1.1.2.cmml" xref="S2.SS3.p5.3.m3.1.1.2">𝑦</ci><ci id="S2.SS3.p5.3.m3.1.1.3.cmml" xref="S2.SS3.p5.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.3.m3.1c">y_{i}</annotation></semantics></math>). We then measure alignment between a language model <math id="S2.SS3.p5.4.m4.1" class="ltx_Math" alttext="f_{\texttt{text}}" display="inline"><semantics id="S2.SS3.p5.4.m4.1a"><msub id="S2.SS3.p5.4.m4.1.1" xref="S2.SS3.p5.4.m4.1.1.cmml"><mi id="S2.SS3.p5.4.m4.1.1.2" xref="S2.SS3.p5.4.m4.1.1.2.cmml">f</mi><mtext class="ltx_mathvariant_monospace" id="S2.SS3.p5.4.m4.1.1.3" xref="S2.SS3.p5.4.m4.1.1.3a.cmml">text</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.4.m4.1b"><apply id="S2.SS3.p5.4.m4.1.1.cmml" xref="S2.SS3.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.4.m4.1.1.1.cmml" xref="S2.SS3.p5.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.p5.4.m4.1.1.2.cmml" xref="S2.SS3.p5.4.m4.1.1.2">𝑓</ci><ci id="S2.SS3.p5.4.m4.1.1.3a.cmml" xref="S2.SS3.p5.4.m4.1.1.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="S2.SS3.p5.4.m4.1.1.3.cmml" xref="S2.SS3.p5.4.m4.1.1.3">text</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.4.m4.1c">f_{\texttt{text}}</annotation></semantics></math> and a vision model <math id="S2.SS3.p5.5.m5.1" class="ltx_Math" alttext="f_{\texttt{img}}" display="inline"><semantics id="S2.SS3.p5.5.m5.1a"><msub id="S2.SS3.p5.5.m5.1.1" xref="S2.SS3.p5.5.m5.1.1.cmml"><mi id="S2.SS3.p5.5.m5.1.1.2" xref="S2.SS3.p5.5.m5.1.1.2.cmml">f</mi><mtext class="ltx_mathvariant_monospace" id="S2.SS3.p5.5.m5.1.1.3" xref="S2.SS3.p5.5.m5.1.1.3a.cmml">img</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.p5.5.m5.1b"><apply id="S2.SS3.p5.5.m5.1.1.cmml" xref="S2.SS3.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS3.p5.5.m5.1.1.1.cmml" xref="S2.SS3.p5.5.m5.1.1">subscript</csymbol><ci id="S2.SS3.p5.5.m5.1.1.2.cmml" xref="S2.SS3.p5.5.m5.1.1.2">𝑓</ci><ci id="S2.SS3.p5.5.m5.1.1.3a.cmml" xref="S2.SS3.p5.5.m5.1.1.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="S2.SS3.p5.5.m5.1.1.3.cmml" xref="S2.SS3.p5.5.m5.1.1.3">img</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p5.5.m5.1c">f_{\texttt{img}}</annotation></semantics></math> as the alignment of the two following kernels:</p>
<table id="A6.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S2.E1.m1.4" class="ltx_Math" alttext="\displaystyle K_{\texttt{img}}(i,j)=\langle f_{\texttt{img}}(x_{i}),f_{\texttt{img}}(x_{j})\rangle" display="inline"><semantics id="S2.E1.m1.4a"><mrow id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml"><mrow id="S2.E1.m1.4.4.4" xref="S2.E1.m1.4.4.4.cmml"><msub id="S2.E1.m1.4.4.4.2" xref="S2.E1.m1.4.4.4.2.cmml"><mi id="S2.E1.m1.4.4.4.2.2" xref="S2.E1.m1.4.4.4.2.2.cmml">K</mi><mtext class="ltx_mathvariant_monospace" id="S2.E1.m1.4.4.4.2.3" xref="S2.E1.m1.4.4.4.2.3a.cmml">img</mtext></msub><mo lspace="0em" rspace="0em" id="S2.E1.m1.4.4.4.1" xref="S2.E1.m1.4.4.4.1.cmml">​</mo><mrow id="S2.E1.m1.4.4.4.3.2" xref="S2.E1.m1.4.4.4.3.1.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.4.3.2.1" xref="S2.E1.m1.4.4.4.3.1.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">i</mi><mo id="S2.E1.m1.4.4.4.3.2.2" xref="S2.E1.m1.4.4.4.3.1.cmml">,</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">j</mi><mo stretchy="false" id="S2.E1.m1.4.4.4.3.2.3" xref="S2.E1.m1.4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.4.3" xref="S2.E1.m1.4.4.3.cmml">=</mo><mrow id="S2.E1.m1.4.4.2.2" xref="S2.E1.m1.4.4.2.3.cmml"><mo stretchy="false" id="S2.E1.m1.4.4.2.2.3" xref="S2.E1.m1.4.4.2.3.cmml">⟨</mo><mrow id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml"><msub id="S2.E1.m1.3.3.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.3.2" xref="S2.E1.m1.3.3.1.1.1.3.2.cmml">f</mi><mtext class="ltx_mathvariant_monospace" id="S2.E1.m1.3.3.1.1.1.3.3" xref="S2.E1.m1.3.3.1.1.1.3.3a.cmml">img</mtext></msub><mo lspace="0em" rspace="0em" id="S2.E1.m1.3.3.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.2.cmml">​</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.3.3.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2" 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xref="S2.E1.m1.4.4.2.2.2.1.1">subscript</csymbol><ci id="S2.E1.m1.4.4.2.2.2.1.1.1.2.cmml" xref="S2.E1.m1.4.4.2.2.2.1.1.1.2">𝑥</ci><ci id="S2.E1.m1.4.4.2.2.2.1.1.1.3.cmml" xref="S2.E1.m1.4.4.2.2.2.1.1.1.3">𝑗</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.4c">\displaystyle K_{\texttt{img}}(i,j)=\langle f_{\texttt{img}}(x_{i}),f_{\texttt{img}}(x_{j})\rangle</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td>
</tr></tbody>
<tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="S2.E2.m1.3.3.1.1.2.2.2.1.1.1.3">𝑗</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.3c">\displaystyle K_{\texttt{text}}(i,j)=\langle f_{\texttt{text}}(y_{i}),f_{\texttt{text}}(y_{j})\rangle.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td>
</tr></tbody>
</table>
<p id="S2.SS3.p5.6" class="ltx_p">Using this analysis, we find that the better an LLM is at language modeling, the more it tends to aligns with vision models, as shown in <a href="#S2.F3" title="In 2.1 Different models, with different architectures and objectives, can have aligned representations ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a>. The converse effect also holds: the better a vision models is, the more it tends to align with LLMs. See <a href="#A3.SS2" title="C.2 Cross-Modal Alignment ‣ Appendix C Experiments on Evaluating Alignment and Convergence ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">C.2</span></a> for more details.</p>
</div>
</section>
<section id="S2.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.4 </span>Models are increasingly aligning to brains</h3>

<div id="S2.SS4.p1" class="ltx_para ltx_noindent">
<p id="S2.SS4.p1.1" class="ltx_p">Neural networks also show substantial alignment with biological representations in the brain <cite class="ltx_cite ltx_citemacro_citep">(Yamins et al., <a href="#bib.bib133" title="" class="ltx_ref">2014</a>)</cite>. This commonality may be due to similarities in the task and data constraints both systems are confronted with.
Even though the mediums may differ – silicon transistors versus biological neurons – the fundamental problem faced by brains and machines is the same: efficiently extracting and understanding the underlying structure in images, text, sounds, etc <cite class="ltx_cite ltx_citemacro_citep">(Barlow et al., <a href="#bib.bib12" title="" class="ltx_ref">1961</a>; Olshausen &amp; Field, <a href="#bib.bib90" title="" class="ltx_ref">1997</a>)</cite>. The tasks that the human visual system has been honed to perform through evolution – like segmentation, detection, and whole-image classification – are also the ones that we train our neural nets to perform. <cite class="ltx_cite ltx_citemacro_citet">Yamins et al. (<a href="#bib.bib133" title="" class="ltx_ref">2014</a>)</cite> went as far as to title their work in the spirit that performance over such tasks implies brain alignment. Further, <cite class="ltx_cite ltx_citemacro_citet">Conwell et al. (<a href="#bib.bib22" title="" class="ltx_ref">2022</a>)</cite> showed that training data plays a large role in alignment. Psychophysical studies have also shown agreement between how humans perceive visual similarity and how models do, even when the models are trained on tasks, such as self-supervised prediction, that are seemingly unrelated to mimicking human perception <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a href="#bib.bib136" title="" class="ltx_ref">2018</a>)</cite>.</p>
</div>
</section>
<section id="S2.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">2.5 </span>Does alignment predict downstream performance?</h3>

<div id="S2.SS5.p1" class="ltx_para ltx_noindent">
<p id="S2.SS5.p1.1" class="ltx_p">If models are converging towards a more accurate representation of reality, we expect that alignment should correspond to improved performance on downstream tasks. <a href="#S2.F4" title="In 2.2 Alignment increases with scale and performance ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">4</span></a> supports this hypothesis by demonstrating improved performance on commonsense reasoning (Hellaswag; <cite class="ltx_cite ltx_citemacro_citet">Zellers et al. (<a href="#bib.bib134" title="" class="ltx_ref">2019</a>)</cite>) and mathematical problem solving (GSM8K; <cite class="ltx_cite ltx_citemacro_citet">Cobbe et al. (<a href="#bib.bib21" title="" class="ltx_ref">2021</a>)</cite>) as alignment improves.</p>
</div>
</section>
</section>
<section id="S3" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">3 </span>Why are representations converging?</h2>

<div id="S3.p1" class="ltx_para ltx_noindent">
<p id="S3.p1.1" class="ltx_p">Modern machine learning models are generally trained to minimize the empirical risk with possible implicit and/or explicit regularization:</p>
<table id="S3.Ex1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="S3.p1.2" class="ltx_p">In the following sections, we lay out how each colored component in this optimization process potentially plays a role in facilitating representational convergence.</p>
</div>
<figure id="S3.F5" class="ltx_figure"><img src="/html/2405.07987/assets/x7.png" id="S3.F5.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="303" height="100" alt="Refer to caption">
<br class="ltx_break ltx_centering">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span><span id="S3.F5.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">The Capacity Hypothesis:<span id="S3.F5.4.2.3" class="ltx_text ltx_font_medium"> If an optimal representation exists in function space, larger hypothesis spaces are more likely to cover it. </span>LEFT:<span id="S3.F5.3.1.1" class="ltx_text ltx_font_medium"> Two small models might not cover the optimum and thus find <span id="S3.F5.3.1.1.2" class="ltx_text ltx_font_italic">different</span> solutions (marked by outlined 
<span id="S3.F5.3.1.1.1" class="ltx_inline-block ltx_transformed_outer" style="width:0.0pt;height:0pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1.25,1.25) ;">
<span id="S3.F5.3.1.1.1.1" class="ltx_p"></span>
</span></span>). </span>RIGHT:<span id="S3.F5.4.2.2" class="ltx_text ltx_font_medium"> As the models become larger, they cover the optimum and converge to the same solution (marked by filled 
<span id="S3.F5.4.2.2.1" class="ltx_inline-block ltx_transformed_outer" style="width:8.9pt;height:8.300000000000001pt;vertical-align:-1.8pt;"><span class="ltx_transformed_inner" style="transform:translate(0.2pt,-0.2pt) scale(1.05,1.05) ;">
<span id="S3.F5.4.2.2.1.1" class="ltx_p"><math id="S3.F5.4.2.2.1.1.m1.1" class="ltx_Math" alttext="\bigstar" display="inline"><semantics id="S3.F5.4.2.2.1.1.m1.1b"><mi mathvariant="normal" id="S3.F5.4.2.2.1.1.m1.1.1" xref="S3.F5.4.2.2.1.1.m1.1.1.cmml">★</mi><annotation-xml encoding="MathML-Content" id="S3.F5.4.2.2.1.1.m1.1c"><ci id="S3.F5.4.2.2.1.1.m1.1.1.cmml" xref="S3.F5.4.2.2.1.1.m1.1.1">★</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F5.4.2.2.1.1.m1.1d">\bigstar</annotation></semantics></math></span>
</span></span>).</span></span></figcaption>
</figure>
<figure id="S3.F6" class="ltx_figure"><img src="/html/2405.07987/assets/x8.png" id="S3.F6.g1" class="ltx_graphics ltx_centering ltx_img_square" width="380" height="331" alt="Refer to caption">
<br class="ltx_break ltx_centering">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span><span id="S3.F6.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">The Multitask Scaling Hypothesis:<span id="S3.F6.2.1.1" class="ltx_text ltx_font_medium"> Models trained with an increasing number of tasks are subjected to pressure to learn a representation that can solve all the tasks.</span></span></figcaption>
</figure>
<section id="S3.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.1 </span>Convergence via  <svg id="S3.SS1.1.pic1" class="ltx_picture" height="17.83" overflow="visible" version="1.1" width="101.84"><g transform="translate(0,17.83) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#B3FFB3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 99.07 17.83 C 100.6 17.83 101.84 16.6 101.84 15.07 L 101.84 2.77 C 101.84 1.24 100.6 0 99.07 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#B3FFB3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 99.07 17.83 C 100.6 17.83 101.84 16.6 101.84 15.07 L 101.84 2.77 C 101.84 1.24 100.6 0 99.07 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 5.46)"><text transform="matrix(1 0 0 -1 0 0)" color="#000000">Task Generality</text></g></g></svg>
</h3>

<div id="S3.SS1.p1" class="ltx_para ltx_noindent">
<p id="S3.SS1.p1.1" class="ltx_p">Each training datapoint and objective (task) places an additional constraint on the model. As data and tasks scale,
the volume of representations that satisfy these constraints must proportionately grow smaller, as visualized in Figure <a href="#S3.F6" title="Figure 6 ‣ 3 Why are representations converging? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a> and stated below:</p>
</div>
<div id="S3.SS1.p2" class="ltx_para ltx_noindent">
<svg id="S3.SS1.p2.pic1" class="ltx_picture" height="119.52" overflow="visible" version="1.1" width="600"><g transform="translate(0,119.52) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#4D4D4D" fill-opacity="1.0"><path d="M 0 5.91 L 0 113.61 C 0 116.87 2.64 119.52 5.91 119.52 L 594.09 119.52 C 597.36 119.52 600 116.87 600 113.61 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#FAFAFA" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 113.61 C 1.97 115.79 3.73 117.55 5.91 117.55 L 594.09 117.55 C 596.27 117.55 598.03 115.79 598.03 113.61 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 19.24 11.36)"><foreignObject width="561.52" height="95.32" transform="matrix(1 0 0 -1 0 16.6)" overflow="visible" color="#000000">
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<span id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.3" class="ltx_p"><span id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.3.1" class="ltx_text ltx_font_bold">The Multitask Scaling Hypothesis
<br class="ltx_break"></span>
<span id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.3.2" class="ltx_text ltx_inline-block" style="width:433.6pt;"><span class="ltx_rule" style="width:433.6pt;height:0.4pt;background:black;display:inline-block;"> </span></span></span>
<span id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2" class="ltx_p">There are fewer representations that are competent for <math id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1" class="ltx_Math" alttext="N" display="inline"><semantics id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">N</annotation></semantics></math> tasks than there are for <math id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1" class="ltx_Math" alttext="M&lt;N" display="inline"><semantics id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1a"><mrow id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.2" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.2.cmml">M</mi><mo id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.1" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.1.cmml">&lt;</mo><mi id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.3" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1b"><apply id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1"><lt id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.1"></lt><ci id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.2">𝑀</ci><ci id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1c">M&lt;N</annotation></semantics></math> tasks. As we train more general models that solve more tasks at once, we should expect fewer possible solutions.</span>
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</div>
<div id="S3.SS1.p3" class="ltx_para ltx_noindent">
<p id="S3.SS1.p3.3" class="ltx_p">This has been previously termed as the Contravariance principle by <cite class="ltx_cite ltx_citemacro_citet">Cao &amp; Yamins (<a href="#bib.bib17" title="" class="ltx_ref">2024</a>)</cite>, which states that the set of solutions to an easy goal is large, while the set of solutions to a challenging goal is comparatively smaller. Moreover, we argue that this narrower solution set also generalizes better. 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<div id="S3.SS1.p4" class="ltx_para ltx_noindent">
<p id="S3.SS1.p4.1" class="ltx_p">Recent work has demonstrated a power law relationship between data scale and model performance <cite class="ltx_cite ltx_citemacro_citep">(Hestness et al., <a href="#bib.bib47" title="" class="ltx_ref">2017</a>)</cite>. This implies that with enough data (<span id="S3.SS1.p4.1.1" class="ltx_text ltx_font_italic">e.g.</span>, consisting of the entire internet and all offline scientific measurements) one ought to converge to a very small solution set with irreducible error – the inherent epistemic uncertainty of the world.
As more models are trained on internet-scale data, the set of solutions that satisfies all data constraints must become relatively small.</p>
</div>
<div id="S3.SS1.p5" class="ltx_para ltx_noindent">
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Contrastive learning finds a distance structure over data samples that optimizes many classification tasks <cite class="ltx_cite ltx_citemacro_citep">(Arora et al., <a href="#bib.bib7" title="" class="ltx_ref">2019b</a>; Wang &amp; Isola, <a href="#bib.bib126" title="" class="ltx_ref">2020</a>; Tian et al., <a href="#bib.bib118" title="" class="ltx_ref">2020b</a>)</cite>. Masked Autoencoders <cite class="ltx_cite ltx_citemacro_citep">(He et al., <a href="#bib.bib45" title="" class="ltx_ref">2021</a>)</cite> optimize randomly sampled reconstruction tasks. In fact, autoregressive language modeling can also be seen as optimizing a diverse set of tasks <cite class="ltx_cite ltx_citemacro_citep">(Radford et al., <a href="#bib.bib98" title="" class="ltx_ref">2019</a>)</cite>. Such multi-task objectives may be more effective than single-task ones (<span id="S3.SS1.p5.1.1" class="ltx_text ltx_font_italic">e.g.</span>, ImageNet classification) due to the fact that they impose more task constraints on the representation, leading to a smaller and higher-quality solution space <cite class="ltx_cite ltx_citemacro_citep">(Chen et al., <a href="#bib.bib20" title="" class="ltx_ref">2020</a>; He et al., <a href="#bib.bib44" title="" class="ltx_ref">2020</a>; Radford et al., <a href="#bib.bib97" title="" class="ltx_ref">2017</a>, <a href="#bib.bib98" title="" class="ltx_ref">2019</a>)</cite>.</p>
</div>
</section>
<section id="S3.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.2 </span>Convergence via  <svg id="S3.SS2.1.pic1" class="ltx_picture" height="17.83" overflow="visible" version="1.1" width="101.63"><g transform="translate(0,17.83) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 98.86 17.83 C 100.39 17.83 101.63 16.6 101.63 15.07 L 101.63 2.77 C 101.63 1.24 100.39 0 98.86 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 98.86 17.83 C 100.39 17.83 101.63 16.6 101.63 15.07 L 101.63 2.77 C 101.63 1.24 100.39 0 98.86 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 5.46)"><text transform="matrix(1 0 0 -1 0 0)" color="#000000">Model Capacity</text></g></g></svg>
</h3>

<div id="S3.SS2.p1" class="ltx_para ltx_noindent">
<p id="S3.SS2.p1.2" class="ltx_p">Suppose there is a globally optimal representation for standard learning objectives. Then, under sufficient data, <span id="S3.SS2.p1.2.1" class="ltx_text ltx_font_italic">scaling</span> a model (<span id="S3.SS2.p1.2.2" class="ltx_text ltx_font_italic">i.e.</span>, using larger function classes
 <svg id="S3.SS2.p1.1.pic1" class="ltx_picture" height="14.99" overflow="visible" version="1.1" width="14.57"><g transform="translate(0,14.99) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 12.22 C 0 13.75 1.24 14.99 2.77 14.99 L 11.8 14.99 C 13.33 14.99 14.57 13.75 14.57 12.22 L 14.57 2.77 C 14.57 1.24 13.33 0 11.8 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 12.22 C 0 13.75 1.24 14.99 2.77 14.99 L 11.8 14.99 C 13.33 14.99 14.57 13.75 14.57 12.22 L 14.57 2.77 C 14.57 1.24 13.33 0 11.8 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 2.77)"><foreignObject width="9.03" height="9.46" transform="matrix(1 0 0 -1 0 16.6)" overflow="visible" color="#000000"><math id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1" class="ltx_Math" alttext="\mathcal{F}" display="inline"><semantics id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\mathcal{F}</annotation></semantics></math></foreignObject></g></g></svg>
), as well as  <svg id="S3.SS2.p1.2.pic2" class="ltx_picture" height="17.83" overflow="visible" version="1.1" width="142.02"><g transform="translate(0,17.83) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 139.25 17.83 C 140.78 17.83 142.02 16.6 142.02 15.07 L 142.02 2.77 C 142.02 1.24 140.78 0 139.25 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#B3B3FF" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 139.25 17.83 C 140.78 17.83 142.02 16.6 142.02 15.07 L 142.02 2.77 C 142.02 1.24 140.78 0 139.25 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 5.46)"><text transform="matrix(1 0 0 -1 0 0)" color="#000000">improved optimization</text></g></g></svg>
, should be more effective at finding better approximations to this optimum, as illustrated in <a href="#S3.F5" title="In 3 Why are representations converging? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">5</span></a>.
With the same training objective, larger models, even of different architectures, will thus tend to converge toward this optimum. When different training objectives share similar minimizers, larger models are better at finding these minimizers,
and will train to similar solutions over the training tasks. We summarize this hypothesis as follows:</p>
</div>
<div id="S3.SS2.p2" class="ltx_para ltx_noindent">
<svg id="S3.SS2.p2.pic1" class="ltx_picture" height="86.31" overflow="visible" version="1.1" width="600"><g transform="translate(0,86.31) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#4D4D4D" fill-opacity="1.0"><path d="M 0 5.91 L 0 80.4 C 0 83.66 2.64 86.31 5.91 86.31 L 594.09 86.31 C 597.36 86.31 600 83.66 600 80.4 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#FAFAFA" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 80.4 C 1.97 82.58 3.73 84.34 5.91 84.34 L 594.09 84.34 C 596.27 84.34 598.03 82.58 598.03 80.4 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 19.24 11.36)"><foreignObject width="561.52" height="62.11" transform="matrix(1 0 0 -1 0 16.6)" overflow="visible" color="#000000">
<span id="S3.SS2.p2.pic1.1.1.1.1.1" class="ltx_inline-block ltx_minipage ltx_align_bottom" style="width:405.8pt;">
<span id="S3.SS2.p2.pic1.1.1.1.1.1.1" class="ltx_p"><span id="S3.SS2.p2.pic1.1.1.1.1.1.1.1" class="ltx_text ltx_font_bold">The Capacity Hypothesis
<br class="ltx_break"></span>
<span id="S3.SS2.p2.pic1.1.1.1.1.1.1.2" class="ltx_text ltx_inline-block" style="width:433.6pt;"><span class="ltx_rule" style="width:433.6pt;height:0.4pt;background:black;display:inline-block;"> </span></span></span>
<span id="S3.SS2.p2.pic1.1.1.1.1.1.2" class="ltx_p">Bigger models are more likely to converge to a shared representation than smaller models.</span>
</span></foreignObject></g></g></svg>
</div>
</section>
<section id="S3.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.3 </span>Convergence via  <svg id="S3.SS3.1.pic1" class="ltx_picture" height="17.83" overflow="visible" version="1.1" width="96.9"><g transform="translate(0,17.83) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 94.13 17.83 C 95.66 17.83 96.9 16.6 96.9 15.07 L 96.9 2.77 C 96.9 1.24 95.66 0 94.13 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 94.13 17.83 C 95.66 17.83 96.9 16.6 96.9 15.07 L 96.9 2.77 C 96.9 1.24 95.66 0 94.13 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 5.46)"><text transform="matrix(1 0 0 -1 0 0)" color="#000000">Simplicity Bias</text></g></g></svg>
</h3>

<div id="S3.SS3.p1" class="ltx_para ltx_noindent">
<p id="S3.SS3.p1.1" class="ltx_p">Arriving at the same mapping on the <span id="S3.SS3.p1.1.1" class="ltx_text ltx_font_italic">training data</span> does not prohibit the models from developing distinct internal representations. It is not unreasonable to posit that the representations used to detect a dog in a 1M parameter model could be quite different than that used by a 1B parameter model. What would stop a billion-parameter (and counting) model from learning an overly complicated and distinct representation? One key factor might be simplicity bias:</p>
</div>
<div id="S3.SS3.p2" class="ltx_para ltx_noindent">
<svg id="S3.SS3.p2.pic1" class="ltx_picture" height="119.52" overflow="visible" version="1.1" width="600"><g transform="translate(0,119.52) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#4D4D4D" fill-opacity="1.0"><path d="M 0 5.91 L 0 113.61 C 0 116.87 2.64 119.52 5.91 119.52 L 594.09 119.52 C 597.36 119.52 600 116.87 600 113.61 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#FAFAFA" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 113.61 C 1.97 115.79 3.73 117.55 5.91 117.55 L 594.09 117.55 C 596.27 117.55 598.03 115.79 598.03 113.61 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 19.24 11.36)"><foreignObject width="561.52" height="95.32" transform="matrix(1 0 0 -1 0 16.6)" overflow="visible" color="#000000">
<span id="S3.SS3.p2.pic1.1.1.1.1.1" class="ltx_inline-block ltx_minipage ltx_align_bottom" style="width:405.8pt;">
<span id="S3.SS3.p2.pic1.1.1.1.1.1.1" class="ltx_p"><span id="S3.SS3.p2.pic1.1.1.1.1.1.1.1" class="ltx_text ltx_font_bold">The Simplicity Bias Hypothesis
<br class="ltx_break"></span>
<span id="S3.SS3.p2.pic1.1.1.1.1.1.1.2" class="ltx_text ltx_inline-block" style="width:433.6pt;"><span class="ltx_rule" style="width:433.6pt;height:0.4pt;background:black;display:inline-block;"> </span></span></span>
<span id="S3.SS3.p2.pic1.1.1.1.1.1.2" class="ltx_p">Deep networks are biased toward finding simple fits to the data, and the bigger the model, the stronger the bias. Therefore, as models get bigger, we should expect convergence to a smaller solution space.</span>
</span></foreignObject></g></g></svg>
</div>
<div id="S3.SS3.p3" class="ltx_para ltx_noindent">
<p id="S3.SS3.p3.2" class="ltx_p">Such simplicity bias could be coming from explicit regularization <svg id="S3.SS3.p3.1.m1.1.1.pic1" class="ltx_picture ltx_markedasmath" height="19.37" overflow="visible" version="1.1" width="34.75"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,19.37) matrix(1 0 0 -1 0 0)"><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 16.6 C 0 18.13 1.24 19.37 2.77 19.37 L 31.98 19.37 C 33.51 19.37 34.75 18.13 34.75 16.6 L 34.75 2.77 C 34.75 1.24 33.51 0 31.98 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 16.6 C 0 18.13 1.24 19.37 2.77 19.37 L 31.98 19.37 C 33.51 19.37 34.75 18.13 34.75 16.6 L 34.75 2.77 C 34.75 1.24 33.51 0 31.98 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 6.23)"><foreignObject height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="29.21" color="#000000"><math id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1" class="ltx_Math" alttext="\mathcal{R}(f)" display="inline"><semantics id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mrow id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.2" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.2.cmml">ℛ</mi><mo lspace="0em" rspace="0em" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.1" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.1.cmml">​</mo><mrow id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.3.2" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.cmml"><mo stretchy="false" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.3.2.1" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.cmml">(</mo><mi id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">f</mi><mo stretchy="false" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.3.2.2" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2"><times id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.1.cmml" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.1"></times><ci id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.2.cmml" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.2.2">ℛ</ci><ci id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\mathcal{R}(f)</annotation></semantics></math></foreignObject></g></g></svg> commonly used in deep learning (<span id="S3.SS3.p3.2.1" class="ltx_text ltx_font_italic">e.g.</span>, weight decay and dropout). However, even in the absence of external influences, deep networks naturally adhere to Occam’s razor,  <svg id="S3.SS3.p3.2.pic1" class="ltx_picture" height="17.83" overflow="visible" version="1.1" width="216.82"><g transform="translate(0,17.83) matrix(1 0 0 -1 0 0)" fill="#000000" stroke="#000000" stroke-width="0.4pt"><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 214.05 17.83 C 215.58 17.83 216.82 16.6 216.82 15.07 L 216.82 2.77 C 216.82 1.24 215.58 0 214.05 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill="#FFB3B3" fill-opacity="1.0"><path d="M 0 2.77 L 0 15.07 C 0 16.6 1.24 17.83 2.77 17.83 L 214.05 17.83 C 215.58 17.83 216.82 16.6 216.82 15.07 L 216.82 2.77 C 216.82 1.24 215.58 0 214.05 0 L 2.77 0 C 1.24 0 0 1.24 0 2.77 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 2.77 5.46)"><text transform="matrix(1 0 0 -1 0 0)" color="#000000">implicitly favoring simple solutions</text></g></g></svg>
that fit the data <cite class="ltx_cite ltx_citemacro_citep">(Solomonoff, <a href="#bib.bib109" title="" class="ltx_ref">1964</a>; Gunasekar et al., <a href="#bib.bib39" title="" class="ltx_ref">2018</a>; Arora et al., <a href="#bib.bib6" title="" class="ltx_ref">2019a</a>; Valle-Perez et al., <a href="#bib.bib125" title="" class="ltx_ref">2019</a>; Huh et al., <a href="#bib.bib49" title="" class="ltx_ref">2023</a>; Dingle et al., <a href="#bib.bib26" title="" class="ltx_ref">2018</a>; Goldblum et al., <a href="#bib.bib36" title="" class="ltx_ref">2023</a>)</cite>.
Figure <a href="#S3.F7" title="Figure 7 ‣ 3.3 Convergence via Simplicity Bias ‣ 3 Why are representations converging? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a> visualizes how simplicity bias can drive convergence.</p>
</div>
<figure id="S3.F7" class="ltx_figure"><img src="/html/2405.07987/assets/x9.png" id="S3.F7.g1" class="ltx_graphics ltx_centering ltx_img_square" width="380" height="331" alt="Refer to caption">
<br class="ltx_break ltx_centering">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span><span id="S3.F7.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">The Simplicity Bias Hypothesis:<span id="S3.F7.2.1.1" class="ltx_text ltx_font_medium"> Larger models have larger coverage of all possible ways to fit the same data. However, the implicit simplicity biases of deep networks encourage larger models to find the simplest of these solutions.</span></span></figcaption>
</figure>
</section>
</section>
<section id="S4" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">4 </span>What representation are we converging to?</h2>

<div id="S4.p1" class="ltx_para ltx_noindent">
<p id="S4.p1.1" class="ltx_p">By now, we hope to have convinced the reader that task and data pressures, combined with increasing model capacity, can lead to convergence. We next turn our attention to <span id="S4.p1.1.1" class="ltx_text ltx_font_italic">what</span> exactly is the endpoint of all this convergence.</p>
</div>
<div id="S4.p2" class="ltx_para ltx_noindent">
<p id="S4.p2.1" class="ltx_p">Our central hypothesis, stated in <a href="#S1.F1" title="In 1 Introduction ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>, is that the representation we are converging toward is a statistical model of the underlying reality that generates our observations. Consistent with the multitask scaling hypothesis, such a representation would naturally be useful toward many tasks (or at least toward any task grounded in reality). Additionally, this representation might be relatively simple, assuming that scientists
are correct in suggesting that the fundamental laws of nature are indeed simple functions <cite class="ltx_cite ltx_citemacro_citep">(Gell-Mann, <a href="#bib.bib33" title="" class="ltx_ref">1995</a>)</cite>, in line with the simplicity bias hypothesis.</p>
</div>
<div id="S4.p3" class="ltx_para ltx_noindent">
<p id="S4.p3.1" class="ltx_p">But what exactly do we mean by “a statistical model of the underlying reality.” In this section, we formalize one definition with concrete mathematical statements. <em id="S4.p3.1.1" class="ltx_emph ltx_font_italic">Importantly</em>, this section should be read as just one concrete candidate for the form of the platonic representation; other candidates could be arrived at from other modeling assumptions.</p>
</div>
<section id="S4.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.1 </span>An idealized world</h3>

<div id="S4.SS1.p1" class="ltx_para ltx_noindent">
<p id="S4.SS1.p1.4" class="ltx_p">We consider a world that works as follows, consistent with the cartoon in Figure <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>. The world consists of a sequence of <math id="S4.SS1.p1.1.m1.1" class="ltx_Math" alttext="T" display="inline"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">T</annotation></semantics></math> discrete events, denoted as <math id="S4.SS1.p1.2.m2.3" class="ltx_Math" alttext="\mathbf{Z}\triangleq[z_{1},\ldots,z_{T}]" display="inline"><semantics id="S4.SS1.p1.2.m2.3a"><mrow id="S4.SS1.p1.2.m2.3.3" xref="S4.SS1.p1.2.m2.3.3.cmml"><mi id="S4.SS1.p1.2.m2.3.3.4" xref="S4.SS1.p1.2.m2.3.3.4.cmml">𝐙</mi><mo id="S4.SS1.p1.2.m2.3.3.3" xref="S4.SS1.p1.2.m2.3.3.3.cmml">≜</mo><mrow id="S4.SS1.p1.2.m2.3.3.2.2" xref="S4.SS1.p1.2.m2.3.3.2.3.cmml"><mo stretchy="false" id="S4.SS1.p1.2.m2.3.3.2.2.3" xref="S4.SS1.p1.2.m2.3.3.2.3.cmml">[</mo><msub id="S4.SS1.p1.2.m2.2.2.1.1.1" xref="S4.SS1.p1.2.m2.2.2.1.1.1.cmml"><mi id="S4.SS1.p1.2.m2.2.2.1.1.1.2" xref="S4.SS1.p1.2.m2.2.2.1.1.1.2.cmml">z</mi><mn id="S4.SS1.p1.2.m2.2.2.1.1.1.3" xref="S4.SS1.p1.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p1.2.m2.3.3.2.2.4" xref="S4.SS1.p1.2.m2.3.3.2.3.cmml">,</mo><mi mathvariant="normal" id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml">…</mi><mo id="S4.SS1.p1.2.m2.3.3.2.2.5" xref="S4.SS1.p1.2.m2.3.3.2.3.cmml">,</mo><msub id="S4.SS1.p1.2.m2.3.3.2.2.2" xref="S4.SS1.p1.2.m2.3.3.2.2.2.cmml"><mi id="S4.SS1.p1.2.m2.3.3.2.2.2.2" xref="S4.SS1.p1.2.m2.3.3.2.2.2.2.cmml">z</mi><mi id="S4.SS1.p1.2.m2.3.3.2.2.2.3" xref="S4.SS1.p1.2.m2.3.3.2.2.2.3.cmml">T</mi></msub><mo stretchy="false" id="S4.SS1.p1.2.m2.3.3.2.2.6" xref="S4.SS1.p1.2.m2.3.3.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.3b"><apply id="S4.SS1.p1.2.m2.3.3.cmml" xref="S4.SS1.p1.2.m2.3.3"><ci id="S4.SS1.p1.2.m2.3.3.3.cmml" xref="S4.SS1.p1.2.m2.3.3.3">≜</ci><ci id="S4.SS1.p1.2.m2.3.3.4.cmml" xref="S4.SS1.p1.2.m2.3.3.4">𝐙</ci><list id="S4.SS1.p1.2.m2.3.3.2.3.cmml" xref="S4.SS1.p1.2.m2.3.3.2.2"><apply id="S4.SS1.p1.2.m2.2.2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.p1.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S4.SS1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S4.SS1.p1.2.m2.2.2.1.1.1.2">𝑧</ci><cn type="integer" id="S4.SS1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S4.SS1.p1.2.m2.2.2.1.1.1.3">1</cn></apply><ci id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">…</ci><apply id="S4.SS1.p1.2.m2.3.3.2.2.2.cmml" xref="S4.SS1.p1.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.3.3.2.2.2.1.cmml" xref="S4.SS1.p1.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S4.SS1.p1.2.m2.3.3.2.2.2.2.cmml" xref="S4.SS1.p1.2.m2.3.3.2.2.2.2">𝑧</ci><ci id="S4.SS1.p1.2.m2.3.3.2.2.2.3.cmml" xref="S4.SS1.p1.2.m2.3.3.2.2.2.3">𝑇</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.3c">\mathbf{Z}\triangleq[z_{1},\ldots,z_{T}]</annotation></semantics></math>, sampled from some unknown distribution <math id="S4.SS1.p1.3.m3.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S4.SS1.p1.3.m3.1a"><mrow id="S4.SS1.p1.3.m3.1.2" xref="S4.SS1.p1.3.m3.1.2.cmml"><mi mathvariant="normal" id="S4.SS1.p1.3.m3.1.2.2" xref="S4.SS1.p1.3.m3.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S4.SS1.p1.3.m3.1.2.1" xref="S4.SS1.p1.3.m3.1.2.1.cmml">​</mo><mrow id="S4.SS1.p1.3.m3.1.2.3.2" xref="S4.SS1.p1.3.m3.1.2.cmml"><mo stretchy="false" id="S4.SS1.p1.3.m3.1.2.3.2.1" xref="S4.SS1.p1.3.m3.1.2.cmml">(</mo><mi id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml">𝐙</mi><mo stretchy="false" id="S4.SS1.p1.3.m3.1.2.3.2.2" xref="S4.SS1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.2.cmml" xref="S4.SS1.p1.3.m3.1.2"><times id="S4.SS1.p1.3.m3.1.2.1.cmml" xref="S4.SS1.p1.3.m3.1.2.1"></times><ci id="S4.SS1.p1.3.m3.1.2.2.cmml" xref="S4.SS1.p1.3.m3.1.2.2">ℙ</ci><ci id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">\mathbb{P}(\mathbf{Z})</annotation></semantics></math>. Each event can be observed in various ways. An observation is a bijective, deterministic function <math id="S4.SS1.p1.4.m4.1" class="ltx_Math" alttext="\texttt{obs}:\mathcal{Z}\rightarrow\cdot{}\," display="inline"><semantics id="S4.SS1.p1.4.m4.1a"><mrow id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S4.SS1.p1.4.m4.1.1.2" xref="S4.SS1.p1.4.m4.1.1.2a.cmml">obs</mtext><mo lspace="0.278em" rspace="0.278em" id="S4.SS1.p1.4.m4.1.1.1" xref="S4.SS1.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p1.4.m4.1.1.3.2" xref="S4.SS1.p1.4.m4.1.1.3.2.cmml">𝒵</mi><mo rspace="0em" stretchy="false" id="S4.SS1.p1.4.m4.1.1.3.1" xref="S4.SS1.p1.4.m4.1.1.3.1.cmml">→</mo><mo lspace="0em" id="S4.SS1.p1.4.m4.1.1.3.3" xref="S4.SS1.p1.4.m4.1.1.3.3.cmml">⋅</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><ci id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1.1">:</ci><ci id="S4.SS1.p1.4.m4.1.1.2a.cmml" xref="S4.SS1.p1.4.m4.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2">obs</mtext></ci><apply id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3"><ci id="S4.SS1.p1.4.m4.1.1.3.1.cmml" xref="S4.SS1.p1.4.m4.1.1.3.1">→</ci><ci id="S4.SS1.p1.4.m4.1.1.3.2.cmml" xref="S4.SS1.p1.4.m4.1.1.3.2">𝒵</ci><ci id="S4.SS1.p1.4.m4.1.1.3.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3.3">⋅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">\texttt{obs}:\mathcal{Z}\rightarrow\cdot{}\,</annotation></semantics></math> that maps events to an arbitrary measurement space, such as pixels, sounds, mass, force, torque, words, etc.
Later, in <a href="#S6" title="6 Counterexamples and limitations ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a>, we discuss limitations and potential extensions to continuous and unbounded worlds, and stochastic observations, that could yield a model that better reflects real learning scenarios.</p>
</div>
<div id="S4.SS1.p2" class="ltx_para ltx_noindent">
<p id="S4.SS1.p2.1" class="ltx_p">One can think of an event as corresponding to the state of the world at some point in time<span id="footnote3" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>Here we only analyze temporal sequences, but note that the same could be done with respect to events laid out in space instead.</span></span></span>, but it is also fine to simply consider an event as any variable that indexes observations, with no further physical meaning<span id="footnote4" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>This latter interpretation may be more consistent with Plato’s intent. Scholars have argued that his allegory of the cave rejects any notion of a true world state <cite class="ltx_cite ltx_citemacro_citep">(Nettleship, <a href="#bib.bib85" title="" class="ltx_ref">1897</a>)</cite>. Instead, we could say that the joint distribution of observation indices is <span id="footnote4.1" class="ltx_text ltx_font_italic">itself</span> the platonic reality.</span></span></span>.</p>
</div>
<div id="S4.SS1.p3" class="ltx_para ltx_noindent">
<p id="S4.SS1.p3.2" class="ltx_p">In this idealized world, knowing <math id="S4.SS1.p3.1.m1.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S4.SS1.p3.1.m1.1a"><mrow id="S4.SS1.p3.1.m1.1.2" xref="S4.SS1.p3.1.m1.1.2.cmml"><mi mathvariant="normal" id="S4.SS1.p3.1.m1.1.2.2" xref="S4.SS1.p3.1.m1.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S4.SS1.p3.1.m1.1.2.1" xref="S4.SS1.p3.1.m1.1.2.1.cmml">​</mo><mrow id="S4.SS1.p3.1.m1.1.2.3.2" xref="S4.SS1.p3.1.m1.1.2.cmml"><mo stretchy="false" id="S4.SS1.p3.1.m1.1.2.3.2.1" xref="S4.SS1.p3.1.m1.1.2.cmml">(</mo><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">𝐙</mi><mo stretchy="false" id="S4.SS1.p3.1.m1.1.2.3.2.2" xref="S4.SS1.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.2"><times id="S4.SS1.p3.1.m1.1.2.1.cmml" xref="S4.SS1.p3.1.m1.1.2.1"></times><ci id="S4.SS1.p3.1.m1.1.2.2.cmml" xref="S4.SS1.p3.1.m1.1.2.2">ℙ</ci><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\mathbb{P}(\mathbf{Z})</annotation></semantics></math> would be useful for many kinds of predictions; this would constitute a world model over the events that cause our observations <cite class="ltx_cite ltx_citemacro_citep">(Werbos, <a href="#bib.bib127" title="" class="ltx_ref">1987</a>; Ha &amp; Schmidhuber, <a href="#bib.bib41" title="" class="ltx_ref">2018</a>; Richens &amp; Everitt, <a href="#bib.bib101" title="" class="ltx_ref">2024</a>)</cite>. We will next show that a particular representation of <math id="S4.SS1.p3.2.m2.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S4.SS1.p3.2.m2.1a"><mrow id="S4.SS1.p3.2.m2.1.2" xref="S4.SS1.p3.2.m2.1.2.cmml"><mi mathvariant="normal" id="S4.SS1.p3.2.m2.1.2.2" xref="S4.SS1.p3.2.m2.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S4.SS1.p3.2.m2.1.2.1" xref="S4.SS1.p3.2.m2.1.2.1.cmml">​</mo><mrow id="S4.SS1.p3.2.m2.1.2.3.2" xref="S4.SS1.p3.2.m2.1.2.cmml"><mo stretchy="false" id="S4.SS1.p3.2.m2.1.2.3.2.1" xref="S4.SS1.p3.2.m2.1.2.cmml">(</mo><mi id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">𝐙</mi><mo stretchy="false" id="S4.SS1.p3.2.m2.1.2.3.2.2" xref="S4.SS1.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.2.cmml" xref="S4.SS1.p3.2.m2.1.2"><times id="S4.SS1.p3.2.m2.1.2.1.cmml" xref="S4.SS1.p3.2.m2.1.2.1"></times><ci id="S4.SS1.p3.2.m2.1.2.2.cmml" xref="S4.SS1.p3.2.m2.1.2.2">ℙ</ci><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\mathbb{P}(\mathbf{Z})</annotation></semantics></math> is recovered by certain contrastive learners.</p>
</div>
<figure id="S4.F8" class="ltx_figure"><img src="/html/2405.07987/assets/x10.png" id="S4.F8.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="328" height="92" alt="Refer to caption">
<br class="ltx_break ltx_centering">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span><span id="S4.F8.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Color cooccurrence in VISION and LANGUAGE yields perceptual organization:<span id="S4.F8.2.1.1" class="ltx_text ltx_font_medium"> Similar representations of color are obtained via, </span>from LEFT to RIGHT<span id="S4.F8.2.1.2" class="ltx_text ltx_font_medium">, the perceptual layout from CIELAB color space, cooccurrence in CIFAR-10 images, and language cooccurrence modeling (<cite class="ltx_cite ltx_citemacro_citet">Gao et al. (<a href="#bib.bib31" title="" class="ltx_ref">2021</a>); Liu et al. (<a href="#bib.bib72" title="" class="ltx_ref">2019</a>)</cite>; computed roughly following <cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite>). Details in <a href="#A4" title="Appendix D Color Cooccurrence Experiment ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">D</span></a>.
</span></span></figcaption>
</figure>
</section>
<section id="S4.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.2 </span>A family of contrastive learners converge to a representation of <math id="S4.SS2.1.m1.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S4.SS2.1.m1.1b"><mrow id="S4.SS2.1.m1.1.2" xref="S4.SS2.1.m1.1.2.cmml"><mi mathvariant="normal" id="S4.SS2.1.m1.1.2.2" xref="S4.SS2.1.m1.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S4.SS2.1.m1.1.2.1" xref="S4.SS2.1.m1.1.2.1.cmml">​</mo><mrow id="S4.SS2.1.m1.1.2.3.2" xref="S4.SS2.1.m1.1.2.cmml"><mo stretchy="false" id="S4.SS2.1.m1.1.2.3.2.1" xref="S4.SS2.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.1.m1.1.1" xref="S4.SS2.1.m1.1.1.cmml">𝐙</mi><mo stretchy="false" id="S4.SS2.1.m1.1.2.3.2.2" xref="S4.SS2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.m1.1c"><apply id="S4.SS2.1.m1.1.2.cmml" xref="S4.SS2.1.m1.1.2"><times id="S4.SS2.1.m1.1.2.1.cmml" xref="S4.SS2.1.m1.1.2.1"></times><ci id="S4.SS2.1.m1.1.2.2.cmml" xref="S4.SS2.1.m1.1.2.2">ℙ</ci><ci id="S4.SS2.1.m1.1.1.cmml" xref="S4.SS2.1.m1.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.m1.1d">\mathbb{P}(\mathbf{Z})</annotation></semantics></math>
</h3>

<div id="S4.SS2.p1" class="ltx_para ltx_noindent">
<p id="S4.SS2.p1.4" class="ltx_p">Consider a contrastive learner that models observations that <span id="S4.SS2.p1.4.1" class="ltx_text ltx_font_italic">cooccur</span> together. For simplicity, we ground our discussion with the following definition of the <span id="S4.SS2.p1.4.2" class="ltx_text ltx_font_italic">cooccurrence probability</span>, <math id="S4.SS2.p1.1.m1.1" class="ltx_Math" alttext="{P}_{\mathsf{coor}}" display="inline"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">P</mi><mi id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑃</ci><ci id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3">𝖼𝗈𝗈𝗋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">{P}_{\mathsf{coor}}</annotation></semantics></math>, of two observations <math id="S4.SS2.p1.2.m2.1" class="ltx_Math" alttext="x_{a}" display="inline"><semantics id="S4.SS2.p1.2.m2.1a"><msub id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml">x</mi><mi id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝑥</ci><ci id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">x_{a}</annotation></semantics></math> and <math id="S4.SS2.p1.3.m3.1" class="ltx_Math" alttext="x_{b}" display="inline"><semantics id="S4.SS2.p1.3.m3.1a"><msub id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml">x</mi><mi id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2">𝑥</ci><ci id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">x_{b}</annotation></semantics></math> both occurring within some window <math id="S4.SS2.p1.4.m4.1" class="ltx_Math" alttext="T_{\mathsf{window}}" display="inline"><semantics id="S4.SS2.p1.4.m4.1a"><msub id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">T</mi><mi id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml">𝗐𝗂𝗇𝖽𝗈𝗐</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2">𝑇</ci><ci id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3">𝗐𝗂𝗇𝖽𝗈𝗐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">T_{\mathsf{window}}</annotation></semantics></math>:</p>
<table id="A6.EGx2" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.Ex2.m1.4" class="ltx_Math" alttext="\displaystyle{P}_{\mathsf{coor}}(x_{a},x_{b})\hskip 7.22743pt\propto\sum_{(t,t^{\prime})\colon\left\lvert t-t^{\prime}\right\rvert\leq T_{\mathsf{window}}}\mathbb{P}(X_{t}=x_{a},X_{t^{\prime}}=x_{b})." display="inline"><semantics id="S4.Ex2.m1.4a"><mrow id="S4.Ex2.m1.4.4.1" xref="S4.Ex2.m1.4.4.1.1.cmml"><mrow id="S4.Ex2.m1.4.4.1.1" xref="S4.Ex2.m1.4.4.1.1.cmml"><mrow id="S4.Ex2.m1.4.4.1.1.2" xref="S4.Ex2.m1.4.4.1.1.2.cmml"><msub id="S4.Ex2.m1.4.4.1.1.2.4" xref="S4.Ex2.m1.4.4.1.1.2.4.cmml"><mi id="S4.Ex2.m1.4.4.1.1.2.4.2" xref="S4.Ex2.m1.4.4.1.1.2.4.2.cmml">P</mi><mi id="S4.Ex2.m1.4.4.1.1.2.4.3" xref="S4.Ex2.m1.4.4.1.1.2.4.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="S4.Ex2.m1.4.4.1.1.2.3" xref="S4.Ex2.m1.4.4.1.1.2.3.cmml">​</mo><mrow id="S4.Ex2.m1.4.4.1.1.2.2.2" xref="S4.Ex2.m1.4.4.1.1.2.2.3.cmml"><mo stretchy="false" id="S4.Ex2.m1.4.4.1.1.2.2.2.3" 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xref="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3.1.cmml" xref="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3">subscript</csymbol><ci id="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3.2.cmml" xref="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3.2">𝑥</ci><ci id="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3.3.cmml" xref="S4.Ex2.m1.4.4.1.1.3.1.1.1.1.2.2.3.3">𝑏</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.4c">\displaystyle{P}_{\mathsf{coor}}(x_{a},x_{b})\hskip 7.22743pt\propto\sum_{(t,t^{\prime})\colon\left\lvert t-t^{\prime}\right\rvert\leq T_{\mathsf{window}}}\mathbb{P}(X_{t}=x_{a},X_{t^{\prime}}=x_{b}).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="S4.SS2.p1.7" class="ltx_p">Analogously, we can define <math id="S4.SS2.p1.5.m1.1" class="ltx_Math" alttext="{P}_{\mathsf{coor}}" display="inline"><semantics id="S4.SS2.p1.5.m1.1a"><msub id="S4.SS2.p1.5.m1.1.1" xref="S4.SS2.p1.5.m1.1.1.cmml"><mi id="S4.SS2.p1.5.m1.1.1.2" xref="S4.SS2.p1.5.m1.1.1.2.cmml">P</mi><mi id="S4.SS2.p1.5.m1.1.1.3" xref="S4.SS2.p1.5.m1.1.1.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m1.1b"><apply id="S4.SS2.p1.5.m1.1.1.cmml" xref="S4.SS2.p1.5.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.5.m1.1.1.1.cmml" xref="S4.SS2.p1.5.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.5.m1.1.1.2.cmml" xref="S4.SS2.p1.5.m1.1.1.2">𝑃</ci><ci id="S4.SS2.p1.5.m1.1.1.3.cmml" xref="S4.SS2.p1.5.m1.1.1.3">𝖼𝗈𝗈𝗋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m1.1c">{P}_{\mathsf{coor}}</annotation></semantics></math> for <math id="S4.SS2.p1.6.m2.1" class="ltx_Math" alttext="\mathbf{Z}" display="inline"><semantics id="S4.SS2.p1.6.m2.1a"><mi id="S4.SS2.p1.6.m2.1.1" xref="S4.SS2.p1.6.m2.1.1.cmml">𝐙</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m2.1b"><ci id="S4.SS2.p1.6.m2.1.1.cmml" xref="S4.SS2.p1.6.m2.1.1">𝐙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m2.1c">\mathbf{Z}</annotation></semantics></math> and other observation modalities. Note that <math id="S4.SS2.p1.7.m3.1" class="ltx_Math" alttext="{P}_{\mathsf{coor}}" display="inline"><semantics id="S4.SS2.p1.7.m3.1a"><msub id="S4.SS2.p1.7.m3.1.1" xref="S4.SS2.p1.7.m3.1.1.cmml"><mi id="S4.SS2.p1.7.m3.1.1.2" xref="S4.SS2.p1.7.m3.1.1.2.cmml">P</mi><mi id="S4.SS2.p1.7.m3.1.1.3" xref="S4.SS2.p1.7.m3.1.1.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m3.1b"><apply id="S4.SS2.p1.7.m3.1.1.cmml" xref="S4.SS2.p1.7.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m3.1.1.1.cmml" xref="S4.SS2.p1.7.m3.1.1">subscript</csymbol><ci id="S4.SS2.p1.7.m3.1.1.2.cmml" xref="S4.SS2.p1.7.m3.1.1.2">𝑃</ci><ci id="S4.SS2.p1.7.m3.1.1.3.cmml" xref="S4.SS2.p1.7.m3.1.1.3">𝖼𝗈𝗈𝗋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m3.1c">{P}_{\mathsf{coor}}</annotation></semantics></math> is symmetric.</p>
</div>
<div id="S4.SS2.p2" class="ltx_para ltx_noindent">
<p id="S4.SS2.p2.2" class="ltx_p">Consider <em id="S4.SS2.p2.2.1" class="ltx_emph ltx_font_italic">positive pairs</em> as two observations nearby in time (sampled from <math id="S4.SS2.p2.1.m1.1" class="ltx_Math" alttext="{P}_{\mathsf{coor}}" display="inline"><semantics id="S4.SS2.p2.1.m1.1a"><msub id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">P</mi><mi id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑃</ci><ci id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3">𝖼𝗈𝗈𝗋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">{P}_{\mathsf{coor}}</annotation></semantics></math>) and <em id="S4.SS2.p2.2.2" class="ltx_emph ltx_font_italic">negative pairs</em> as observations drawn from any point in time (sampled independently from the marginal). Our contrastive learner tries to classify if a pair is positive or negative by learning a representation <math id="S4.SS2.p2.2.m2.1" class="ltx_Math" alttext="f_{X}\colon X\rightarrow\mathbb{R}^{d}" display="inline"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><msub id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2.2" xref="S4.SS2.p2.2.m2.1.1.2.2.cmml">f</mi><mi id="S4.SS2.p2.2.m2.1.1.2.3" xref="S4.SS2.p2.2.m2.1.1.2.3.cmml">X</mi></msub><mo lspace="0.278em" rspace="0.278em" id="S4.SS2.p2.2.m2.1.1.1" xref="S4.SS2.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml"><mi id="S4.SS2.p2.2.m2.1.1.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.cmml">X</mi><mo stretchy="false" id="S4.SS2.p2.2.m2.1.1.3.1" xref="S4.SS2.p2.2.m2.1.1.3.1.cmml">→</mo><msup id="S4.SS2.p2.2.m2.1.1.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.cmml"><mi mathvariant="normal" id="S4.SS2.p2.2.m2.1.1.3.3.2" xref="S4.SS2.p2.2.m2.1.1.3.3.2.cmml">ℝ</mi><mi id="S4.SS2.p2.2.m2.1.1.3.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.3.cmml">d</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><ci id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1">:</ci><apply id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.2.1.cmml" xref="S4.SS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.2.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2.2">𝑓</ci><ci id="S4.SS2.p2.2.m2.1.1.2.3.cmml" xref="S4.SS2.p2.2.m2.1.1.2.3">𝑋</ci></apply><apply id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3"><ci id="S4.SS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.1">→</ci><ci id="S4.SS2.p2.2.m2.1.1.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.2">𝑋</ci><apply id="S4.SS2.p2.2.m2.1.1.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.3.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3">superscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.3.3.2.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3.2">ℝ</ci><ci id="S4.SS2.p2.2.m2.1.1.3.3.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3.3.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">f_{X}\colon X\rightarrow\mathbb{R}^{d}</annotation></semantics></math> such that the dot-product kernel approximates the log odds ratio up to some offset:</p>
<table id="A6.EGx3" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.E3.m1.2" class="ltx_Math" alttext="\displaystyle\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle" display="inline"><semantics id="S4.E3.m1.2a"><mrow id="S4.E3.m1.2.2.2" xref="S4.E3.m1.2.2.3.cmml"><mo stretchy="false" id="S4.E3.m1.2.2.2.3" xref="S4.E3.m1.2.2.3.cmml">⟨</mo><mrow id="S4.E3.m1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.cmml"><msub id="S4.E3.m1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.3.cmml"><mi id="S4.E3.m1.1.1.1.1.3.2" xref="S4.E3.m1.1.1.1.1.3.2.cmml">f</mi><mi id="S4.E3.m1.1.1.1.1.3.3" xref="S4.E3.m1.1.1.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E3.m1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.2.cmml">​</mo><mrow id="S4.E3.m1.1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.E3.m1.1.1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E3.m1.1.1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E3.m1.1.1.1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.E3.m1.1.1.1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="S4.E3.m1.1.1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E3.m1.2.2.2.4" xref="S4.E3.m1.2.2.3.cmml">,</mo><mrow id="S4.E3.m1.2.2.2.2" xref="S4.E3.m1.2.2.2.2.cmml"><msub id="S4.E3.m1.2.2.2.2.3" xref="S4.E3.m1.2.2.2.2.3.cmml"><mi id="S4.E3.m1.2.2.2.2.3.2" xref="S4.E3.m1.2.2.2.2.3.2.cmml">f</mi><mi id="S4.E3.m1.2.2.2.2.3.3" xref="S4.E3.m1.2.2.2.2.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E3.m1.2.2.2.2.2" xref="S4.E3.m1.2.2.2.2.2.cmml">​</mo><mrow id="S4.E3.m1.2.2.2.2.1.1" xref="S4.E3.m1.2.2.2.2.1.1.1.cmml"><mo stretchy="false" id="S4.E3.m1.2.2.2.2.1.1.2" xref="S4.E3.m1.2.2.2.2.1.1.1.cmml">(</mo><msub id="S4.E3.m1.2.2.2.2.1.1.1" xref="S4.E3.m1.2.2.2.2.1.1.1.cmml"><mi id="S4.E3.m1.2.2.2.2.1.1.1.2" xref="S4.E3.m1.2.2.2.2.1.1.1.2.cmml">x</mi><mi id="S4.E3.m1.2.2.2.2.1.1.1.3" xref="S4.E3.m1.2.2.2.2.1.1.1.3.cmml">b</mi></msub><mo stretchy="false" id="S4.E3.m1.2.2.2.2.1.1.3" xref="S4.E3.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S4.E3.m1.2.2.2.5" xref="S4.E3.m1.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E3.m1.2b"><list id="S4.E3.m1.2.2.3.cmml" xref="S4.E3.m1.2.2.2"><apply id="S4.E3.m1.1.1.1.1.cmml" xref="S4.E3.m1.1.1.1.1"><times id="S4.E3.m1.1.1.1.1.2.cmml" xref="S4.E3.m1.1.1.1.1.2"></times><apply id="S4.E3.m1.1.1.1.1.3.cmml" xref="S4.E3.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E3.m1.1.1.1.1.3.1.cmml" xref="S4.E3.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.E3.m1.1.1.1.1.3.2.cmml" xref="S4.E3.m1.1.1.1.1.3.2">𝑓</ci><ci id="S4.E3.m1.1.1.1.1.3.3.cmml" xref="S4.E3.m1.1.1.1.1.3.3">𝑋</ci></apply><apply id="S4.E3.m1.1.1.1.1.1.1.1.cmml" xref="S4.E3.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E3.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E3.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E3.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E3.m1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.E3.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E3.m1.1.1.1.1.1.1.1.3">𝑎</ci></apply></apply><apply id="S4.E3.m1.2.2.2.2.cmml" xref="S4.E3.m1.2.2.2.2"><times id="S4.E3.m1.2.2.2.2.2.cmml" xref="S4.E3.m1.2.2.2.2.2"></times><apply id="S4.E3.m1.2.2.2.2.3.cmml" xref="S4.E3.m1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.2.2.3.1.cmml" xref="S4.E3.m1.2.2.2.2.3">subscript</csymbol><ci id="S4.E3.m1.2.2.2.2.3.2.cmml" xref="S4.E3.m1.2.2.2.2.3.2">𝑓</ci><ci id="S4.E3.m1.2.2.2.2.3.3.cmml" xref="S4.E3.m1.2.2.2.2.3.3">𝑋</ci></apply><apply id="S4.E3.m1.2.2.2.2.1.1.1.cmml" xref="S4.E3.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E3.m1.2.2.2.2.1.1.1.1.cmml" xref="S4.E3.m1.2.2.2.2.1.1">subscript</csymbol><ci id="S4.E3.m1.2.2.2.2.1.1.1.2.cmml" xref="S4.E3.m1.2.2.2.2.1.1.1.2">𝑥</ci><ci id="S4.E3.m1.2.2.2.2.1.1.1.3.cmml" xref="S4.E3.m1.2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.2c">\displaystyle\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.E3.m2.3" class="ltx_Math" alttext="\displaystyle\approx\log\frac{\mathbb{P}(\texttt{pos}\mathrel{|}x_{a},x_{b})}{\mathbb{P}(\texttt{neg}\mathrel{|}x_{a},x_{b})}+\tilde{c}_{X}(x_{a})" display="inline"><semantics id="S4.E3.m2.3a"><mrow id="S4.E3.m2.3.3" xref="S4.E3.m2.3.3.cmml"><mi id="S4.E3.m2.3.3.3" xref="S4.E3.m2.3.3.3.cmml"></mi><mo id="S4.E3.m2.3.3.2" xref="S4.E3.m2.3.3.2.cmml">≈</mo><mrow id="S4.E3.m2.3.3.1" xref="S4.E3.m2.3.3.1.cmml"><mrow id="S4.E3.m2.3.3.1.3" xref="S4.E3.m2.3.3.1.3.cmml"><mi id="S4.E3.m2.3.3.1.3.1" xref="S4.E3.m2.3.3.1.3.1.cmml">log</mi><mo lspace="0.167em" id="S4.E3.m2.3.3.1.3a" xref="S4.E3.m2.3.3.1.3.cmml">⁡</mo><mstyle displaystyle="true" id="S4.E3.m2.2.2" xref="S4.E3.m2.2.2.cmml"><mfrac id="S4.E3.m2.2.2a" xref="S4.E3.m2.2.2.cmml"><mrow id="S4.E3.m2.1.1.1" xref="S4.E3.m2.1.1.1.cmml"><mi mathvariant="normal" id="S4.E3.m2.1.1.1.3" xref="S4.E3.m2.1.1.1.3.cmml">ℙ</mi><mo lspace="0em" rspace="0em" 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td>
</tr></tbody>
<tbody id="S4.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td>
</tr></tbody>
<tbody id="S4.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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id="S4.E5.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.E5.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E5.m1.1.1.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.E5.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S4.E5.m1.1.1.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.1.1.2.2.2.2.2.1.cmml" xref="S4.E5.m1.1.1.1.1.2.2.2.2.2">subscript</csymbol><ci id="S4.E5.m1.1.1.1.1.2.2.2.2.2.2.cmml" xref="S4.E5.m1.1.1.1.1.2.2.2.2.2.2">𝑥</ci><ci id="S4.E5.m1.1.1.1.1.2.2.2.2.2.3.cmml" xref="S4.E5.m1.1.1.1.1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply><apply id="S4.E5.m1.1.1.1.1.3.3.cmml" xref="S4.E5.m1.1.1.1.1.3.3"><times id="S4.E5.m1.1.1.1.1.3.3.2.cmml" xref="S4.E5.m1.1.1.1.1.3.3.2"></times><apply id="S4.E5.m1.1.1.1.1.3.3.3.cmml" xref="S4.E5.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.1.1.3.3.3.1.cmml" xref="S4.E5.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S4.E5.m1.1.1.1.1.3.3.3.2.cmml" xref="S4.E5.m1.1.1.1.1.3.3.3.2">𝑐</ci><ci id="S4.E5.m1.1.1.1.1.3.3.3.3.cmml" xref="S4.E5.m1.1.1.1.1.3.3.3.3">𝑋</ci></apply><apply id="S4.E5.m1.1.1.1.1.3.3.1.1.1.cmml" xref="S4.E5.m1.1.1.1.1.3.3.1.1"><csymbol cd="ambiguous" id="S4.E5.m1.1.1.1.1.3.3.1.1.1.1.cmml" xref="S4.E5.m1.1.1.1.1.3.3.1.1">subscript</csymbol><ci id="S4.E5.m1.1.1.1.1.3.3.1.1.1.2.cmml" xref="S4.E5.m1.1.1.1.1.3.3.1.1.1.2">𝑥</ci><ci id="S4.E5.m1.1.1.1.1.3.3.1.1.1.3.cmml" xref="S4.E5.m1.1.1.1.1.3.3.1.1.1.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.1c">\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X}(x_{a}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td>
</tr></tbody>
</table>
<p id="S4.SS2.p2.5" class="ltx_p">where <math id="S4.SS2.p2.3.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="S4.SS2.p2.3.m1.1a"><msub id="S4.SS2.p2.3.m1.1.1" xref="S4.SS2.p2.3.m1.1.1.cmml"><mi id="S4.SS2.p2.3.m1.1.1.2" xref="S4.SS2.p2.3.m1.1.1.2.cmml">K</mi><mi id="S4.SS2.p2.3.m1.1.1.3" xref="S4.SS2.p2.3.m1.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m1.1b"><apply id="S4.SS2.p2.3.m1.1.1.cmml" xref="S4.SS2.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m1.1.1.1.cmml" xref="S4.SS2.p2.3.m1.1.1">subscript</csymbol><ci id="S4.SS2.p2.3.m1.1.1.2.cmml" xref="S4.SS2.p2.3.m1.1.1.2">𝐾</ci><ci id="S4.SS2.p2.3.m1.1.1.3.cmml" xref="S4.SS2.p2.3.m1.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m1.1c">K_{\mathsf{PMI}}</annotation></semantics></math> is the pointwise mutual information (PMI) kernel, and <math id="S4.SS2.p2.4.m2.1" class="ltx_Math" alttext="c_{X}(x_{a})" display="inline"><semantics id="S4.SS2.p2.4.m2.1a"><mrow id="S4.SS2.p2.4.m2.1.1" xref="S4.SS2.p2.4.m2.1.1.cmml"><msub id="S4.SS2.p2.4.m2.1.1.3" xref="S4.SS2.p2.4.m2.1.1.3.cmml"><mi id="S4.SS2.p2.4.m2.1.1.3.2" xref="S4.SS2.p2.4.m2.1.1.3.2.cmml">c</mi><mi id="S4.SS2.p2.4.m2.1.1.3.3" xref="S4.SS2.p2.4.m2.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.SS2.p2.4.m2.1.1.2" xref="S4.SS2.p2.4.m2.1.1.2.cmml">​</mo><mrow id="S4.SS2.p2.4.m2.1.1.1.1" xref="S4.SS2.p2.4.m2.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.SS2.p2.4.m2.1.1.1.1.2" xref="S4.SS2.p2.4.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p2.4.m2.1.1.1.1.1" xref="S4.SS2.p2.4.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.p2.4.m2.1.1.1.1.1.2" xref="S4.SS2.p2.4.m2.1.1.1.1.1.2.cmml">x</mi><mi id="S4.SS2.p2.4.m2.1.1.1.1.1.3" xref="S4.SS2.p2.4.m2.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="S4.SS2.p2.4.m2.1.1.1.1.3" xref="S4.SS2.p2.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m2.1b"><apply id="S4.SS2.p2.4.m2.1.1.cmml" xref="S4.SS2.p2.4.m2.1.1"><times id="S4.SS2.p2.4.m2.1.1.2.cmml" xref="S4.SS2.p2.4.m2.1.1.2"></times><apply id="S4.SS2.p2.4.m2.1.1.3.cmml" xref="S4.SS2.p2.4.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m2.1.1.3.1.cmml" xref="S4.SS2.p2.4.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.p2.4.m2.1.1.3.2.cmml" xref="S4.SS2.p2.4.m2.1.1.3.2">𝑐</ci><ci id="S4.SS2.p2.4.m2.1.1.3.3.cmml" xref="S4.SS2.p2.4.m2.1.1.3.3">𝑋</ci></apply><apply id="S4.SS2.p2.4.m2.1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.p2.4.m2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p2.4.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.p2.4.m2.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS2.p2.4.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.p2.4.m2.1.1.1.1.1.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m2.1c">c_{X}(x_{a})</annotation></semantics></math> is constant in <math id="S4.SS2.p2.5.m3.1" class="ltx_Math" alttext="x_{b}" display="inline"><semantics id="S4.SS2.p2.5.m3.1a"><msub id="S4.SS2.p2.5.m3.1.1" xref="S4.SS2.p2.5.m3.1.1.cmml"><mi id="S4.SS2.p2.5.m3.1.1.2" xref="S4.SS2.p2.5.m3.1.1.2.cmml">x</mi><mi id="S4.SS2.p2.5.m3.1.1.3" xref="S4.SS2.p2.5.m3.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m3.1b"><apply id="S4.SS2.p2.5.m3.1.1.cmml" xref="S4.SS2.p2.5.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.5.m3.1.1.1.cmml" xref="S4.SS2.p2.5.m3.1.1">subscript</csymbol><ci id="S4.SS2.p2.5.m3.1.1.2.cmml" xref="S4.SS2.p2.5.m3.1.1.2">𝑥</ci><ci id="S4.SS2.p2.5.m3.1.1.3.cmml" xref="S4.SS2.p2.5.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m3.1c">x_{b}</annotation></semantics></math>. We note that is a common setting for self-supervised contrastive learners with NCE objectives <cite class="ltx_cite ltx_citemacro_citep">(Gutmann &amp; Hyvärinen, <a href="#bib.bib40" title="" class="ltx_ref">2010</a>; Oord et al., <a href="#bib.bib91" title="" class="ltx_ref">2018</a>)</cite>, such as SimCLR <cite class="ltx_cite ltx_citemacro_citep">(Chen et al., <a href="#bib.bib20" title="" class="ltx_ref">2020</a>)</cite> and SimCSE <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib31" title="" class="ltx_ref">2021</a>)</cite>. (See <cite class="ltx_cite ltx_citemacro_citet">Oord et al. (<a href="#bib.bib91" title="" class="ltx_ref">2018</a>)</cite> and <a href="#A6.SS1" title="F.1 Contrastive objectives learn pointwise mutual information ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">F.1</span></a> for details.)</p>
</div>
<div id="S4.SS2.p3" class="ltx_para ltx_noindent">
<p id="S4.SS2.p3.2" class="ltx_p">Under mild conditions that the world is smooth enough (see <a href="#A6.SS2" title="F.2 Contrastive learners can represent 𝐾_𝖯𝖬𝖨 exactly under smoothness conditions ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">F.2</span></a>), a choice of <math id="S4.SS2.p3.1.m1.1" class="ltx_Math" alttext="f_{X}" display="inline"><semantics id="S4.SS2.p3.1.m1.1a"><msub id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">f</mi><mi id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2">𝑓</ci><ci id="S4.SS2.p3.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">f_{X}</annotation></semantics></math> can exactly represent <math id="S4.SS2.p3.2.m2.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="S4.SS2.p3.2.m2.1a"><msub id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.2" xref="S4.SS2.p3.2.m2.1.1.2.cmml">K</mi><mi id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.2">𝐾</ci><ci id="S4.SS2.p3.2.m2.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">K_{\mathsf{PMI}}</annotation></semantics></math>:</p>
<table id="A6.EGx4" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.E6.m1.2" class="ltx_Math" alttext="\displaystyle\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle" display="inline"><semantics id="S4.E6.m1.2a"><mrow id="S4.E6.m1.2.2.2" xref="S4.E6.m1.2.2.3.cmml"><mo stretchy="false" id="S4.E6.m1.2.2.2.3" xref="S4.E6.m1.2.2.3.cmml">⟨</mo><mrow id="S4.E6.m1.1.1.1.1" xref="S4.E6.m1.1.1.1.1.cmml"><msub id="S4.E6.m1.1.1.1.1.3" xref="S4.E6.m1.1.1.1.1.3.cmml"><mi id="S4.E6.m1.1.1.1.1.3.2" xref="S4.E6.m1.1.1.1.1.3.2.cmml">f</mi><mi id="S4.E6.m1.1.1.1.1.3.3" xref="S4.E6.m1.1.1.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E6.m1.1.1.1.1.2" xref="S4.E6.m1.1.1.1.1.2.cmml">​</mo><mrow id="S4.E6.m1.1.1.1.1.1.1" xref="S4.E6.m1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.E6.m1.1.1.1.1.1.1.2" xref="S4.E6.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E6.m1.1.1.1.1.1.1.1" xref="S4.E6.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.E6.m1.1.1.1.1.1.1.1.2" xref="S4.E6.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.E6.m1.1.1.1.1.1.1.1.3" xref="S4.E6.m1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="S4.E6.m1.1.1.1.1.1.1.3" xref="S4.E6.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E6.m1.2.2.2.4" xref="S4.E6.m1.2.2.3.cmml">,</mo><mrow id="S4.E6.m1.2.2.2.2" xref="S4.E6.m1.2.2.2.2.cmml"><msub id="S4.E6.m1.2.2.2.2.3" xref="S4.E6.m1.2.2.2.2.3.cmml"><mi id="S4.E6.m1.2.2.2.2.3.2" xref="S4.E6.m1.2.2.2.2.3.2.cmml">f</mi><mi id="S4.E6.m1.2.2.2.2.3.3" xref="S4.E6.m1.2.2.2.2.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E6.m1.2.2.2.2.2" xref="S4.E6.m1.2.2.2.2.2.cmml">​</mo><mrow id="S4.E6.m1.2.2.2.2.1.1" xref="S4.E6.m1.2.2.2.2.1.1.1.cmml"><mo stretchy="false" id="S4.E6.m1.2.2.2.2.1.1.2" xref="S4.E6.m1.2.2.2.2.1.1.1.cmml">(</mo><msub id="S4.E6.m1.2.2.2.2.1.1.1" xref="S4.E6.m1.2.2.2.2.1.1.1.cmml"><mi id="S4.E6.m1.2.2.2.2.1.1.1.2" xref="S4.E6.m1.2.2.2.2.1.1.1.2.cmml">x</mi><mi id="S4.E6.m1.2.2.2.2.1.1.1.3" xref="S4.E6.m1.2.2.2.2.1.1.1.3.cmml">b</mi></msub><mo stretchy="false" id="S4.E6.m1.2.2.2.2.1.1.3" xref="S4.E6.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S4.E6.m1.2.2.2.5" xref="S4.E6.m1.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.2b"><list id="S4.E6.m1.2.2.3.cmml" xref="S4.E6.m1.2.2.2"><apply id="S4.E6.m1.1.1.1.1.cmml" xref="S4.E6.m1.1.1.1.1"><times id="S4.E6.m1.1.1.1.1.2.cmml" xref="S4.E6.m1.1.1.1.1.2"></times><apply id="S4.E6.m1.1.1.1.1.3.cmml" xref="S4.E6.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.1.1.3.1.cmml" xref="S4.E6.m1.1.1.1.1.3">subscript</csymbol><ci id="S4.E6.m1.1.1.1.1.3.2.cmml" xref="S4.E6.m1.1.1.1.1.3.2">𝑓</ci><ci id="S4.E6.m1.1.1.1.1.3.3.cmml" xref="S4.E6.m1.1.1.1.1.3.3">𝑋</ci></apply><apply id="S4.E6.m1.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E6.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E6.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E6.m1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.E6.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.E6.m1.1.1.1.1.1.1.1.3">𝑎</ci></apply></apply><apply id="S4.E6.m1.2.2.2.2.cmml" xref="S4.E6.m1.2.2.2.2"><times id="S4.E6.m1.2.2.2.2.2.cmml" xref="S4.E6.m1.2.2.2.2.2"></times><apply id="S4.E6.m1.2.2.2.2.3.cmml" xref="S4.E6.m1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.2.2.3.1.cmml" xref="S4.E6.m1.2.2.2.2.3">subscript</csymbol><ci id="S4.E6.m1.2.2.2.2.3.2.cmml" xref="S4.E6.m1.2.2.2.2.3.2">𝑓</ci><ci id="S4.E6.m1.2.2.2.2.3.3.cmml" xref="S4.E6.m1.2.2.2.2.3.3">𝑋</ci></apply><apply id="S4.E6.m1.2.2.2.2.1.1.1.cmml" xref="S4.E6.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.2.2.2.2.1.1.1.1.cmml" xref="S4.E6.m1.2.2.2.2.1.1">subscript</csymbol><ci id="S4.E6.m1.2.2.2.2.1.1.1.2.cmml" xref="S4.E6.m1.2.2.2.2.1.1.1.2">𝑥</ci><ci id="S4.E6.m1.2.2.2.2.1.1.1.3.cmml" xref="S4.E6.m1.2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.2c">\displaystyle\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.E6.m2.1" class="ltx_Math" alttext="\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X}," display="inline"><semantics id="S4.E6.m2.1a"><mrow id="S4.E6.m2.1.1.1" xref="S4.E6.m2.1.1.1.1.cmml"><mrow id="S4.E6.m2.1.1.1.1" xref="S4.E6.m2.1.1.1.1.cmml"><mi id="S4.E6.m2.1.1.1.1.4" xref="S4.E6.m2.1.1.1.1.4.cmml"></mi><mo id="S4.E6.m2.1.1.1.1.3" xref="S4.E6.m2.1.1.1.1.3.cmml">=</mo><mrow id="S4.E6.m2.1.1.1.1.2" xref="S4.E6.m2.1.1.1.1.2.cmml"><mrow id="S4.E6.m2.1.1.1.1.2.2" xref="S4.E6.m2.1.1.1.1.2.2.cmml"><msub id="S4.E6.m2.1.1.1.1.2.2.4" xref="S4.E6.m2.1.1.1.1.2.2.4.cmml"><mi id="S4.E6.m2.1.1.1.1.2.2.4.2" xref="S4.E6.m2.1.1.1.1.2.2.4.2.cmml">K</mi><mi id="S4.E6.m2.1.1.1.1.2.2.4.3" xref="S4.E6.m2.1.1.1.1.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="S4.E6.m2.1.1.1.1.2.2.3" xref="S4.E6.m2.1.1.1.1.2.2.3.cmml">​</mo><mrow id="S4.E6.m2.1.1.1.1.2.2.2.2" xref="S4.E6.m2.1.1.1.1.2.2.2.3.cmml"><mo stretchy="false" id="S4.E6.m2.1.1.1.1.2.2.2.2.3" xref="S4.E6.m2.1.1.1.1.2.2.2.3.cmml">(</mo><msub id="S4.E6.m2.1.1.1.1.1.1.1.1.1" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E6.m2.1.1.1.1.1.1.1.1.1.2" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.E6.m2.1.1.1.1.1.1.1.1.1.3" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.E6.m2.1.1.1.1.2.2.2.2.4" xref="S4.E6.m2.1.1.1.1.2.2.2.3.cmml">,</mo><msub id="S4.E6.m2.1.1.1.1.2.2.2.2.2" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2.cmml"><mi id="S4.E6.m2.1.1.1.1.2.2.2.2.2.2" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2.2.cmml">x</mi><mi id="S4.E6.m2.1.1.1.1.2.2.2.2.2.3" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.E6.m2.1.1.1.1.2.2.2.2.5" xref="S4.E6.m2.1.1.1.1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.E6.m2.1.1.1.1.2.3" xref="S4.E6.m2.1.1.1.1.2.3.cmml">+</mo><msub id="S4.E6.m2.1.1.1.1.2.4" xref="S4.E6.m2.1.1.1.1.2.4.cmml"><mi id="S4.E6.m2.1.1.1.1.2.4.2" xref="S4.E6.m2.1.1.1.1.2.4.2.cmml">c</mi><mi id="S4.E6.m2.1.1.1.1.2.4.3" xref="S4.E6.m2.1.1.1.1.2.4.3.cmml">X</mi></msub></mrow></mrow><mo id="S4.E6.m2.1.1.1.2" xref="S4.E6.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m2.1b"><apply id="S4.E6.m2.1.1.1.1.cmml" xref="S4.E6.m2.1.1.1"><eq id="S4.E6.m2.1.1.1.1.3.cmml" xref="S4.E6.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S4.E6.m2.1.1.1.1.4.cmml" xref="S4.E6.m2.1.1.1.1.4">absent</csymbol><apply id="S4.E6.m2.1.1.1.1.2.cmml" xref="S4.E6.m2.1.1.1.1.2"><plus id="S4.E6.m2.1.1.1.1.2.3.cmml" xref="S4.E6.m2.1.1.1.1.2.3"></plus><apply id="S4.E6.m2.1.1.1.1.2.2.cmml" xref="S4.E6.m2.1.1.1.1.2.2"><times id="S4.E6.m2.1.1.1.1.2.2.3.cmml" xref="S4.E6.m2.1.1.1.1.2.2.3"></times><apply id="S4.E6.m2.1.1.1.1.2.2.4.cmml" xref="S4.E6.m2.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="S4.E6.m2.1.1.1.1.2.2.4.1.cmml" xref="S4.E6.m2.1.1.1.1.2.2.4">subscript</csymbol><ci id="S4.E6.m2.1.1.1.1.2.2.4.2.cmml" xref="S4.E6.m2.1.1.1.1.2.2.4.2">𝐾</ci><ci id="S4.E6.m2.1.1.1.1.2.2.4.3.cmml" xref="S4.E6.m2.1.1.1.1.2.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="S4.E6.m2.1.1.1.1.2.2.2.3.cmml" xref="S4.E6.m2.1.1.1.1.2.2.2.2"><apply id="S4.E6.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E6.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.E6.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E6.m2.1.1.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.E6.m2.1.1.1.1.2.2.2.2.2.cmml" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E6.m2.1.1.1.1.2.2.2.2.2.1.cmml" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2">subscript</csymbol><ci id="S4.E6.m2.1.1.1.1.2.2.2.2.2.2.cmml" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2.2">𝑥</ci><ci id="S4.E6.m2.1.1.1.1.2.2.2.2.2.3.cmml" xref="S4.E6.m2.1.1.1.1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply><apply id="S4.E6.m2.1.1.1.1.2.4.cmml" xref="S4.E6.m2.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S4.E6.m2.1.1.1.1.2.4.1.cmml" xref="S4.E6.m2.1.1.1.1.2.4">subscript</csymbol><ci id="S4.E6.m2.1.1.1.1.2.4.2.cmml" xref="S4.E6.m2.1.1.1.1.2.4.2">𝑐</ci><ci id="S4.E6.m2.1.1.1.1.2.4.3.cmml" xref="S4.E6.m2.1.1.1.1.2.4.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m2.1c">\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td>
</tr></tbody>
</table>
<p id="S4.SS2.p3.3" class="ltx_p">where we observed that <math id="S4.SS2.p3.3.m1.1" class="ltx_Math" alttext="c_{X}(x_{a})" display="inline"><semantics id="S4.SS2.p3.3.m1.1a"><mrow id="S4.SS2.p3.3.m1.1.1" xref="S4.SS2.p3.3.m1.1.1.cmml"><msub id="S4.SS2.p3.3.m1.1.1.3" xref="S4.SS2.p3.3.m1.1.1.3.cmml"><mi id="S4.SS2.p3.3.m1.1.1.3.2" xref="S4.SS2.p3.3.m1.1.1.3.2.cmml">c</mi><mi id="S4.SS2.p3.3.m1.1.1.3.3" xref="S4.SS2.p3.3.m1.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.SS2.p3.3.m1.1.1.2" xref="S4.SS2.p3.3.m1.1.1.2.cmml">​</mo><mrow id="S4.SS2.p3.3.m1.1.1.1.1" xref="S4.SS2.p3.3.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.SS2.p3.3.m1.1.1.1.1.2" xref="S4.SS2.p3.3.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p3.3.m1.1.1.1.1.1" xref="S4.SS2.p3.3.m1.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.3.m1.1.1.1.1.1.2" xref="S4.SS2.p3.3.m1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.SS2.p3.3.m1.1.1.1.1.1.3" xref="S4.SS2.p3.3.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="S4.SS2.p3.3.m1.1.1.1.1.3" xref="S4.SS2.p3.3.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m1.1b"><apply id="S4.SS2.p3.3.m1.1.1.cmml" xref="S4.SS2.p3.3.m1.1.1"><times id="S4.SS2.p3.3.m1.1.1.2.cmml" xref="S4.SS2.p3.3.m1.1.1.2"></times><apply id="S4.SS2.p3.3.m1.1.1.3.cmml" xref="S4.SS2.p3.3.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m1.1.1.3.1.cmml" xref="S4.SS2.p3.3.m1.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.3.m1.1.1.3.2.cmml" xref="S4.SS2.p3.3.m1.1.1.3.2">𝑐</ci><ci id="S4.SS2.p3.3.m1.1.1.3.3.cmml" xref="S4.SS2.p3.3.m1.1.1.3.3">𝑋</ci></apply><apply id="S4.SS2.p3.3.m1.1.1.1.1.1.cmml" xref="S4.SS2.p3.3.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.3.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.3.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.3.m1.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS2.p3.3.m1.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.3.m1.1.1.1.1.1.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m1.1c">c_{X}(x_{a})</annotation></semantics></math> from <a href="#S4.E5" title="In 4.2 A family of contrastive learners converge to a representation of ℙ⁢(𝐙) ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">5</span></a> must be a constant since both sides are symmetric.</p>
</div>
<div id="S4.SS2.p4" class="ltx_para ltx_noindent">
<p id="S4.SS2.p4.2" class="ltx_p">Therefore, the contrastive learners we consider are minimized by a representation <math id="S4.SS2.p4.1.m1.1" class="ltx_Math" alttext="f_{X}" display="inline"><semantics id="S4.SS2.p4.1.m1.1a"><msub id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">f</mi><mi id="S4.SS2.p4.1.m1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2">𝑓</ci><ci id="S4.SS2.p4.1.m1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">f_{X}</annotation></semantics></math> whose kernel is <math id="S4.SS2.p4.2.m2.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="S4.SS2.p4.2.m2.1a"><msub id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml"><mi id="S4.SS2.p4.2.m2.1.1.2" xref="S4.SS2.p4.2.m2.1.1.2.cmml">K</mi><mi id="S4.SS2.p4.2.m2.1.1.3" xref="S4.SS2.p4.2.m2.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><apply id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.2.m2.1.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p4.2.m2.1.1.2.cmml" xref="S4.SS2.p4.2.m2.1.1.2">𝐾</ci><ci id="S4.SS2.p4.2.m2.1.1.3.cmml" xref="S4.SS2.p4.2.m2.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">K_{\mathsf{PMI}}</annotation></semantics></math> (up to a constant offset). With sufficient data and optimization, we will observe convergence to this point.</p>
</div>
<div id="S4.SS2.p5" class="ltx_para ltx_noindent">
<p id="S4.SS2.p5.2" class="ltx_p">Thus we have convergence to a representation of the statistics of <math id="S4.SS2.p5.1.m1.1" class="ltx_Math" alttext="X" display="inline"><semantics id="S4.SS2.p5.1.m1.1a"><mi id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><ci id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">X</annotation></semantics></math>, but what about <math id="S4.SS2.p5.2.m2.1" class="ltx_Math" alttext="Z" display="inline"><semantics id="S4.SS2.p5.2.m2.1a"><mi id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><ci id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">Z</annotation></semantics></math>? Recall that our idealized world consists of <span id="S4.SS2.p5.2.1" class="ltx_text ltx_font_italic">bijective</span> observation functions, which, over discrete random variables, preserve probabilities. So we have:</p>
<table id="A6.EGx5" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.Ex3.m1.2" class="ltx_Math" alttext="\displaystyle{P}_{\mathsf{coor}}(x_{a},x_{b})" display="inline"><semantics id="S4.Ex3.m1.2a"><mrow id="S4.Ex3.m1.2.2" xref="S4.Ex3.m1.2.2.cmml"><msub id="S4.Ex3.m1.2.2.4" xref="S4.Ex3.m1.2.2.4.cmml"><mi id="S4.Ex3.m1.2.2.4.2" xref="S4.Ex3.m1.2.2.4.2.cmml">P</mi><mi id="S4.Ex3.m1.2.2.4.3" xref="S4.Ex3.m1.2.2.4.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="S4.Ex3.m1.2.2.3" xref="S4.Ex3.m1.2.2.3.cmml">​</mo><mrow id="S4.Ex3.m1.2.2.2.2" xref="S4.Ex3.m1.2.2.2.3.cmml"><mo stretchy="false" id="S4.Ex3.m1.2.2.2.2.3" xref="S4.Ex3.m1.2.2.2.3.cmml">(</mo><msub id="S4.Ex3.m1.1.1.1.1.1" xref="S4.Ex3.m1.1.1.1.1.1.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.2" xref="S4.Ex3.m1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.Ex3.m1.1.1.1.1.1.3" xref="S4.Ex3.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.Ex3.m1.2.2.2.2.4" xref="S4.Ex3.m1.2.2.2.3.cmml">,</mo><msub id="S4.Ex3.m1.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.cmml"><mi id="S4.Ex3.m1.2.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.2.cmml">x</mi><mi id="S4.Ex3.m1.2.2.2.2.2.3" xref="S4.Ex3.m1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.Ex3.m1.2.2.2.2.5" xref="S4.Ex3.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex3.m1.2b"><apply id="S4.Ex3.m1.2.2.cmml" xref="S4.Ex3.m1.2.2"><times id="S4.Ex3.m1.2.2.3.cmml" xref="S4.Ex3.m1.2.2.3"></times><apply id="S4.Ex3.m1.2.2.4.cmml" xref="S4.Ex3.m1.2.2.4"><csymbol cd="ambiguous" id="S4.Ex3.m1.2.2.4.1.cmml" xref="S4.Ex3.m1.2.2.4">subscript</csymbol><ci id="S4.Ex3.m1.2.2.4.2.cmml" xref="S4.Ex3.m1.2.2.4.2">𝑃</ci><ci id="S4.Ex3.m1.2.2.4.3.cmml" xref="S4.Ex3.m1.2.2.4.3">𝖼𝗈𝗈𝗋</ci></apply><interval closure="open" id="S4.Ex3.m1.2.2.2.3.cmml" xref="S4.Ex3.m1.2.2.2.2"><apply id="S4.Ex3.m1.1.1.1.1.1.cmml" xref="S4.Ex3.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex3.m1.1.1.1.1.1.1.cmml" xref="S4.Ex3.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex3.m1.1.1.1.1.1.2.cmml" xref="S4.Ex3.m1.1.1.1.1.1.2">𝑥</ci><ci id="S4.Ex3.m1.1.1.1.1.1.3.cmml" xref="S4.Ex3.m1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.Ex3.m1.2.2.2.2.2.cmml" xref="S4.Ex3.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex3.m1.2.2.2.2.2.1.cmml" xref="S4.Ex3.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex3.m1.2.2.2.2.2.2.cmml" xref="S4.Ex3.m1.2.2.2.2.2.2">𝑥</ci><ci id="S4.Ex3.m1.2.2.2.2.2.3.cmml" xref="S4.Ex3.m1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex3.m1.2c">\displaystyle{P}_{\mathsf{coor}}(x_{a},x_{b})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.Ex3.m2.2" class="ltx_Math" alttext="\displaystyle={P}_{\mathsf{coor}}(z_{a},z_{b})" display="inline"><semantics id="S4.Ex3.m2.2a"><mrow id="S4.Ex3.m2.2.2" xref="S4.Ex3.m2.2.2.cmml"><mi id="S4.Ex3.m2.2.2.4" xref="S4.Ex3.m2.2.2.4.cmml"></mi><mo id="S4.Ex3.m2.2.2.3" xref="S4.Ex3.m2.2.2.3.cmml">=</mo><mrow id="S4.Ex3.m2.2.2.2" xref="S4.Ex3.m2.2.2.2.cmml"><msub id="S4.Ex3.m2.2.2.2.4" xref="S4.Ex3.m2.2.2.2.4.cmml"><mi id="S4.Ex3.m2.2.2.2.4.2" xref="S4.Ex3.m2.2.2.2.4.2.cmml">P</mi><mi id="S4.Ex3.m2.2.2.2.4.3" xref="S4.Ex3.m2.2.2.2.4.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="S4.Ex3.m2.2.2.2.3" xref="S4.Ex3.m2.2.2.2.3.cmml">​</mo><mrow id="S4.Ex3.m2.2.2.2.2.2" xref="S4.Ex3.m2.2.2.2.2.3.cmml"><mo stretchy="false" id="S4.Ex3.m2.2.2.2.2.2.3" xref="S4.Ex3.m2.2.2.2.2.3.cmml">(</mo><msub id="S4.Ex3.m2.1.1.1.1.1.1" xref="S4.Ex3.m2.1.1.1.1.1.1.cmml"><mi id="S4.Ex3.m2.1.1.1.1.1.1.2" xref="S4.Ex3.m2.1.1.1.1.1.1.2.cmml">z</mi><mi id="S4.Ex3.m2.1.1.1.1.1.1.3" xref="S4.Ex3.m2.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.Ex3.m2.2.2.2.2.2.4" xref="S4.Ex3.m2.2.2.2.2.3.cmml">,</mo><msub id="S4.Ex3.m2.2.2.2.2.2.2" xref="S4.Ex3.m2.2.2.2.2.2.2.cmml"><mi id="S4.Ex3.m2.2.2.2.2.2.2.2" xref="S4.Ex3.m2.2.2.2.2.2.2.2.cmml">z</mi><mi id="S4.Ex3.m2.2.2.2.2.2.2.3" xref="S4.Ex3.m2.2.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.Ex3.m2.2.2.2.2.2.5" xref="S4.Ex3.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex3.m2.2b"><apply id="S4.Ex3.m2.2.2.cmml" xref="S4.Ex3.m2.2.2"><eq id="S4.Ex3.m2.2.2.3.cmml" xref="S4.Ex3.m2.2.2.3"></eq><csymbol cd="latexml" id="S4.Ex3.m2.2.2.4.cmml" xref="S4.Ex3.m2.2.2.4">absent</csymbol><apply id="S4.Ex3.m2.2.2.2.cmml" xref="S4.Ex3.m2.2.2.2"><times id="S4.Ex3.m2.2.2.2.3.cmml" xref="S4.Ex3.m2.2.2.2.3"></times><apply id="S4.Ex3.m2.2.2.2.4.cmml" xref="S4.Ex3.m2.2.2.2.4"><csymbol cd="ambiguous" id="S4.Ex3.m2.2.2.2.4.1.cmml" xref="S4.Ex3.m2.2.2.2.4">subscript</csymbol><ci id="S4.Ex3.m2.2.2.2.4.2.cmml" xref="S4.Ex3.m2.2.2.2.4.2">𝑃</ci><ci id="S4.Ex3.m2.2.2.2.4.3.cmml" xref="S4.Ex3.m2.2.2.2.4.3">𝖼𝗈𝗈𝗋</ci></apply><interval closure="open" id="S4.Ex3.m2.2.2.2.2.3.cmml" xref="S4.Ex3.m2.2.2.2.2.2"><apply id="S4.Ex3.m2.1.1.1.1.1.1.cmml" xref="S4.Ex3.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex3.m2.1.1.1.1.1.1.1.cmml" xref="S4.Ex3.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex3.m2.1.1.1.1.1.1.2.cmml" xref="S4.Ex3.m2.1.1.1.1.1.1.2">𝑧</ci><ci id="S4.Ex3.m2.1.1.1.1.1.1.3.cmml" xref="S4.Ex3.m2.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.Ex3.m2.2.2.2.2.2.2.cmml" xref="S4.Ex3.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex3.m2.2.2.2.2.2.2.1.cmml" xref="S4.Ex3.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex3.m2.2.2.2.2.2.2.2.cmml" xref="S4.Ex3.m2.2.2.2.2.2.2.2">𝑧</ci><ci id="S4.Ex3.m2.2.2.2.2.2.2.3.cmml" xref="S4.Ex3.m2.2.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex3.m2.2c">\displaystyle={P}_{\mathsf{coor}}(z_{a},z_{b})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="S4.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.Ex4.m1.2" class="ltx_Math" alttext="\displaystyle K_{\mathsf{PMI}}(x_{a},x_{b})" display="inline"><semantics id="S4.Ex4.m1.2a"><mrow id="S4.Ex4.m1.2.2" xref="S4.Ex4.m1.2.2.cmml"><msub id="S4.Ex4.m1.2.2.4" xref="S4.Ex4.m1.2.2.4.cmml"><mi id="S4.Ex4.m1.2.2.4.2" xref="S4.Ex4.m1.2.2.4.2.cmml">K</mi><mi id="S4.Ex4.m1.2.2.4.3" xref="S4.Ex4.m1.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="S4.Ex4.m1.2.2.3" xref="S4.Ex4.m1.2.2.3.cmml">​</mo><mrow id="S4.Ex4.m1.2.2.2.2" xref="S4.Ex4.m1.2.2.2.3.cmml"><mo stretchy="false" id="S4.Ex4.m1.2.2.2.2.3" xref="S4.Ex4.m1.2.2.2.3.cmml">(</mo><msub id="S4.Ex4.m1.1.1.1.1.1" xref="S4.Ex4.m1.1.1.1.1.1.cmml"><mi id="S4.Ex4.m1.1.1.1.1.1.2" xref="S4.Ex4.m1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.Ex4.m1.1.1.1.1.1.3" xref="S4.Ex4.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.Ex4.m1.2.2.2.2.4" xref="S4.Ex4.m1.2.2.2.3.cmml">,</mo><msub id="S4.Ex4.m1.2.2.2.2.2" xref="S4.Ex4.m1.2.2.2.2.2.cmml"><mi id="S4.Ex4.m1.2.2.2.2.2.2" xref="S4.Ex4.m1.2.2.2.2.2.2.cmml">x</mi><mi id="S4.Ex4.m1.2.2.2.2.2.3" xref="S4.Ex4.m1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.Ex4.m1.2.2.2.2.5" xref="S4.Ex4.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex4.m1.2b"><apply id="S4.Ex4.m1.2.2.cmml" xref="S4.Ex4.m1.2.2"><times id="S4.Ex4.m1.2.2.3.cmml" xref="S4.Ex4.m1.2.2.3"></times><apply id="S4.Ex4.m1.2.2.4.cmml" xref="S4.Ex4.m1.2.2.4"><csymbol cd="ambiguous" id="S4.Ex4.m1.2.2.4.1.cmml" xref="S4.Ex4.m1.2.2.4">subscript</csymbol><ci id="S4.Ex4.m1.2.2.4.2.cmml" xref="S4.Ex4.m1.2.2.4.2">𝐾</ci><ci id="S4.Ex4.m1.2.2.4.3.cmml" xref="S4.Ex4.m1.2.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="S4.Ex4.m1.2.2.2.3.cmml" xref="S4.Ex4.m1.2.2.2.2"><apply id="S4.Ex4.m1.1.1.1.1.1.cmml" xref="S4.Ex4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex4.m1.1.1.1.1.1.1.cmml" xref="S4.Ex4.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex4.m1.1.1.1.1.1.2.cmml" xref="S4.Ex4.m1.1.1.1.1.1.2">𝑥</ci><ci id="S4.Ex4.m1.1.1.1.1.1.3.cmml" xref="S4.Ex4.m1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.Ex4.m1.2.2.2.2.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex4.m1.2.2.2.2.2.1.cmml" xref="S4.Ex4.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.Ex4.m1.2.2.2.2.2.2.cmml" xref="S4.Ex4.m1.2.2.2.2.2.2">𝑥</ci><ci id="S4.Ex4.m1.2.2.2.2.2.3.cmml" xref="S4.Ex4.m1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m1.2c">\displaystyle K_{\mathsf{PMI}}(x_{a},x_{b})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.Ex4.m2.1" class="ltx_Math" alttext="\displaystyle=K_{\mathsf{PMI}}(z_{a},z_{b})," display="inline"><semantics id="S4.Ex4.m2.1a"><mrow id="S4.Ex4.m2.1.1.1" xref="S4.Ex4.m2.1.1.1.1.cmml"><mrow id="S4.Ex4.m2.1.1.1.1" xref="S4.Ex4.m2.1.1.1.1.cmml"><mi id="S4.Ex4.m2.1.1.1.1.4" xref="S4.Ex4.m2.1.1.1.1.4.cmml"></mi><mo id="S4.Ex4.m2.1.1.1.1.3" xref="S4.Ex4.m2.1.1.1.1.3.cmml">=</mo><mrow id="S4.Ex4.m2.1.1.1.1.2" xref="S4.Ex4.m2.1.1.1.1.2.cmml"><msub id="S4.Ex4.m2.1.1.1.1.2.4" xref="S4.Ex4.m2.1.1.1.1.2.4.cmml"><mi id="S4.Ex4.m2.1.1.1.1.2.4.2" xref="S4.Ex4.m2.1.1.1.1.2.4.2.cmml">K</mi><mi id="S4.Ex4.m2.1.1.1.1.2.4.3" xref="S4.Ex4.m2.1.1.1.1.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="S4.Ex4.m2.1.1.1.1.2.3" xref="S4.Ex4.m2.1.1.1.1.2.3.cmml">​</mo><mrow id="S4.Ex4.m2.1.1.1.1.2.2.2" xref="S4.Ex4.m2.1.1.1.1.2.2.3.cmml"><mo stretchy="false" id="S4.Ex4.m2.1.1.1.1.2.2.2.3" xref="S4.Ex4.m2.1.1.1.1.2.2.3.cmml">(</mo><msub id="S4.Ex4.m2.1.1.1.1.1.1.1.1" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex4.m2.1.1.1.1.1.1.1.1.2" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1.2.cmml">z</mi><mi id="S4.Ex4.m2.1.1.1.1.1.1.1.1.3" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.Ex4.m2.1.1.1.1.2.2.2.4" xref="S4.Ex4.m2.1.1.1.1.2.2.3.cmml">,</mo><msub id="S4.Ex4.m2.1.1.1.1.2.2.2.2" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2.cmml"><mi id="S4.Ex4.m2.1.1.1.1.2.2.2.2.2" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2.2.cmml">z</mi><mi id="S4.Ex4.m2.1.1.1.1.2.2.2.2.3" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.Ex4.m2.1.1.1.1.2.2.2.5" xref="S4.Ex4.m2.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex4.m2.1.1.1.2" xref="S4.Ex4.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex4.m2.1b"><apply id="S4.Ex4.m2.1.1.1.1.cmml" xref="S4.Ex4.m2.1.1.1"><eq id="S4.Ex4.m2.1.1.1.1.3.cmml" xref="S4.Ex4.m2.1.1.1.1.3"></eq><csymbol cd="latexml" id="S4.Ex4.m2.1.1.1.1.4.cmml" xref="S4.Ex4.m2.1.1.1.1.4">absent</csymbol><apply id="S4.Ex4.m2.1.1.1.1.2.cmml" xref="S4.Ex4.m2.1.1.1.1.2"><times id="S4.Ex4.m2.1.1.1.1.2.3.cmml" xref="S4.Ex4.m2.1.1.1.1.2.3"></times><apply id="S4.Ex4.m2.1.1.1.1.2.4.cmml" xref="S4.Ex4.m2.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S4.Ex4.m2.1.1.1.1.2.4.1.cmml" xref="S4.Ex4.m2.1.1.1.1.2.4">subscript</csymbol><ci id="S4.Ex4.m2.1.1.1.1.2.4.2.cmml" xref="S4.Ex4.m2.1.1.1.1.2.4.2">𝐾</ci><ci id="S4.Ex4.m2.1.1.1.1.2.4.3.cmml" xref="S4.Ex4.m2.1.1.1.1.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="S4.Ex4.m2.1.1.1.1.2.2.3.cmml" xref="S4.Ex4.m2.1.1.1.1.2.2.2"><apply id="S4.Ex4.m2.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex4.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex4.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1.2">𝑧</ci><ci id="S4.Ex4.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex4.m2.1.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.Ex4.m2.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex4.m2.1.1.1.1.2.2.2.2.1.cmml" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S4.Ex4.m2.1.1.1.1.2.2.2.2.2.cmml" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2.2">𝑧</ci><ci id="S4.Ex4.m2.1.1.1.1.2.2.2.2.3.cmml" xref="S4.Ex4.m2.1.1.1.1.2.2.2.2.3">𝑏</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m2.1c">\displaystyle=K_{\mathsf{PMI}}(z_{a},z_{b}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="S4.SS2.p5.4" class="ltx_p">where we use <math id="S4.SS2.p5.3.m1.1" class="ltx_Math" alttext="{P}_{\mathsf{coor}}" display="inline"><semantics id="S4.SS2.p5.3.m1.1a"><msub id="S4.SS2.p5.3.m1.1.1" xref="S4.SS2.p5.3.m1.1.1.cmml"><mi id="S4.SS2.p5.3.m1.1.1.2" xref="S4.SS2.p5.3.m1.1.1.2.cmml">P</mi><mi id="S4.SS2.p5.3.m1.1.1.3" xref="S4.SS2.p5.3.m1.1.1.3.cmml">𝖼𝗈𝗈𝗋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m1.1b"><apply id="S4.SS2.p5.3.m1.1.1.cmml" xref="S4.SS2.p5.3.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m1.1.1.1.cmml" xref="S4.SS2.p5.3.m1.1.1">subscript</csymbol><ci id="S4.SS2.p5.3.m1.1.1.2.cmml" xref="S4.SS2.p5.3.m1.1.1.2">𝑃</ci><ci id="S4.SS2.p5.3.m1.1.1.3.cmml" xref="S4.SS2.p5.3.m1.1.1.3">𝖼𝗈𝗈𝗋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m1.1c">{P}_{\mathsf{coor}}</annotation></semantics></math> and <math id="S4.SS2.p5.4.m2.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="S4.SS2.p5.4.m2.1a"><msub id="S4.SS2.p5.4.m2.1.1" xref="S4.SS2.p5.4.m2.1.1.cmml"><mi id="S4.SS2.p5.4.m2.1.1.2" xref="S4.SS2.p5.4.m2.1.1.2.cmml">K</mi><mi id="S4.SS2.p5.4.m2.1.1.3" xref="S4.SS2.p5.4.m2.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m2.1b"><apply id="S4.SS2.p5.4.m2.1.1.cmml" xref="S4.SS2.p5.4.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m2.1.1.1.cmml" xref="S4.SS2.p5.4.m2.1.1">subscript</csymbol><ci id="S4.SS2.p5.4.m2.1.1.2.cmml" xref="S4.SS2.p5.4.m2.1.1.2">𝐾</ci><ci id="S4.SS2.p5.4.m2.1.1.3.cmml" xref="S4.SS2.p5.4.m2.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m2.1c">K_{\mathsf{PMI}}</annotation></semantics></math> in a modality-agnostic way to emphasize that different modalities share the same such quantities.</p>
</div>
<div id="S4.SS2.p6" class="ltx_para ltx_noindent">
<p id="S4.SS2.p6.2" class="ltx_p">All these arguments hold not just for <math id="S4.SS2.p6.1.m1.1" class="ltx_Math" alttext="X" display="inline"><semantics id="S4.SS2.p6.1.m1.1a"><mi id="S4.SS2.p6.1.m1.1.1" xref="S4.SS2.p6.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.1.m1.1b"><ci id="S4.SS2.p6.1.m1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.1.m1.1c">X</annotation></semantics></math> but also for <math id="S4.SS2.p6.2.m2.1" class="ltx_Math" alttext="Y" display="inline"><semantics id="S4.SS2.p6.2.m2.1a"><mi id="S4.SS2.p6.2.m2.1.1" xref="S4.SS2.p6.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.2.m2.1b"><ci id="S4.SS2.p6.2.m2.1.1.cmml" xref="S4.SS2.p6.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.2.m2.1c">Y</annotation></semantics></math> (or any other bijective, discrete modality), implying:</p>
<table id="A6.EGx6" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.E7.m1.2" class="ltx_Math" alttext="\displaystyle K_{\mathsf{PMI}}(z_{a},z_{b})" display="inline"><semantics id="S4.E7.m1.2a"><mrow id="S4.E7.m1.2.2" xref="S4.E7.m1.2.2.cmml"><msub id="S4.E7.m1.2.2.4" xref="S4.E7.m1.2.2.4.cmml"><mi id="S4.E7.m1.2.2.4.2" xref="S4.E7.m1.2.2.4.2.cmml">K</mi><mi id="S4.E7.m1.2.2.4.3" xref="S4.E7.m1.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="S4.E7.m1.2.2.3" xref="S4.E7.m1.2.2.3.cmml">​</mo><mrow id="S4.E7.m1.2.2.2.2" xref="S4.E7.m1.2.2.2.3.cmml"><mo stretchy="false" id="S4.E7.m1.2.2.2.2.3" xref="S4.E7.m1.2.2.2.3.cmml">(</mo><msub id="S4.E7.m1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.cmml"><mi id="S4.E7.m1.1.1.1.1.1.2" xref="S4.E7.m1.1.1.1.1.1.2.cmml">z</mi><mi id="S4.E7.m1.1.1.1.1.1.3" xref="S4.E7.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S4.E7.m1.2.2.2.2.4" xref="S4.E7.m1.2.2.2.3.cmml">,</mo><msub id="S4.E7.m1.2.2.2.2.2" xref="S4.E7.m1.2.2.2.2.2.cmml"><mi id="S4.E7.m1.2.2.2.2.2.2" xref="S4.E7.m1.2.2.2.2.2.2.cmml">z</mi><mi id="S4.E7.m1.2.2.2.2.2.3" xref="S4.E7.m1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="S4.E7.m1.2.2.2.2.5" xref="S4.E7.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m1.2b"><apply id="S4.E7.m1.2.2.cmml" xref="S4.E7.m1.2.2"><times id="S4.E7.m1.2.2.3.cmml" xref="S4.E7.m1.2.2.3"></times><apply id="S4.E7.m1.2.2.4.cmml" xref="S4.E7.m1.2.2.4"><csymbol cd="ambiguous" id="S4.E7.m1.2.2.4.1.cmml" xref="S4.E7.m1.2.2.4">subscript</csymbol><ci id="S4.E7.m1.2.2.4.2.cmml" xref="S4.E7.m1.2.2.4.2">𝐾</ci><ci id="S4.E7.m1.2.2.4.3.cmml" xref="S4.E7.m1.2.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="S4.E7.m1.2.2.2.3.cmml" xref="S4.E7.m1.2.2.2.2"><apply id="S4.E7.m1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E7.m1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.E7.m1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.1.1.1.1.1.2">𝑧</ci><ci id="S4.E7.m1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.1.1.1.1.1.3">𝑎</ci></apply><apply id="S4.E7.m1.2.2.2.2.2.cmml" xref="S4.E7.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.E7.m1.2.2.2.2.2.1.cmml" xref="S4.E7.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.E7.m1.2.2.2.2.2.2.cmml" xref="S4.E7.m1.2.2.2.2.2.2">𝑧</ci><ci id="S4.E7.m1.2.2.2.2.2.3.cmml" xref="S4.E7.m1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.2c">\displaystyle K_{\mathsf{PMI}}(z_{a},z_{b})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.E7.m2.2" class="ltx_Math" alttext="\displaystyle=\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle-c_{X}" display="inline"><semantics id="S4.E7.m2.2a"><mrow id="S4.E7.m2.2.2" xref="S4.E7.m2.2.2.cmml"><mi id="S4.E7.m2.2.2.4" xref="S4.E7.m2.2.2.4.cmml"></mi><mo id="S4.E7.m2.2.2.3" xref="S4.E7.m2.2.2.3.cmml">=</mo><mrow id="S4.E7.m2.2.2.2" xref="S4.E7.m2.2.2.2.cmml"><mrow id="S4.E7.m2.2.2.2.2.2" xref="S4.E7.m2.2.2.2.2.3.cmml"><mo stretchy="false" id="S4.E7.m2.2.2.2.2.2.3" xref="S4.E7.m2.2.2.2.2.3.cmml">⟨</mo><mrow id="S4.E7.m2.1.1.1.1.1.1" xref="S4.E7.m2.1.1.1.1.1.1.cmml"><msub id="S4.E7.m2.1.1.1.1.1.1.3" xref="S4.E7.m2.1.1.1.1.1.1.3.cmml"><mi id="S4.E7.m2.1.1.1.1.1.1.3.2" xref="S4.E7.m2.1.1.1.1.1.1.3.2.cmml">f</mi><mi id="S4.E7.m2.1.1.1.1.1.1.3.3" xref="S4.E7.m2.1.1.1.1.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E7.m2.1.1.1.1.1.1.2" xref="S4.E7.m2.1.1.1.1.1.1.2.cmml">​</mo><mrow id="S4.E7.m2.1.1.1.1.1.1.1.1" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.E7.m2.1.1.1.1.1.1.1.1.2" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.E7.m2.1.1.1.1.1.1.1.1.1" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E7.m2.1.1.1.1.1.1.1.1.1.2" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S4.E7.m2.1.1.1.1.1.1.1.1.1.3" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="S4.E7.m2.1.1.1.1.1.1.1.1.3" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E7.m2.2.2.2.2.2.4" xref="S4.E7.m2.2.2.2.2.3.cmml">,</mo><mrow id="S4.E7.m2.2.2.2.2.2.2" xref="S4.E7.m2.2.2.2.2.2.2.cmml"><msub id="S4.E7.m2.2.2.2.2.2.2.3" xref="S4.E7.m2.2.2.2.2.2.2.3.cmml"><mi id="S4.E7.m2.2.2.2.2.2.2.3.2" xref="S4.E7.m2.2.2.2.2.2.2.3.2.cmml">f</mi><mi id="S4.E7.m2.2.2.2.2.2.2.3.3" xref="S4.E7.m2.2.2.2.2.2.2.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="S4.E7.m2.2.2.2.2.2.2.2" xref="S4.E7.m2.2.2.2.2.2.2.2.cmml">​</mo><mrow id="S4.E7.m2.2.2.2.2.2.2.1.1" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.cmml"><mo stretchy="false" id="S4.E7.m2.2.2.2.2.2.2.1.1.2" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.cmml">(</mo><msub id="S4.E7.m2.2.2.2.2.2.2.1.1.1" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.cmml"><mi id="S4.E7.m2.2.2.2.2.2.2.1.1.1.2" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.2.cmml">x</mi><mi id="S4.E7.m2.2.2.2.2.2.2.1.1.1.3" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.3.cmml">b</mi></msub><mo stretchy="false" id="S4.E7.m2.2.2.2.2.2.2.1.1.3" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S4.E7.m2.2.2.2.2.2.5" xref="S4.E7.m2.2.2.2.2.3.cmml">⟩</mo></mrow><mo id="S4.E7.m2.2.2.2.3" xref="S4.E7.m2.2.2.2.3.cmml">−</mo><msub id="S4.E7.m2.2.2.2.4" xref="S4.E7.m2.2.2.2.4.cmml"><mi id="S4.E7.m2.2.2.2.4.2" xref="S4.E7.m2.2.2.2.4.2.cmml">c</mi><mi id="S4.E7.m2.2.2.2.4.3" xref="S4.E7.m2.2.2.2.4.3.cmml">X</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m2.2b"><apply id="S4.E7.m2.2.2.cmml" xref="S4.E7.m2.2.2"><eq id="S4.E7.m2.2.2.3.cmml" xref="S4.E7.m2.2.2.3"></eq><csymbol cd="latexml" id="S4.E7.m2.2.2.4.cmml" xref="S4.E7.m2.2.2.4">absent</csymbol><apply id="S4.E7.m2.2.2.2.cmml" xref="S4.E7.m2.2.2.2"><minus id="S4.E7.m2.2.2.2.3.cmml" xref="S4.E7.m2.2.2.2.3"></minus><list id="S4.E7.m2.2.2.2.2.3.cmml" xref="S4.E7.m2.2.2.2.2.2"><apply id="S4.E7.m2.1.1.1.1.1.1.cmml" xref="S4.E7.m2.1.1.1.1.1.1"><times id="S4.E7.m2.1.1.1.1.1.1.2.cmml" xref="S4.E7.m2.1.1.1.1.1.1.2"></times><apply id="S4.E7.m2.1.1.1.1.1.1.3.cmml" xref="S4.E7.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.E7.m2.1.1.1.1.1.1.3.1.cmml" xref="S4.E7.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.E7.m2.1.1.1.1.1.1.3.2.cmml" xref="S4.E7.m2.1.1.1.1.1.1.3.2">𝑓</ci><ci id="S4.E7.m2.1.1.1.1.1.1.3.3.cmml" xref="S4.E7.m2.1.1.1.1.1.1.3.3">𝑋</ci></apply><apply id="S4.E7.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E7.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E7.m2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E7.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.E7.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.E7.m2.1.1.1.1.1.1.1.1.1.3">𝑎</ci></apply></apply><apply id="S4.E7.m2.2.2.2.2.2.2.cmml" xref="S4.E7.m2.2.2.2.2.2.2"><times id="S4.E7.m2.2.2.2.2.2.2.2.cmml" xref="S4.E7.m2.2.2.2.2.2.2.2"></times><apply id="S4.E7.m2.2.2.2.2.2.2.3.cmml" xref="S4.E7.m2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E7.m2.2.2.2.2.2.2.3.1.cmml" xref="S4.E7.m2.2.2.2.2.2.2.3">subscript</csymbol><ci id="S4.E7.m2.2.2.2.2.2.2.3.2.cmml" xref="S4.E7.m2.2.2.2.2.2.2.3.2">𝑓</ci><ci id="S4.E7.m2.2.2.2.2.2.2.3.3.cmml" xref="S4.E7.m2.2.2.2.2.2.2.3.3">𝑋</ci></apply><apply id="S4.E7.m2.2.2.2.2.2.2.1.1.1.cmml" xref="S4.E7.m2.2.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E7.m2.2.2.2.2.2.2.1.1.1.1.cmml" xref="S4.E7.m2.2.2.2.2.2.2.1.1">subscript</csymbol><ci id="S4.E7.m2.2.2.2.2.2.2.1.1.1.2.cmml" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.2">𝑥</ci><ci id="S4.E7.m2.2.2.2.2.2.2.1.1.1.3.cmml" xref="S4.E7.m2.2.2.2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></list><apply id="S4.E7.m2.2.2.2.4.cmml" xref="S4.E7.m2.2.2.2.4"><csymbol cd="ambiguous" id="S4.E7.m2.2.2.2.4.1.cmml" xref="S4.E7.m2.2.2.2.4">subscript</csymbol><ci id="S4.E7.m2.2.2.2.4.2.cmml" xref="S4.E7.m2.2.2.2.4.2">𝑐</ci><ci id="S4.E7.m2.2.2.2.4.3.cmml" xref="S4.E7.m2.2.2.2.4.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m2.2c">\displaystyle=\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle-c_{X}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td>
</tr></tbody>
<tbody id="S4.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="S4.E8.m1.1.1.1.1.2.3.cmml">−</mo><msub id="S4.E8.m1.1.1.1.1.2.4" xref="S4.E8.m1.1.1.1.1.2.4.cmml"><mi id="S4.E8.m1.1.1.1.1.2.4.2" xref="S4.E8.m1.1.1.1.1.2.4.2.cmml">c</mi><mi id="S4.E8.m1.1.1.1.1.2.4.3" xref="S4.E8.m1.1.1.1.1.2.4.3.cmml">Y</mi></msub></mrow></mrow><mo lspace="0em" id="S4.E8.m1.1.1.1.2" xref="S4.E8.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E8.m1.1b"><apply id="S4.E8.m1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1"><eq id="S4.E8.m1.1.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.3"></eq><csymbol cd="latexml" id="S4.E8.m1.1.1.1.1.4.cmml" xref="S4.E8.m1.1.1.1.1.4">absent</csymbol><apply id="S4.E8.m1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.2"><minus id="S4.E8.m1.1.1.1.1.2.3.cmml" xref="S4.E8.m1.1.1.1.1.2.3"></minus><list id="S4.E8.m1.1.1.1.1.2.2.3.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2"><apply id="S4.E8.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.1"><times id="S4.E8.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.1.1.1.1.2"></times><apply 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xref="S4.E8.m1.1.1.1.1.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.2.2.2.2.3.1.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.3">subscript</csymbol><ci id="S4.E8.m1.1.1.1.1.2.2.2.2.3.2.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.3.2">𝑓</ci><ci id="S4.E8.m1.1.1.1.1.2.2.2.2.3.3.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.3.3">𝑌</ci></apply><apply id="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.1.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.1.1">subscript</csymbol><ci id="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.2.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.2">𝑦</ci><ci id="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.3.cmml" xref="S4.E8.m1.1.1.1.1.2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></list><apply id="S4.E8.m1.1.1.1.1.2.4.cmml" xref="S4.E8.m1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="S4.E8.m1.1.1.1.1.2.4.1.cmml" xref="S4.E8.m1.1.1.1.1.2.4">subscript</csymbol><ci id="S4.E8.m1.1.1.1.1.2.4.2.cmml" xref="S4.E8.m1.1.1.1.1.2.4.2">𝑐</ci><ci id="S4.E8.m1.1.1.1.1.2.4.3.cmml" xref="S4.E8.m1.1.1.1.1.2.4.3">𝑌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.1c">\displaystyle=\langle f_{Y}(y_{a}),f_{Y}(y_{b})\rangle-c_{Y}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td>
</tr></tbody>
</table>
<p id="S4.SS2.p6.3" class="ltx_p">Therefore, for any modality in our idealized world, we observe representational convergence to the same kernel, which represents certain pairwise statistics of <math id="S4.SS2.p6.3.m1.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S4.SS2.p6.3.m1.1a"><mrow id="S4.SS2.p6.3.m1.1.2" xref="S4.SS2.p6.3.m1.1.2.cmml"><mi mathvariant="normal" id="S4.SS2.p6.3.m1.1.2.2" xref="S4.SS2.p6.3.m1.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S4.SS2.p6.3.m1.1.2.1" xref="S4.SS2.p6.3.m1.1.2.1.cmml">​</mo><mrow id="S4.SS2.p6.3.m1.1.2.3.2" xref="S4.SS2.p6.3.m1.1.2.cmml"><mo stretchy="false" id="S4.SS2.p6.3.m1.1.2.3.2.1" xref="S4.SS2.p6.3.m1.1.2.cmml">(</mo><mi id="S4.SS2.p6.3.m1.1.1" xref="S4.SS2.p6.3.m1.1.1.cmml">𝐙</mi><mo stretchy="false" id="S4.SS2.p6.3.m1.1.2.3.2.2" xref="S4.SS2.p6.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.3.m1.1b"><apply id="S4.SS2.p6.3.m1.1.2.cmml" xref="S4.SS2.p6.3.m1.1.2"><times id="S4.SS2.p6.3.m1.1.2.1.cmml" xref="S4.SS2.p6.3.m1.1.2.1"></times><ci id="S4.SS2.p6.3.m1.1.2.2.cmml" xref="S4.SS2.p6.3.m1.1.2.2">ℙ</ci><ci id="S4.SS2.p6.3.m1.1.1.cmml" xref="S4.SS2.p6.3.m1.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.3.m1.1c">\mathbb{P}(\mathbf{Z})</annotation></semantics></math>.</p>
</div>
<div id="S4.SS2.p7" class="ltx_para ltx_noindent">
<p id="S4.SS2.p7.1" class="ltx_p">This analysis suggests that certain representation learning algorithms may boil down to a simple rule: <span id="S4.SS2.p7.1.1" class="ltx_text ltx_font_italic">find an embedding in which similarity equals PMI</span>. We note that this idea is consistent with prior works that have used PMI as a similarity measure for clustering in vision and language (<span id="S4.SS2.p7.1.2" class="ltx_text ltx_font_italic">e.g.</span>,  <cite class="ltx_cite ltx_citemacro_citet">Isola et al. (<a href="#bib.bib51" title="" class="ltx_ref">2014</a>); Isola (<a href="#bib.bib50" title="" class="ltx_ref">2015</a>); Isola et al. (<a href="#bib.bib52" title="" class="ltx_ref">2016</a>); Chambers &amp; Jurafsky (<a href="#bib.bib19" title="" class="ltx_ref">2008</a>)</cite>).</p>
</div>
<section id="S4.SS2.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">A study in color</h4>

<div id="S4.SS2.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="S4.SS2.SSS0.Px1.p1.1" class="ltx_p">We conduct a case study to verify that convergence does happen on real data.
<cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite> discovered that color distances in learned language representations, when trained to predict cooccurrences in <em id="S4.SS2.SSS0.Px1.p1.1.1" class="ltx_emph ltx_font_italic">text</em> <cite class="ltx_cite ltx_citemacro_citep">(Devlin et al., <a href="#bib.bib24" title="" class="ltx_ref">2018</a>)</cite>, closely mirror human perception of these distances, which we reproduce in <a href="#S4.F8" title="In 4.1 An idealized world ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">8</span></a> with both contrastive and predictive models. Interestingly, they noted an increasing similarity as models scale larger and become better at modeling <em id="S4.SS2.SSS0.Px1.p1.1.2" class="ltx_emph ltx_font_italic">text</em> cooccurrences. In <a href="#S4.F8" title="In 4.1 An idealized world ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">8</span></a>, we also learn representations of color based on <math id="S4.SS2.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="S4.SS2.SSS0.Px1.p1.1.m1.1a"><msub id="S4.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.2" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml">K</mi><mi id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.3" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.1.m1.1b"><apply id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.2">𝐾</ci><ci id="S4.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.1.m1.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.1.m1.1c">K_{\mathsf{PMI}}</annotation></semantics></math> from cooccurrences in <em id="S4.SS2.SSS0.Px1.p1.1.3" class="ltx_emph ltx_font_italic">images</em>. Indeed, learning cooccurrence statistics in either domain recovers roughly the <em id="S4.SS2.SSS0.Px1.p1.1.4" class="ltx_emph ltx_font_italic">same</em> perceptual representation. Details of this experiment are described in Appendix <a href="#A4" title="Appendix D Color Cooccurrence Experiment ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">D</span></a>.</p>
</div>
<div id="S4.SS2.SSS0.Px1.p2" class="ltx_para ltx_noindent">
<p id="S4.SS2.SSS0.Px1.p2.1" class="ltx_p">We believe that our simple model encapsulates essential aspects of complex real-world systems, and offers a path toward understanding the representation that models are converging to—a unified model that is proficient across various domains and modalities, grounded in the statistical properties of the underlying world. <a href="#S6" title="6 Counterexamples and limitations ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a> further elaborates some limitations.</p>
</div>
</section>
</section>
</section>
<section id="S5" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">5 </span>What are the implications of convergence?</h2>

<section id="S5.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Scaling is sufficient, but not necessarily efficient</h4>

<div id="S5.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="S5.SS0.SSS0.Px1.p1.1" class="ltx_p">Our arguments are roughly in line with the claim that “scale is all you need” to reach high levels of intelligence. We have argued that as resources are scaled (# parameters, # datapoints, # flops), representations are converging, regardless of other modeling choices and even data modality. Does this mean that scale is all that matters? Not quite: different methods can scale with different levels of <span id="S5.SS0.SSS0.Px1.p1.1.1" class="ltx_text ltx_font_italic">efficiency</span> <cite class="ltx_cite ltx_citemacro_citep">(Hestness et al., <a href="#bib.bib47" title="" class="ltx_ref">2017</a>; Kaplan et al., <a href="#bib.bib58" title="" class="ltx_ref">2020</a>)</cite>, and successful methods must still satisfy some general requirements (<span id="S5.SS0.SSS0.Px1.p1.1.2" class="ltx_text ltx_font_italic">e.g.</span>, be a consistent estimator, model pairwise statistics of <math id="S5.SS0.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\mathbb{P}(\mathbf{Z})" display="inline"><semantics id="S5.SS0.SSS0.Px1.p1.1.m1.1a"><mrow id="S5.SS0.SSS0.Px1.p1.1.m1.1.2" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.cmml"><mi mathvariant="normal" id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.2" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.2.cmml">ℙ</mi><mo lspace="0em" rspace="0em" id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.1" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.1.cmml">​</mo><mrow id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.3.2" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.cmml"><mo stretchy="false" id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.3.2.1" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.cmml">(</mo><mi id="S5.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">𝐙</mi><mo stretchy="false" id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.3.2.2" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px1.p1.1.m1.1b"><apply id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.cmml" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2"><times id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.1.cmml" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.1"></times><ci id="S5.SS0.SSS0.Px1.p1.1.m1.1.2.2.cmml" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.2.2">ℙ</ci><ci id="S5.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S5.SS0.SSS0.Px1.p1.1.m1.1.1">𝐙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px1.p1.1.m1.1c">\mathbb{P}(\mathbf{Z})</annotation></semantics></math>).</p>
</div>
</section>
<section id="S5.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Training data can be shared across modalities</h4>

<div id="S5.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="S5.SS0.SSS0.Px2.p1.6" class="ltx_p">Suppose you have access to <math id="S5.SS0.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="N" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.1.m1.1a"><mi id="S5.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S5.SS0.SSS0.Px2.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.1.m1.1b"><ci id="S5.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.1.m1.1c">N</annotation></semantics></math> images and <math id="S5.SS0.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="M" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.2.m2.1a"><mi id="S5.SS0.SSS0.Px2.p1.2.m2.1.1" xref="S5.SS0.SSS0.Px2.p1.2.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.2.m2.1b"><ci id="S5.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.2.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.2.m2.1c">M</annotation></semantics></math> sentences, and want to learn the best representation. If there is indeed a modality-agnostic platonic representation, then the image data should help find it, and so should the language data. The implication is that if you want to train the best vision model, you should train not just on <math id="S5.SS0.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="N" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.3.m3.1a"><mi id="S5.SS0.SSS0.Px2.p1.3.m3.1.1" xref="S5.SS0.SSS0.Px2.p1.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.3.m3.1b"><ci id="S5.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.3.m3.1c">N</annotation></semantics></math> images but also on <math id="S5.SS0.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="M" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.4.m4.1a"><mi id="S5.SS0.SSS0.Px2.p1.4.m4.1.1" xref="S5.SS0.SSS0.Px2.p1.4.m4.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.4.m4.1b"><ci id="S5.SS0.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.4.m4.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.4.m4.1c">M</annotation></semantics></math> sentences. This is already becoming common practice <cite class="ltx_cite ltx_citemacro_citep">(Achiam et al., <a href="#bib.bib2" title="" class="ltx_ref">2023</a>; Radford et al., <a href="#bib.bib99" title="" class="ltx_ref">2021</a>)</cite>. Many vision models are finetuned from pre-trained LLMs. The other direction is less common, but also is implied by our hypothesis: if you want to build the best LLM, <span id="S5.SS0.SSS0.Px2.p1.6.1" class="ltx_text ltx_font_italic">you should also train it on image data</span>. Indeed, <cite class="ltx_cite ltx_citemacro_citet">Achiam et al. (<a href="#bib.bib2" title="" class="ltx_ref">2023</a>)</cite> claim evidence that this is true, where training on images improved performance on text. In theory, there should be some conversion ratio: a pixel is worth <math id="S5.SS0.SSS0.Px2.p1.5.m5.1" class="ltx_Math" alttext="a" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.5.m5.1a"><mi id="S5.SS0.SSS0.Px2.p1.5.m5.1.1" xref="S5.SS0.SSS0.Px2.p1.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.5.m5.1b"><ci id="S5.SS0.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.5.m5.1c">a</annotation></semantics></math> words for training LLMs, and a word is worth <math id="S5.SS0.SSS0.Px2.p1.6.m6.1" class="ltx_Math" alttext="b" display="inline"><semantics id="S5.SS0.SSS0.Px2.p1.6.m6.1a"><mi id="S5.SS0.SSS0.Px2.p1.6.m6.1.1" xref="S5.SS0.SSS0.Px2.p1.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S5.SS0.SSS0.Px2.p1.6.m6.1b"><ci id="S5.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S5.SS0.SSS0.Px2.p1.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS0.SSS0.Px2.p1.6.m6.1c">b</annotation></semantics></math> pixels for training vision models.</p>
</div>
</section>
<section id="S5.SS0.SSS0.Px3" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Ease of translation and adaptation across modalities</h4>

<div id="S5.SS0.SSS0.Px3.p1" class="ltx_para ltx_noindent">
<p id="S5.SS0.SSS0.Px3.p1.1" class="ltx_p">When two representations are aligned, transitioning from one to the other should be a simple function that’s easily obtained. Our hypothesis could explain the phenomenon that conditional generation is easier than unconditional <cite class="ltx_cite ltx_citemacro_citep">(Mirza &amp; Osindero, <a href="#bib.bib82" title="" class="ltx_ref">2014</a>; Liu et al., <a href="#bib.bib71" title="" class="ltx_ref">2020</a>; Sauer et al., <a href="#bib.bib103" title="" class="ltx_ref">2022</a>)</cite>, as the data we condition on may have the same platonic structure as the data we are generating. In line with this, recent work has found that representation-conditioning is even easier <cite class="ltx_cite ltx_citemacro_citep">(Li et al., <a href="#bib.bib66" title="" class="ltx_ref">2023</a>)</cite>. Similarly, representational convergence could act as a bridge that lets us find mappings between domains even without paired data; this may underlie the success of unpaired translation in vision <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a href="#bib.bib138" title="" class="ltx_ref">2017</a>; Shi et al., <a href="#bib.bib107" title="" class="ltx_ref">2024</a>; Xie et al., <a href="#bib.bib132" title="" class="ltx_ref">2022</a>)</cite> and language <cite class="ltx_cite ltx_citemacro_citep">(Tran et al., <a href="#bib.bib122" title="" class="ltx_ref">2017</a>; Lample et al., <a href="#bib.bib64" title="" class="ltx_ref">2018</a>)</cite>.
We emphasize that this doesn’t mean that models trained on a single modality (<span id="S5.SS0.SSS0.Px3.p1.1.1" class="ltx_text ltx_font_italic">e.g.</span>, language) can immediately process raw data from another (<span id="S5.SS0.SSS0.Px3.p1.1.2" class="ltx_text ltx_font_italic">e.g.</span>, vision). What makes them adaptable to the new modalities is that they share a common modality-agnostic representation, and can readily process <em id="S5.SS0.SSS0.Px3.p1.1.3" class="ltx_emph ltx_font_italic">representations</em> of new modalities.
Furthermore, this implies that language models would achieve some notion of grounding in the visual domain even in the absence of cross-modal data.<span id="footnote5" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>In 1688, William Molyneux posed the question: could someone born blind, upon being given sight, be able to distinguish shapes by vision alone? <cite class="ltx_cite ltx_citemacro_citep">(Locke, <a href="#bib.bib73" title="" class="ltx_ref">1690</a>)</cite> Our arguments suggest an answer: not immediately, but after a bit of visual experience (to form a visual representation) it should be easy (by mapping to prior touch-based representations). Empirical data shows that indeed congentially blind children given sight can quickly learn such abilities <cite class="ltx_cite ltx_citemacro_citep">(Held et al., <a href="#bib.bib46" title="" class="ltx_ref">2011</a>)</cite>.</span></span></span>
The primary advantage of cross-modal data could then simply be sample efficiency.</p>
</div>
</section>
<section id="S5.SS0.SSS0.Px4" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Scaling may reduce hallucination and bias</h4>

<div id="S5.SS0.SSS0.Px4.p1" class="ltx_para ltx_noindent">
<p id="S5.SS0.SSS0.Px4.p1.1" class="ltx_p">A prominent shortcoming of current LLMs is their propensity to hallucinate, or output false statements. If models are indeed converging toward an accurate model of reality, and scale powers this convergence, then we may expect hallucinations to decrease with scale. Of course, our hypothesis is conditioned on the training data for future models constituting a sufficiently lossless and diverse set of measurements. This may not come to pass, but it is an implication of our hypothesis worth pointing out. A similar argument can be made about certain kinds of bias. It has been shown that large models can exacerbate existing biases present in their training data <cite class="ltx_cite ltx_citemacro_citep">(Hall et al., <a href="#bib.bib42" title="" class="ltx_ref">2022</a>)</cite>. Our hypothesis implies that, while this may be true, we should expect <span id="S5.SS0.SSS0.Px4.p1.1.1" class="ltx_text ltx_font_italic">larger</span> models to amplify bias <span id="S5.SS0.SSS0.Px4.p1.1.2" class="ltx_text ltx_font_italic">less</span>. This does not mean bias will be removed, rather that the model’s biases will more accurately reflect the data’s biases, rather than exacerbating them.</p>
</div>
</section>
</section>
<section id="S6" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">6 </span>Counterexamples and limitations</h2>

<section id="S6.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Different modalities may contain different information</h4>

<div id="S6.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px1.p1.1" class="ltx_p">One immediate objection to our hypothesis is: what about the information that is unique to a given modality? Can language really describe the ineffable experience of watching a total solar eclipse? Or, how could an image convey the a concept like “I believe in the freedom of speech,” which is easy to write in English? Two different models cannot converge to the same representation if they have access to fundamentally different information.</p>
</div>
<div id="S6.SS0.SSS0.Px1.p2" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px1.p2.1" class="ltx_p">More precisely, our mathematical argument in <a href="#S4" title="4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a> only strictly holds for bijective projections of <math id="S6.SS0.SSS0.Px1.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{Z}" display="inline"><semantics id="S6.SS0.SSS0.Px1.p2.1.m1.1a"><mi id="S6.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S6.SS0.SSS0.Px1.p2.1.m1.1.1.cmml">𝐙</mi><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px1.p2.1.m1.1b"><ci id="S6.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px1.p2.1.m1.1.1">𝐙</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px1.p2.1.m1.1c">\mathbf{Z}</annotation></semantics></math>, so that the information in all the projections is equivalent to the information in the underlying world. This will not hold true for either lossy or stochastic observation functions. Nonetheless, similar arguments have been made theoretically and empirically that cooccurrence relations are learned by practical contrastive <cite class="ltx_cite ltx_citemacro_citep">(Wang &amp; Isola, <a href="#bib.bib126" title="" class="ltx_ref">2020</a>; Zimmermann et al., <a href="#bib.bib139" title="" class="ltx_ref">2021</a>)</cite> and predictive learners <cite class="ltx_cite ltx_citemacro_citep">(Papyan et al., <a href="#bib.bib94" title="" class="ltx_ref">2020</a>)</cite>.
<cite class="ltx_cite ltx_citemacro_citet">Lu et al. (<a href="#bib.bib75" title="" class="ltx_ref">2021</a>)</cite> and <cite class="ltx_cite ltx_citemacro_citet">Mirchandani et al. (<a href="#bib.bib81" title="" class="ltx_ref">2023</a>)</cite> also showed that models trained to autoregressively generate text also capture statistical relations in many other modalities, including symbolic reasoning,
vision, protein folding, and robotics.</p>
</div>
<figure id="S6.F9" class="ltx_figure"><img src="/html/2405.07987/assets/x11.png" id="S6.F9.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="438" height="290" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 9: </span><span id="S6.F9.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Increasing caption density improves alignment:<span id="S6.F9.2.1.1" class="ltx_text ltx_font_medium"> We vary caption length using the Densely-Captioned-Images (DCI) dataset <cite class="ltx_cite ltx_citemacro_citep">(Urbanek et al., <a href="#bib.bib124" title="" class="ltx_ref">2023</a>)</cite>. Starting from a dense caption, we used LLaMA3-8B-Instruct <cite class="ltx_cite ltx_citemacro_citep">(Meta, <a href="#bib.bib80" title="" class="ltx_ref">2024</a>)</cite> to summarize and generate coarse-grained captions. We compute the average alignment score across all vision and language models with standard deviation measured over the language models we evaluated. With denser captions, the mapping may become more bijective, leading to improved language-vision alignment scores.</span></span></figcaption>
</figure>
<div id="S6.SS0.SSS0.Px1.p3" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px1.p3.1" class="ltx_p">A more nuanced version of our hypothesis will need to be developed to handle the case of non-bijective observations and abstract concepts. A starting point could be: different models will converge to the same representation <span id="S6.SS0.SSS0.Px1.p3.1.1" class="ltx_text ltx_font_italic">when the input signals are sufficiently high information and the models are sufficiently high capacity</span>; when they are not, the lower-information representation will only align with the higher-information one up to a level capped by the mutual information between the input signals and by the capacity of each model. This cap might or might not be practically important. Popular representations like CLIP are explicitly optimized to only capture the shared information between vision and language, yet are highly successful on many pure vision tasks.
We perform a preliminary test of the effect of information level in Figure <a href="#S6.F9" title="Figure 9 ‣ Different modalities may contain different information ‣ 6 Counterexamples and limitations ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">9</span></a> (detailed in Appendix <a href="#A5" title="Appendix E Caption Density Experiments ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">E</span></a>), and find that the more descriptive (higher information) a caption is, the better its LLM representation aligns with the visual representation of the corresponding image.</p>
</div>
</section>
<section id="S6.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Not all representations are presently converging</h4>

<div id="S6.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px2.p1.1" class="ltx_p">Our argument has mainly focused on two modalities: vision and language. While we do expect other modalities will follow similar trends, we have yet to see the same level of convergence across all domains. For example, in robotics there is not yet a standardized approach to representing world states in the same way as there is for representing images and text. One limitation lies in the hardware used in robotics, which is often expensive and slow. This creates a bottleneck in the quantity and diversity of training data.</p>
</div>
</section>
<section id="S6.SS0.SSS0.Px3" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Sociological bias in producing AI models</h4>

<div id="S6.SS0.SSS0.Px3.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px3.p1.1" class="ltx_p">Researcher bias and collective preferences within the AI community have shaped the trajectory of model development. There is often an explicit or implicit goal of designing AI systems that mimic human reasoning and performance, and this could lead to convergence toward human-like representations even if other kinds of intelligence are in fact possible.
Additionally, the “hardware lottery” <cite class="ltx_cite ltx_citemacro_citep">(Hooker, <a href="#bib.bib48" title="" class="ltx_ref">2021</a>)</cite> suggests that the success of AI models can also depend on the compatibility of their design with available computational architectures, further contributing to convergent trends.</p>
</div>
</section>
<section id="S6.SS0.SSS0.Px4" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Special-purpose intelligences might not converge</h4>

<div id="S6.SS0.SSS0.Px4.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px4.p1.1" class="ltx_p">Different intelligent systems can be designed to accomplish different tasks. For instance:
A bioinformatics systems might predict protein structure;
an autonomous vehicle might follow lanes on highways. It’s possible that not much is shared between these two narrow tasks. Our argument only holds for intelligences that are optimized to perform well on <span id="S6.SS0.SSS0.Px4.p1.1.1" class="ltx_text ltx_font_italic">many</span> tasks. We have argued that a representation of <span id="S6.SS0.SSS0.Px4.p1.1.2" class="ltx_text ltx_font_italic">reality</span> is a structure that is useful across many tasks, but for any special purpose there may be shortcuts, or even effective representations detached from reality. Such shortcuts may be more efficient and necessary for continued improvements in specific domains. This will become more relevant if continued scaling comes up against boundary conditions around resources like energy and compute.</p>
</div>
</section>
<section id="S6.SS0.SSS0.Px5" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">
How do we measure alignment?
</h4>

<div id="S6.SS0.SSS0.Px5.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px5.p1.1" class="ltx_p">We focused on one particular alignment measure, mutual nearest-neighbor, in our experiments, and cited experiments using several others. However, there is active debate on the merits and deficiencies of all these ways of measuring alignment <cite class="ltx_cite ltx_citemacro_citep">(Bansal et al., <a href="#bib.bib9" title="" class="ltx_ref">2021</a>; Sucholutsky et al., <a href="#bib.bib115" title="" class="ltx_ref">2023</a>)</cite>. We discuss our choice and show results for other alignment metrics in Appendix <a href="#A1" title="Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">A</span></a>.</p>
</div>
</section>
<section id="S6.SS0.SSS0.Px6" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Lots left to explain</h4>

<div id="S6.SS0.SSS0.Px6.p1" class="ltx_para ltx_noindent">
<p id="S6.SS0.SSS0.Px6.p1.3" class="ltx_p">We have shown results where different models arrive at <span id="S6.SS0.SSS0.Px6.p1.3.1" class="ltx_text ltx_font_italic">similar</span> but not the <span id="S6.SS0.SSS0.Px6.p1.3.2" class="ltx_text ltx_font_italic">same</span> representations. For example, in <a href="#S2.F3" title="In 2.1 Different models, with different architectures and objectives, can have aligned representations ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a>, alignment clearly increases but only reaches a score of <math id="S6.SS0.SSS0.Px6.p1.1.m1.1" class="ltx_Math" alttext="0.16" display="inline"><semantics id="S6.SS0.SSS0.Px6.p1.1.m1.1a"><mn id="S6.SS0.SSS0.Px6.p1.1.m1.1.1" xref="S6.SS0.SSS0.Px6.p1.1.m1.1.1.cmml">0.16</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px6.p1.1.m1.1b"><cn type="float" id="S6.SS0.SSS0.Px6.p1.1.m1.1.1.cmml" xref="S6.SS0.SSS0.Px6.p1.1.m1.1.1">0.16</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px6.p1.1.m1.1c">0.16</annotation></semantics></math>, according to our mutual nearest-neighbor metric. The maximum theoretical value for this metric is <math id="S6.SS0.SSS0.Px6.p1.2.m2.1" class="ltx_Math" alttext="1" display="inline"><semantics id="S6.SS0.SSS0.Px6.p1.2.m2.1a"><mn id="S6.SS0.SSS0.Px6.p1.2.m2.1.1" xref="S6.SS0.SSS0.Px6.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px6.p1.2.m2.1b"><cn type="integer" id="S6.SS0.SSS0.Px6.p1.2.m2.1.1.cmml" xref="S6.SS0.SSS0.Px6.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px6.p1.2.m2.1c">1</annotation></semantics></math>. Is a score of <math id="S6.SS0.SSS0.Px6.p1.3.m3.1" class="ltx_Math" alttext="0.16" display="inline"><semantics id="S6.SS0.SSS0.Px6.p1.3.m3.1a"><mn id="S6.SS0.SSS0.Px6.p1.3.m3.1.1" xref="S6.SS0.SSS0.Px6.p1.3.m3.1.1.cmml">0.16</mn><annotation-xml encoding="MathML-Content" id="S6.SS0.SSS0.Px6.p1.3.m3.1b"><cn type="float" id="S6.SS0.SSS0.Px6.p1.3.m3.1.1.cmml" xref="S6.SS0.SSS0.Px6.p1.3.m3.1.1">0.16</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.SS0.SSS0.Px6.p1.3.m3.1c">0.16</annotation></semantics></math> indicative of strong alignment with the remaining gap being “noise” or does it signify poor alignment with major differences left to explain? We leave this as an open question.</p>
</div>
</section>
</section>
<section id="Sx1" class="ltx_section">
<h2 class="ltx_title ltx_title_section">Acknowledgements</h2>

<div id="Sx1.p1" class="ltx_para ltx_noindent">
<p id="Sx1.p1.1" class="ltx_p">We thank <cite class="ltx_cite ltx_citemacro_citeauthor"><a href="#bib.bib69" title="" class="ltx_ref">Lindsey &amp; Brown</a></cite> for sharing their data for our experiments shown in <a href="#S4.F8" title="In 4.1 An idealized world ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">8</span></a>. We thank the anonymous reviewers for helpful feedback, and for providing the counterexample on how to visually convey “I believe in the freedom of speech.” Thanks for Yonglong Tian, Dilip Krishnan, Anna Decker, Yoon Kim, Jyo Pari, Ani Nrusimha, Dave Epstein, Victor Butoi, and Seungwook Han for helpful discussions and suggestions. This work was supported by a Packard Fellowship and a Sloan Research Fellowship to P.I., by the MIT-IBM Watson AI Lab, by ONR MURI grant N00014-22-1-2740, by the Center for Brains, Minds, and Machines, the MIT Quest for Intelligence, NSF STC award CCF-1231216, the DARPA Knowledge Management at Scale and Speed (KMASS) program, and the DARPA Machine Common Sense (MCS) program.</p>
</div>
</section>
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<div class="ltx_pagination ltx_role_newpage"></div>
<div class="ltx_pagination ltx_role_newpage"></div>
<section id="A1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix A </span>Mutual <math id="A1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.1.m1.1b"><mi id="A1.1.m1.1.1" xref="A1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.1.m1.1c"><ci id="A1.1.m1.1.1.cmml" xref="A1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.1.m1.1d">k</annotation></semantics></math>-Nearest Neighbor Alignment Metric</h2>

<div id="A1.p1" class="ltx_para ltx_noindent">
<p id="A1.p1.4" class="ltx_p">For two models with representations <math id="A1.p1.1.m1.1" class="ltx_Math" alttext="f" display="inline"><semantics id="A1.p1.1.m1.1a"><mi id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><ci id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">f</annotation></semantics></math>, <math id="A1.p1.2.m2.1" class="ltx_Math" alttext="g" display="inline"><semantics id="A1.p1.2.m2.1a"><mi id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><ci id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">g</annotation></semantics></math> the mutual <math id="A1.p1.3.m3.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.p1.3.m3.1a"><mi id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.1b"><ci id="A1.p1.3.m3.1.1.cmml" xref="A1.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.1c">k</annotation></semantics></math>-nearest neighbor metric measures the average overlap of their respective nearest neighbor sets. In this section, we refer to this metric as <math id="A1.p1.4.m4.1" class="ltx_Math" alttext="m_{\texttt{NN}}" display="inline"><semantics id="A1.p1.4.m4.1a"><msub id="A1.p1.4.m4.1.1" xref="A1.p1.4.m4.1.1.cmml"><mi id="A1.p1.4.m4.1.1.2" xref="A1.p1.4.m4.1.1.2.cmml">m</mi><mtext class="ltx_mathvariant_monospace" id="A1.p1.4.m4.1.1.3" xref="A1.p1.4.m4.1.1.3a.cmml">NN</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.p1.4.m4.1b"><apply id="A1.p1.4.m4.1.1.cmml" xref="A1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A1.p1.4.m4.1.1.1.cmml" xref="A1.p1.4.m4.1.1">subscript</csymbol><ci id="A1.p1.4.m4.1.1.2.cmml" xref="A1.p1.4.m4.1.1.2">𝑚</ci><ci id="A1.p1.4.m4.1.1.3a.cmml" xref="A1.p1.4.m4.1.1.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="A1.p1.4.m4.1.1.3.cmml" xref="A1.p1.4.m4.1.1.3">NN</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.4.m4.1c">m_{\texttt{NN}}</annotation></semantics></math>, which we will formally define below.</p>
</div>
<div id="A1.p2" class="ltx_para ltx_noindent">
<p id="A1.p2.13" class="ltx_p">For cross-modal domains, define <math id="A1.p2.1.m1.2" class="ltx_Math" alttext="(x_{i},y_{i})\in\mathcal{X}" display="inline"><semantics id="A1.p2.1.m1.2a"><mrow id="A1.p2.1.m1.2.2" xref="A1.p2.1.m1.2.2.cmml"><mrow id="A1.p2.1.m1.2.2.2.2" xref="A1.p2.1.m1.2.2.2.3.cmml"><mo stretchy="false" id="A1.p2.1.m1.2.2.2.2.3" xref="A1.p2.1.m1.2.2.2.3.cmml">(</mo><msub id="A1.p2.1.m1.1.1.1.1.1" xref="A1.p2.1.m1.1.1.1.1.1.cmml"><mi id="A1.p2.1.m1.1.1.1.1.1.2" xref="A1.p2.1.m1.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.1.m1.1.1.1.1.1.3" xref="A1.p2.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.p2.1.m1.2.2.2.2.4" xref="A1.p2.1.m1.2.2.2.3.cmml">,</mo><msub id="A1.p2.1.m1.2.2.2.2.2" xref="A1.p2.1.m1.2.2.2.2.2.cmml"><mi id="A1.p2.1.m1.2.2.2.2.2.2" xref="A1.p2.1.m1.2.2.2.2.2.2.cmml">y</mi><mi id="A1.p2.1.m1.2.2.2.2.2.3" xref="A1.p2.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.1.m1.2.2.2.2.5" xref="A1.p2.1.m1.2.2.2.3.cmml">)</mo></mrow><mo id="A1.p2.1.m1.2.2.3" xref="A1.p2.1.m1.2.2.3.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="A1.p2.1.m1.2.2.4" xref="A1.p2.1.m1.2.2.4.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.2b"><apply id="A1.p2.1.m1.2.2.cmml" xref="A1.p2.1.m1.2.2"><in id="A1.p2.1.m1.2.2.3.cmml" xref="A1.p2.1.m1.2.2.3"></in><interval closure="open" id="A1.p2.1.m1.2.2.2.3.cmml" xref="A1.p2.1.m1.2.2.2.2"><apply id="A1.p2.1.m1.1.1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.1.m1.1.1.1.1.1.1.cmml" xref="A1.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.1.m1.1.1.1.1.1.2.cmml" xref="A1.p2.1.m1.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.1.m1.1.1.1.1.1.3.cmml" xref="A1.p2.1.m1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.p2.1.m1.2.2.2.2.2.cmml" xref="A1.p2.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.p2.1.m1.2.2.2.2.2.1.cmml" xref="A1.p2.1.m1.2.2.2.2.2">subscript</csymbol><ci id="A1.p2.1.m1.2.2.2.2.2.2.cmml" xref="A1.p2.1.m1.2.2.2.2.2.2">𝑦</ci><ci id="A1.p2.1.m1.2.2.2.2.2.3.cmml" xref="A1.p2.1.m1.2.2.2.2.2.3">𝑖</ci></apply></interval><ci id="A1.p2.1.m1.2.2.4.cmml" xref="A1.p2.1.m1.2.2.4">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.2c">(x_{i},y_{i})\in\mathcal{X}</annotation></semantics></math> as a sample from the data distribution <math id="A1.p2.2.m2.1" class="ltx_Math" alttext="\mathcal{X}" display="inline"><semantics id="A1.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><ci id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">\mathcal{X}</annotation></semantics></math> (<span id="A1.p2.13.1" class="ltx_text ltx_font_italic">e.g.</span> image-caption dataset). For the single domain alignment measurements, the samples are equivalent <math id="A1.p2.3.m3.1" class="ltx_Math" alttext="x_{i}=y_{i}" display="inline"><semantics id="A1.p2.3.m3.1a"><mrow id="A1.p2.3.m3.1.1" xref="A1.p2.3.m3.1.1.cmml"><msub id="A1.p2.3.m3.1.1.2" xref="A1.p2.3.m3.1.1.2.cmml"><mi id="A1.p2.3.m3.1.1.2.2" xref="A1.p2.3.m3.1.1.2.2.cmml">x</mi><mi id="A1.p2.3.m3.1.1.2.3" xref="A1.p2.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="A1.p2.3.m3.1.1.1" xref="A1.p2.3.m3.1.1.1.cmml">=</mo><msub id="A1.p2.3.m3.1.1.3" xref="A1.p2.3.m3.1.1.3.cmml"><mi id="A1.p2.3.m3.1.1.3.2" xref="A1.p2.3.m3.1.1.3.2.cmml">y</mi><mi id="A1.p2.3.m3.1.1.3.3" xref="A1.p2.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.3.m3.1b"><apply id="A1.p2.3.m3.1.1.cmml" xref="A1.p2.3.m3.1.1"><eq id="A1.p2.3.m3.1.1.1.cmml" xref="A1.p2.3.m3.1.1.1"></eq><apply id="A1.p2.3.m3.1.1.2.cmml" xref="A1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.2.1.cmml" xref="A1.p2.3.m3.1.1.2">subscript</csymbol><ci id="A1.p2.3.m3.1.1.2.2.cmml" xref="A1.p2.3.m3.1.1.2.2">𝑥</ci><ci id="A1.p2.3.m3.1.1.2.3.cmml" xref="A1.p2.3.m3.1.1.2.3">𝑖</ci></apply><apply id="A1.p2.3.m3.1.1.3.cmml" xref="A1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.3.1.cmml" xref="A1.p2.3.m3.1.1.3">subscript</csymbol><ci id="A1.p2.3.m3.1.1.3.2.cmml" xref="A1.p2.3.m3.1.1.3.2">𝑦</ci><ci id="A1.p2.3.m3.1.1.3.3.cmml" xref="A1.p2.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m3.1c">x_{i}=y_{i}</annotation></semantics></math> (<span id="A1.p2.13.2" class="ltx_text ltx_font_italic">e.g.</span>, images for vision, and text for language).
Let <math id="A1.p2.4.m4.2" class="ltx_Math" alttext="\{x_{i},y_{i}\}_{i=1}^{b}" display="inline"><semantics id="A1.p2.4.m4.2a"><msubsup id="A1.p2.4.m4.2.2" xref="A1.p2.4.m4.2.2.cmml"><mrow id="A1.p2.4.m4.2.2.2.2.2" xref="A1.p2.4.m4.2.2.2.2.3.cmml"><mo stretchy="false" id="A1.p2.4.m4.2.2.2.2.2.3" xref="A1.p2.4.m4.2.2.2.2.3.cmml">{</mo><msub id="A1.p2.4.m4.1.1.1.1.1.1" xref="A1.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="A1.p2.4.m4.1.1.1.1.1.1.2" xref="A1.p2.4.m4.1.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.4.m4.1.1.1.1.1.1.3" xref="A1.p2.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.p2.4.m4.2.2.2.2.2.4" xref="A1.p2.4.m4.2.2.2.2.3.cmml">,</mo><msub id="A1.p2.4.m4.2.2.2.2.2.2" xref="A1.p2.4.m4.2.2.2.2.2.2.cmml"><mi id="A1.p2.4.m4.2.2.2.2.2.2.2" xref="A1.p2.4.m4.2.2.2.2.2.2.2.cmml">y</mi><mi id="A1.p2.4.m4.2.2.2.2.2.2.3" xref="A1.p2.4.m4.2.2.2.2.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.4.m4.2.2.2.2.2.5" xref="A1.p2.4.m4.2.2.2.2.3.cmml">}</mo></mrow><mrow id="A1.p2.4.m4.2.2.2.4" xref="A1.p2.4.m4.2.2.2.4.cmml"><mi id="A1.p2.4.m4.2.2.2.4.2" xref="A1.p2.4.m4.2.2.2.4.2.cmml">i</mi><mo id="A1.p2.4.m4.2.2.2.4.1" xref="A1.p2.4.m4.2.2.2.4.1.cmml">=</mo><mn id="A1.p2.4.m4.2.2.2.4.3" xref="A1.p2.4.m4.2.2.2.4.3.cmml">1</mn></mrow><mi id="A1.p2.4.m4.2.2.4" xref="A1.p2.4.m4.2.2.4.cmml">b</mi></msubsup><annotation-xml encoding="MathML-Content" id="A1.p2.4.m4.2b"><apply id="A1.p2.4.m4.2.2.cmml" xref="A1.p2.4.m4.2.2"><csymbol cd="ambiguous" id="A1.p2.4.m4.2.2.3.cmml" xref="A1.p2.4.m4.2.2">superscript</csymbol><apply id="A1.p2.4.m4.2.2.2.cmml" xref="A1.p2.4.m4.2.2"><csymbol cd="ambiguous" id="A1.p2.4.m4.2.2.2.3.cmml" xref="A1.p2.4.m4.2.2">subscript</csymbol><set id="A1.p2.4.m4.2.2.2.2.3.cmml" xref="A1.p2.4.m4.2.2.2.2.2"><apply id="A1.p2.4.m4.1.1.1.1.1.1.cmml" xref="A1.p2.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.p2.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="A1.p2.4.m4.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="A1.p2.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.p2.4.m4.2.2.2.2.2.2.cmml" xref="A1.p2.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.p2.4.m4.2.2.2.2.2.2.1.cmml" xref="A1.p2.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="A1.p2.4.m4.2.2.2.2.2.2.2.cmml" xref="A1.p2.4.m4.2.2.2.2.2.2.2">𝑦</ci><ci id="A1.p2.4.m4.2.2.2.2.2.2.3.cmml" xref="A1.p2.4.m4.2.2.2.2.2.2.3">𝑖</ci></apply></set><apply id="A1.p2.4.m4.2.2.2.4.cmml" xref="A1.p2.4.m4.2.2.2.4"><eq id="A1.p2.4.m4.2.2.2.4.1.cmml" xref="A1.p2.4.m4.2.2.2.4.1"></eq><ci id="A1.p2.4.m4.2.2.2.4.2.cmml" xref="A1.p2.4.m4.2.2.2.4.2">𝑖</ci><cn type="integer" id="A1.p2.4.m4.2.2.2.4.3.cmml" xref="A1.p2.4.m4.2.2.2.4.3">1</cn></apply></apply><ci id="A1.p2.4.m4.2.2.4.cmml" xref="A1.p2.4.m4.2.2.4">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.4.m4.2c">\{x_{i},y_{i}\}_{i=1}^{b}</annotation></semantics></math> be the corresponding mini-batch sampled from this data distribution. Then given two model representations <math id="A1.p2.5.m5.1" class="ltx_Math" alttext="f" display="inline"><semantics id="A1.p2.5.m5.1a"><mi id="A1.p2.5.m5.1.1" xref="A1.p2.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A1.p2.5.m5.1b"><ci id="A1.p2.5.m5.1.1.cmml" xref="A1.p2.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.5.m5.1c">f</annotation></semantics></math> and <math id="A1.p2.6.m6.1" class="ltx_Math" alttext="g" display="inline"><semantics id="A1.p2.6.m6.1a"><mi id="A1.p2.6.m6.1.1" xref="A1.p2.6.m6.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="A1.p2.6.m6.1b"><ci id="A1.p2.6.m6.1.1.cmml" xref="A1.p2.6.m6.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.6.m6.1c">g</annotation></semantics></math> the corresponding features are:
<math id="A1.p2.7.m7.1" class="ltx_Math" alttext="\phi_{i}=f(x_{i})" display="inline"><semantics id="A1.p2.7.m7.1a"><mrow id="A1.p2.7.m7.1.1" xref="A1.p2.7.m7.1.1.cmml"><msub id="A1.p2.7.m7.1.1.3" xref="A1.p2.7.m7.1.1.3.cmml"><mi id="A1.p2.7.m7.1.1.3.2" xref="A1.p2.7.m7.1.1.3.2.cmml">ϕ</mi><mi id="A1.p2.7.m7.1.1.3.3" xref="A1.p2.7.m7.1.1.3.3.cmml">i</mi></msub><mo id="A1.p2.7.m7.1.1.2" xref="A1.p2.7.m7.1.1.2.cmml">=</mo><mrow id="A1.p2.7.m7.1.1.1" xref="A1.p2.7.m7.1.1.1.cmml"><mi id="A1.p2.7.m7.1.1.1.3" xref="A1.p2.7.m7.1.1.1.3.cmml">f</mi><mo lspace="0em" rspace="0em" id="A1.p2.7.m7.1.1.1.2" xref="A1.p2.7.m7.1.1.1.2.cmml">​</mo><mrow id="A1.p2.7.m7.1.1.1.1.1" xref="A1.p2.7.m7.1.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.p2.7.m7.1.1.1.1.1.2" xref="A1.p2.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.7.m7.1.1.1.1.1.1" xref="A1.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="A1.p2.7.m7.1.1.1.1.1.1.2" xref="A1.p2.7.m7.1.1.1.1.1.1.2.cmml">x</mi><mi id="A1.p2.7.m7.1.1.1.1.1.1.3" xref="A1.p2.7.m7.1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.7.m7.1.1.1.1.1.3" xref="A1.p2.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.7.m7.1b"><apply id="A1.p2.7.m7.1.1.cmml" xref="A1.p2.7.m7.1.1"><eq id="A1.p2.7.m7.1.1.2.cmml" xref="A1.p2.7.m7.1.1.2"></eq><apply id="A1.p2.7.m7.1.1.3.cmml" xref="A1.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="A1.p2.7.m7.1.1.3.1.cmml" xref="A1.p2.7.m7.1.1.3">subscript</csymbol><ci id="A1.p2.7.m7.1.1.3.2.cmml" xref="A1.p2.7.m7.1.1.3.2">italic-ϕ</ci><ci id="A1.p2.7.m7.1.1.3.3.cmml" xref="A1.p2.7.m7.1.1.3.3">𝑖</ci></apply><apply id="A1.p2.7.m7.1.1.1.cmml" xref="A1.p2.7.m7.1.1.1"><times id="A1.p2.7.m7.1.1.1.2.cmml" xref="A1.p2.7.m7.1.1.1.2"></times><ci id="A1.p2.7.m7.1.1.1.3.cmml" xref="A1.p2.7.m7.1.1.1.3">𝑓</ci><apply id="A1.p2.7.m7.1.1.1.1.1.1.cmml" xref="A1.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="A1.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="A1.p2.7.m7.1.1.1.1.1.1.2">𝑥</ci><ci id="A1.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="A1.p2.7.m7.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.7.m7.1c">\phi_{i}=f(x_{i})</annotation></semantics></math> and <math id="A1.p2.8.m8.1" class="ltx_Math" alttext="\psi_{i}=f(y_{i})" display="inline"><semantics id="A1.p2.8.m8.1a"><mrow id="A1.p2.8.m8.1.1" xref="A1.p2.8.m8.1.1.cmml"><msub id="A1.p2.8.m8.1.1.3" xref="A1.p2.8.m8.1.1.3.cmml"><mi id="A1.p2.8.m8.1.1.3.2" xref="A1.p2.8.m8.1.1.3.2.cmml">ψ</mi><mi id="A1.p2.8.m8.1.1.3.3" xref="A1.p2.8.m8.1.1.3.3.cmml">i</mi></msub><mo id="A1.p2.8.m8.1.1.2" xref="A1.p2.8.m8.1.1.2.cmml">=</mo><mrow id="A1.p2.8.m8.1.1.1" xref="A1.p2.8.m8.1.1.1.cmml"><mi id="A1.p2.8.m8.1.1.1.3" xref="A1.p2.8.m8.1.1.1.3.cmml">f</mi><mo lspace="0em" rspace="0em" id="A1.p2.8.m8.1.1.1.2" xref="A1.p2.8.m8.1.1.1.2.cmml">​</mo><mrow id="A1.p2.8.m8.1.1.1.1.1" xref="A1.p2.8.m8.1.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.p2.8.m8.1.1.1.1.1.2" xref="A1.p2.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.8.m8.1.1.1.1.1.1" xref="A1.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="A1.p2.8.m8.1.1.1.1.1.1.2" xref="A1.p2.8.m8.1.1.1.1.1.1.2.cmml">y</mi><mi id="A1.p2.8.m8.1.1.1.1.1.1.3" xref="A1.p2.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.8.m8.1.1.1.1.1.3" xref="A1.p2.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.8.m8.1b"><apply id="A1.p2.8.m8.1.1.cmml" xref="A1.p2.8.m8.1.1"><eq id="A1.p2.8.m8.1.1.2.cmml" xref="A1.p2.8.m8.1.1.2"></eq><apply id="A1.p2.8.m8.1.1.3.cmml" xref="A1.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="A1.p2.8.m8.1.1.3.1.cmml" xref="A1.p2.8.m8.1.1.3">subscript</csymbol><ci id="A1.p2.8.m8.1.1.3.2.cmml" xref="A1.p2.8.m8.1.1.3.2">𝜓</ci><ci id="A1.p2.8.m8.1.1.3.3.cmml" xref="A1.p2.8.m8.1.1.3.3">𝑖</ci></apply><apply id="A1.p2.8.m8.1.1.1.cmml" xref="A1.p2.8.m8.1.1.1"><times id="A1.p2.8.m8.1.1.1.2.cmml" xref="A1.p2.8.m8.1.1.1.2"></times><ci id="A1.p2.8.m8.1.1.1.3.cmml" xref="A1.p2.8.m8.1.1.1.3">𝑓</ci><apply id="A1.p2.8.m8.1.1.1.1.1.1.cmml" xref="A1.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="A1.p2.8.m8.1.1.1.1.1">subscript</csymbol><ci id="A1.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="A1.p2.8.m8.1.1.1.1.1.1.2">𝑦</ci><ci id="A1.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="A1.p2.8.m8.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.8.m8.1c">\psi_{i}=f(y_{i})</annotation></semantics></math>, where the collection of these features are denoted as <math id="A1.p2.9.m9.3" class="ltx_Math" alttext="\Phi=\{\phi_{1},\dots,\phi_{b}\}" display="inline"><semantics id="A1.p2.9.m9.3a"><mrow id="A1.p2.9.m9.3.3" xref="A1.p2.9.m9.3.3.cmml"><mi mathvariant="normal" id="A1.p2.9.m9.3.3.4" xref="A1.p2.9.m9.3.3.4.cmml">Φ</mi><mo id="A1.p2.9.m9.3.3.3" xref="A1.p2.9.m9.3.3.3.cmml">=</mo><mrow id="A1.p2.9.m9.3.3.2.2" xref="A1.p2.9.m9.3.3.2.3.cmml"><mo stretchy="false" id="A1.p2.9.m9.3.3.2.2.3" xref="A1.p2.9.m9.3.3.2.3.cmml">{</mo><msub id="A1.p2.9.m9.2.2.1.1.1" xref="A1.p2.9.m9.2.2.1.1.1.cmml"><mi id="A1.p2.9.m9.2.2.1.1.1.2" xref="A1.p2.9.m9.2.2.1.1.1.2.cmml">ϕ</mi><mn id="A1.p2.9.m9.2.2.1.1.1.3" xref="A1.p2.9.m9.2.2.1.1.1.3.cmml">1</mn></msub><mo id="A1.p2.9.m9.3.3.2.2.4" xref="A1.p2.9.m9.3.3.2.3.cmml">,</mo><mi mathvariant="normal" id="A1.p2.9.m9.1.1" xref="A1.p2.9.m9.1.1.cmml">…</mi><mo id="A1.p2.9.m9.3.3.2.2.5" xref="A1.p2.9.m9.3.3.2.3.cmml">,</mo><msub id="A1.p2.9.m9.3.3.2.2.2" xref="A1.p2.9.m9.3.3.2.2.2.cmml"><mi id="A1.p2.9.m9.3.3.2.2.2.2" xref="A1.p2.9.m9.3.3.2.2.2.2.cmml">ϕ</mi><mi id="A1.p2.9.m9.3.3.2.2.2.3" xref="A1.p2.9.m9.3.3.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A1.p2.9.m9.3.3.2.2.6" xref="A1.p2.9.m9.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.9.m9.3b"><apply id="A1.p2.9.m9.3.3.cmml" xref="A1.p2.9.m9.3.3"><eq id="A1.p2.9.m9.3.3.3.cmml" xref="A1.p2.9.m9.3.3.3"></eq><ci id="A1.p2.9.m9.3.3.4.cmml" xref="A1.p2.9.m9.3.3.4">Φ</ci><set id="A1.p2.9.m9.3.3.2.3.cmml" xref="A1.p2.9.m9.3.3.2.2"><apply id="A1.p2.9.m9.2.2.1.1.1.cmml" xref="A1.p2.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.p2.9.m9.2.2.1.1.1.1.cmml" xref="A1.p2.9.m9.2.2.1.1.1">subscript</csymbol><ci id="A1.p2.9.m9.2.2.1.1.1.2.cmml" xref="A1.p2.9.m9.2.2.1.1.1.2">italic-ϕ</ci><cn type="integer" id="A1.p2.9.m9.2.2.1.1.1.3.cmml" xref="A1.p2.9.m9.2.2.1.1.1.3">1</cn></apply><ci id="A1.p2.9.m9.1.1.cmml" xref="A1.p2.9.m9.1.1">…</ci><apply id="A1.p2.9.m9.3.3.2.2.2.cmml" xref="A1.p2.9.m9.3.3.2.2.2"><csymbol cd="ambiguous" id="A1.p2.9.m9.3.3.2.2.2.1.cmml" xref="A1.p2.9.m9.3.3.2.2.2">subscript</csymbol><ci id="A1.p2.9.m9.3.3.2.2.2.2.cmml" xref="A1.p2.9.m9.3.3.2.2.2.2">italic-ϕ</ci><ci id="A1.p2.9.m9.3.3.2.2.2.3.cmml" xref="A1.p2.9.m9.3.3.2.2.2.3">𝑏</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.9.m9.3c">\Phi=\{\phi_{1},\dots,\phi_{b}\}</annotation></semantics></math> and <math id="A1.p2.10.m10.3" class="ltx_Math" alttext="\Psi=\{\psi_{1},\dots,\psi_{b}\}" display="inline"><semantics id="A1.p2.10.m10.3a"><mrow id="A1.p2.10.m10.3.3" xref="A1.p2.10.m10.3.3.cmml"><mi mathvariant="normal" id="A1.p2.10.m10.3.3.4" xref="A1.p2.10.m10.3.3.4.cmml">Ψ</mi><mo id="A1.p2.10.m10.3.3.3" xref="A1.p2.10.m10.3.3.3.cmml">=</mo><mrow id="A1.p2.10.m10.3.3.2.2" xref="A1.p2.10.m10.3.3.2.3.cmml"><mo stretchy="false" id="A1.p2.10.m10.3.3.2.2.3" xref="A1.p2.10.m10.3.3.2.3.cmml">{</mo><msub id="A1.p2.10.m10.2.2.1.1.1" xref="A1.p2.10.m10.2.2.1.1.1.cmml"><mi id="A1.p2.10.m10.2.2.1.1.1.2" xref="A1.p2.10.m10.2.2.1.1.1.2.cmml">ψ</mi><mn id="A1.p2.10.m10.2.2.1.1.1.3" xref="A1.p2.10.m10.2.2.1.1.1.3.cmml">1</mn></msub><mo id="A1.p2.10.m10.3.3.2.2.4" xref="A1.p2.10.m10.3.3.2.3.cmml">,</mo><mi mathvariant="normal" id="A1.p2.10.m10.1.1" xref="A1.p2.10.m10.1.1.cmml">…</mi><mo id="A1.p2.10.m10.3.3.2.2.5" xref="A1.p2.10.m10.3.3.2.3.cmml">,</mo><msub id="A1.p2.10.m10.3.3.2.2.2" xref="A1.p2.10.m10.3.3.2.2.2.cmml"><mi id="A1.p2.10.m10.3.3.2.2.2.2" xref="A1.p2.10.m10.3.3.2.2.2.2.cmml">ψ</mi><mi id="A1.p2.10.m10.3.3.2.2.2.3" xref="A1.p2.10.m10.3.3.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A1.p2.10.m10.3.3.2.2.6" xref="A1.p2.10.m10.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.10.m10.3b"><apply id="A1.p2.10.m10.3.3.cmml" xref="A1.p2.10.m10.3.3"><eq id="A1.p2.10.m10.3.3.3.cmml" xref="A1.p2.10.m10.3.3.3"></eq><ci id="A1.p2.10.m10.3.3.4.cmml" xref="A1.p2.10.m10.3.3.4">Ψ</ci><set id="A1.p2.10.m10.3.3.2.3.cmml" xref="A1.p2.10.m10.3.3.2.2"><apply id="A1.p2.10.m10.2.2.1.1.1.cmml" xref="A1.p2.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.p2.10.m10.2.2.1.1.1.1.cmml" xref="A1.p2.10.m10.2.2.1.1.1">subscript</csymbol><ci id="A1.p2.10.m10.2.2.1.1.1.2.cmml" xref="A1.p2.10.m10.2.2.1.1.1.2">𝜓</ci><cn type="integer" id="A1.p2.10.m10.2.2.1.1.1.3.cmml" xref="A1.p2.10.m10.2.2.1.1.1.3">1</cn></apply><ci id="A1.p2.10.m10.1.1.cmml" xref="A1.p2.10.m10.1.1">…</ci><apply id="A1.p2.10.m10.3.3.2.2.2.cmml" xref="A1.p2.10.m10.3.3.2.2.2"><csymbol cd="ambiguous" id="A1.p2.10.m10.3.3.2.2.2.1.cmml" xref="A1.p2.10.m10.3.3.2.2.2">subscript</csymbol><ci id="A1.p2.10.m10.3.3.2.2.2.2.cmml" xref="A1.p2.10.m10.3.3.2.2.2.2">𝜓</ci><ci id="A1.p2.10.m10.3.3.2.2.2.3.cmml" xref="A1.p2.10.m10.3.3.2.2.2.3">𝑏</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.10.m10.3c">\Psi=\{\psi_{1},\dots,\psi_{b}\}</annotation></semantics></math>.
Then for each feature pair <math id="A1.p2.11.m11.2" class="ltx_Math" alttext="(\phi_{i},\psi_{i})" display="inline"><semantics id="A1.p2.11.m11.2a"><mrow id="A1.p2.11.m11.2.2.2" xref="A1.p2.11.m11.2.2.3.cmml"><mo stretchy="false" id="A1.p2.11.m11.2.2.2.3" xref="A1.p2.11.m11.2.2.3.cmml">(</mo><msub id="A1.p2.11.m11.1.1.1.1" xref="A1.p2.11.m11.1.1.1.1.cmml"><mi id="A1.p2.11.m11.1.1.1.1.2" xref="A1.p2.11.m11.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.p2.11.m11.1.1.1.1.3" xref="A1.p2.11.m11.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.p2.11.m11.2.2.2.4" xref="A1.p2.11.m11.2.2.3.cmml">,</mo><msub id="A1.p2.11.m11.2.2.2.2" xref="A1.p2.11.m11.2.2.2.2.cmml"><mi id="A1.p2.11.m11.2.2.2.2.2" xref="A1.p2.11.m11.2.2.2.2.2.cmml">ψ</mi><mi id="A1.p2.11.m11.2.2.2.2.3" xref="A1.p2.11.m11.2.2.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.11.m11.2.2.2.5" xref="A1.p2.11.m11.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.11.m11.2b"><interval closure="open" id="A1.p2.11.m11.2.2.3.cmml" xref="A1.p2.11.m11.2.2.2"><apply id="A1.p2.11.m11.1.1.1.1.cmml" xref="A1.p2.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.11.m11.1.1.1.1.1.cmml" xref="A1.p2.11.m11.1.1.1.1">subscript</csymbol><ci id="A1.p2.11.m11.1.1.1.1.2.cmml" xref="A1.p2.11.m11.1.1.1.1.2">italic-ϕ</ci><ci id="A1.p2.11.m11.1.1.1.1.3.cmml" xref="A1.p2.11.m11.1.1.1.1.3">𝑖</ci></apply><apply id="A1.p2.11.m11.2.2.2.2.cmml" xref="A1.p2.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="A1.p2.11.m11.2.2.2.2.1.cmml" xref="A1.p2.11.m11.2.2.2.2">subscript</csymbol><ci id="A1.p2.11.m11.2.2.2.2.2.cmml" xref="A1.p2.11.m11.2.2.2.2.2">𝜓</ci><ci id="A1.p2.11.m11.2.2.2.2.3.cmml" xref="A1.p2.11.m11.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.11.m11.2c">(\phi_{i},\psi_{i})</annotation></semantics></math>, we compute the respective nearest neighbor sets <math id="A1.p2.12.m12.1" class="ltx_Math" alttext="\mathcal{S}(\phi_{i})" display="inline"><semantics id="A1.p2.12.m12.1a"><mrow id="A1.p2.12.m12.1.1" xref="A1.p2.12.m12.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.12.m12.1.1.3" xref="A1.p2.12.m12.1.1.3.cmml">𝒮</mi><mo lspace="0em" rspace="0em" id="A1.p2.12.m12.1.1.2" xref="A1.p2.12.m12.1.1.2.cmml">​</mo><mrow id="A1.p2.12.m12.1.1.1.1" xref="A1.p2.12.m12.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.p2.12.m12.1.1.1.1.2" xref="A1.p2.12.m12.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.12.m12.1.1.1.1.1" xref="A1.p2.12.m12.1.1.1.1.1.cmml"><mi id="A1.p2.12.m12.1.1.1.1.1.2" xref="A1.p2.12.m12.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.p2.12.m12.1.1.1.1.1.3" xref="A1.p2.12.m12.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A1.p2.12.m12.1.1.1.1.3" xref="A1.p2.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.12.m12.1b"><apply id="A1.p2.12.m12.1.1.cmml" xref="A1.p2.12.m12.1.1"><times id="A1.p2.12.m12.1.1.2.cmml" xref="A1.p2.12.m12.1.1.2"></times><ci id="A1.p2.12.m12.1.1.3.cmml" xref="A1.p2.12.m12.1.1.3">𝒮</ci><apply id="A1.p2.12.m12.1.1.1.1.1.cmml" xref="A1.p2.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.12.m12.1.1.1.1.1.1.cmml" xref="A1.p2.12.m12.1.1.1.1">subscript</csymbol><ci id="A1.p2.12.m12.1.1.1.1.1.2.cmml" xref="A1.p2.12.m12.1.1.1.1.1.2">italic-ϕ</ci><ci id="A1.p2.12.m12.1.1.1.1.1.3.cmml" xref="A1.p2.12.m12.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.12.m12.1c">\mathcal{S}(\phi_{i})</annotation></semantics></math> and <math id="A1.p2.13.m13.1" class="ltx_Math" alttext="\mathcal{S}(\psi_{j})" display="inline"><semantics id="A1.p2.13.m13.1a"><mrow id="A1.p2.13.m13.1.1" xref="A1.p2.13.m13.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.p2.13.m13.1.1.3" xref="A1.p2.13.m13.1.1.3.cmml">𝒮</mi><mo lspace="0em" rspace="0em" id="A1.p2.13.m13.1.1.2" xref="A1.p2.13.m13.1.1.2.cmml">​</mo><mrow id="A1.p2.13.m13.1.1.1.1" xref="A1.p2.13.m13.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.p2.13.m13.1.1.1.1.2" xref="A1.p2.13.m13.1.1.1.1.1.cmml">(</mo><msub id="A1.p2.13.m13.1.1.1.1.1" xref="A1.p2.13.m13.1.1.1.1.1.cmml"><mi id="A1.p2.13.m13.1.1.1.1.1.2" xref="A1.p2.13.m13.1.1.1.1.1.2.cmml">ψ</mi><mi id="A1.p2.13.m13.1.1.1.1.1.3" xref="A1.p2.13.m13.1.1.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="A1.p2.13.m13.1.1.1.1.3" xref="A1.p2.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.13.m13.1b"><apply id="A1.p2.13.m13.1.1.cmml" xref="A1.p2.13.m13.1.1"><times id="A1.p2.13.m13.1.1.2.cmml" xref="A1.p2.13.m13.1.1.2"></times><ci id="A1.p2.13.m13.1.1.3.cmml" xref="A1.p2.13.m13.1.1.3">𝒮</ci><apply id="A1.p2.13.m13.1.1.1.1.1.cmml" xref="A1.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="A1.p2.13.m13.1.1.1.1.1.1.cmml" xref="A1.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="A1.p2.13.m13.1.1.1.1.1.2.cmml" xref="A1.p2.13.m13.1.1.1.1.1.2">𝜓</ci><ci id="A1.p2.13.m13.1.1.1.1.1.3.cmml" xref="A1.p2.13.m13.1.1.1.1.1.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.13.m13.1c">\mathcal{S}(\psi_{j})</annotation></semantics></math>.</p>
<table id="A6.EGx7" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E9.m1.3" class="ltx_Math" alttext="\displaystyle d_{\mathsf{knn}}(\phi_{i},\Phi\setminus\phi_{i})=\mathcal{S}(\phi_{i})" display="inline"><semantics id="A1.E9.m1.3a"><mrow id="A1.E9.m1.3.3" xref="A1.E9.m1.3.3.cmml"><mrow id="A1.E9.m1.2.2.2" xref="A1.E9.m1.2.2.2.cmml"><msub id="A1.E9.m1.2.2.2.4" xref="A1.E9.m1.2.2.2.4.cmml"><mi id="A1.E9.m1.2.2.2.4.2" xref="A1.E9.m1.2.2.2.4.2.cmml">d</mi><mi id="A1.E9.m1.2.2.2.4.3" xref="A1.E9.m1.2.2.2.4.3.cmml">𝗄𝗇𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A1.E9.m1.2.2.2.3" xref="A1.E9.m1.2.2.2.3.cmml">​</mo><mrow id="A1.E9.m1.2.2.2.2.2" xref="A1.E9.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="A1.E9.m1.2.2.2.2.2.3" xref="A1.E9.m1.2.2.2.2.3.cmml">(</mo><msub id="A1.E9.m1.1.1.1.1.1.1" xref="A1.E9.m1.1.1.1.1.1.1.cmml"><mi id="A1.E9.m1.1.1.1.1.1.1.2" xref="A1.E9.m1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.E9.m1.1.1.1.1.1.1.3" xref="A1.E9.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E9.m1.2.2.2.2.2.4" xref="A1.E9.m1.2.2.2.2.3.cmml">,</mo><mrow id="A1.E9.m1.2.2.2.2.2.2" xref="A1.E9.m1.2.2.2.2.2.2.cmml"><mi mathvariant="normal" id="A1.E9.m1.2.2.2.2.2.2.2" xref="A1.E9.m1.2.2.2.2.2.2.2.cmml">Φ</mi><mo id="A1.E9.m1.2.2.2.2.2.2.1" xref="A1.E9.m1.2.2.2.2.2.2.1.cmml">∖</mo><msub id="A1.E9.m1.2.2.2.2.2.2.3" xref="A1.E9.m1.2.2.2.2.2.2.3.cmml"><mi id="A1.E9.m1.2.2.2.2.2.2.3.2" xref="A1.E9.m1.2.2.2.2.2.2.3.2.cmml">ϕ</mi><mi id="A1.E9.m1.2.2.2.2.2.2.3.3" xref="A1.E9.m1.2.2.2.2.2.2.3.3.cmml">i</mi></msub></mrow><mo stretchy="false" id="A1.E9.m1.2.2.2.2.2.5" xref="A1.E9.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.E9.m1.3.3.4" xref="A1.E9.m1.3.3.4.cmml">=</mo><mrow id="A1.E9.m1.3.3.3" xref="A1.E9.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.E9.m1.3.3.3.3" xref="A1.E9.m1.3.3.3.3.cmml">𝒮</mi><mo lspace="0em" rspace="0em" id="A1.E9.m1.3.3.3.2" xref="A1.E9.m1.3.3.3.2.cmml">​</mo><mrow id="A1.E9.m1.3.3.3.1.1" xref="A1.E9.m1.3.3.3.1.1.1.cmml"><mo stretchy="false" id="A1.E9.m1.3.3.3.1.1.2" xref="A1.E9.m1.3.3.3.1.1.1.cmml">(</mo><msub id="A1.E9.m1.3.3.3.1.1.1" xref="A1.E9.m1.3.3.3.1.1.1.cmml"><mi id="A1.E9.m1.3.3.3.1.1.1.2" xref="A1.E9.m1.3.3.3.1.1.1.2.cmml">ϕ</mi><mi id="A1.E9.m1.3.3.3.1.1.1.3" xref="A1.E9.m1.3.3.3.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A1.E9.m1.3.3.3.1.1.3" xref="A1.E9.m1.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E9.m1.3b"><apply id="A1.E9.m1.3.3.cmml" xref="A1.E9.m1.3.3"><eq id="A1.E9.m1.3.3.4.cmml" xref="A1.E9.m1.3.3.4"></eq><apply id="A1.E9.m1.2.2.2.cmml" xref="A1.E9.m1.2.2.2"><times id="A1.E9.m1.2.2.2.3.cmml" xref="A1.E9.m1.2.2.2.3"></times><apply id="A1.E9.m1.2.2.2.4.cmml" xref="A1.E9.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.E9.m1.2.2.2.4.1.cmml" xref="A1.E9.m1.2.2.2.4">subscript</csymbol><ci id="A1.E9.m1.2.2.2.4.2.cmml" xref="A1.E9.m1.2.2.2.4.2">𝑑</ci><ci id="A1.E9.m1.2.2.2.4.3.cmml" xref="A1.E9.m1.2.2.2.4.3">𝗄𝗇𝗇</ci></apply><interval closure="open" id="A1.E9.m1.2.2.2.2.3.cmml" xref="A1.E9.m1.2.2.2.2.2"><apply id="A1.E9.m1.1.1.1.1.1.1.cmml" xref="A1.E9.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E9.m1.1.1.1.1.1.1.1.cmml" xref="A1.E9.m1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.E9.m1.1.1.1.1.1.1.2.cmml" xref="A1.E9.m1.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="A1.E9.m1.1.1.1.1.1.1.3.cmml" xref="A1.E9.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.E9.m1.2.2.2.2.2.2.cmml" xref="A1.E9.m1.2.2.2.2.2.2"><setdiff id="A1.E9.m1.2.2.2.2.2.2.1.cmml" xref="A1.E9.m1.2.2.2.2.2.2.1"></setdiff><ci id="A1.E9.m1.2.2.2.2.2.2.2.cmml" xref="A1.E9.m1.2.2.2.2.2.2.2">Φ</ci><apply id="A1.E9.m1.2.2.2.2.2.2.3.cmml" xref="A1.E9.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="A1.E9.m1.2.2.2.2.2.2.3.1.cmml" xref="A1.E9.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="A1.E9.m1.2.2.2.2.2.2.3.2.cmml" xref="A1.E9.m1.2.2.2.2.2.2.3.2">italic-ϕ</ci><ci id="A1.E9.m1.2.2.2.2.2.2.3.3.cmml" xref="A1.E9.m1.2.2.2.2.2.2.3.3">𝑖</ci></apply></apply></interval></apply><apply id="A1.E9.m1.3.3.3.cmml" xref="A1.E9.m1.3.3.3"><times id="A1.E9.m1.3.3.3.2.cmml" xref="A1.E9.m1.3.3.3.2"></times><ci id="A1.E9.m1.3.3.3.3.cmml" xref="A1.E9.m1.3.3.3.3">𝒮</ci><apply id="A1.E9.m1.3.3.3.1.1.1.cmml" xref="A1.E9.m1.3.3.3.1.1"><csymbol cd="ambiguous" id="A1.E9.m1.3.3.3.1.1.1.1.cmml" xref="A1.E9.m1.3.3.3.1.1">subscript</csymbol><ci id="A1.E9.m1.3.3.3.1.1.1.2.cmml" xref="A1.E9.m1.3.3.3.1.1.1.2">italic-ϕ</ci><ci id="A1.E9.m1.3.3.3.1.1.1.3.cmml" xref="A1.E9.m1.3.3.3.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E9.m1.3c">\displaystyle d_{\mathsf{knn}}(\phi_{i},\Phi\setminus\phi_{i})=\mathcal{S}(\phi_{i})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td>
</tr></tbody>
<tbody id="A1.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E10.m1.3" class="ltx_Math" alttext="\displaystyle d_{\mathsf{knn}}(\psi_{i},\Psi\setminus\psi_{i})=\mathcal{S}(\psi_{j})" display="inline"><semantics id="A1.E10.m1.3a"><mrow id="A1.E10.m1.3.3" xref="A1.E10.m1.3.3.cmml"><mrow id="A1.E10.m1.2.2.2" xref="A1.E10.m1.2.2.2.cmml"><msub id="A1.E10.m1.2.2.2.4" xref="A1.E10.m1.2.2.2.4.cmml"><mi id="A1.E10.m1.2.2.2.4.2" xref="A1.E10.m1.2.2.2.4.2.cmml">d</mi><mi id="A1.E10.m1.2.2.2.4.3" xref="A1.E10.m1.2.2.2.4.3.cmml">𝗄𝗇𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A1.E10.m1.2.2.2.3" xref="A1.E10.m1.2.2.2.3.cmml">​</mo><mrow id="A1.E10.m1.2.2.2.2.2" xref="A1.E10.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="A1.E10.m1.2.2.2.2.2.3" xref="A1.E10.m1.2.2.2.2.3.cmml">(</mo><msub id="A1.E10.m1.1.1.1.1.1.1" xref="A1.E10.m1.1.1.1.1.1.1.cmml"><mi id="A1.E10.m1.1.1.1.1.1.1.2" xref="A1.E10.m1.1.1.1.1.1.1.2.cmml">ψ</mi><mi id="A1.E10.m1.1.1.1.1.1.1.3" xref="A1.E10.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E10.m1.2.2.2.2.2.4" xref="A1.E10.m1.2.2.2.2.3.cmml">,</mo><mrow id="A1.E10.m1.2.2.2.2.2.2" xref="A1.E10.m1.2.2.2.2.2.2.cmml"><mi mathvariant="normal" id="A1.E10.m1.2.2.2.2.2.2.2" xref="A1.E10.m1.2.2.2.2.2.2.2.cmml">Ψ</mi><mo id="A1.E10.m1.2.2.2.2.2.2.1" xref="A1.E10.m1.2.2.2.2.2.2.1.cmml">∖</mo><msub id="A1.E10.m1.2.2.2.2.2.2.3" xref="A1.E10.m1.2.2.2.2.2.2.3.cmml"><mi id="A1.E10.m1.2.2.2.2.2.2.3.2" xref="A1.E10.m1.2.2.2.2.2.2.3.2.cmml">ψ</mi><mi id="A1.E10.m1.2.2.2.2.2.2.3.3" xref="A1.E10.m1.2.2.2.2.2.2.3.3.cmml">i</mi></msub></mrow><mo stretchy="false" id="A1.E10.m1.2.2.2.2.2.5" xref="A1.E10.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.E10.m1.3.3.4" xref="A1.E10.m1.3.3.4.cmml">=</mo><mrow id="A1.E10.m1.3.3.3" xref="A1.E10.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.E10.m1.3.3.3.3" xref="A1.E10.m1.3.3.3.3.cmml">𝒮</mi><mo lspace="0em" rspace="0em" id="A1.E10.m1.3.3.3.2" xref="A1.E10.m1.3.3.3.2.cmml">​</mo><mrow id="A1.E10.m1.3.3.3.1.1" xref="A1.E10.m1.3.3.3.1.1.1.cmml"><mo stretchy="false" id="A1.E10.m1.3.3.3.1.1.2" xref="A1.E10.m1.3.3.3.1.1.1.cmml">(</mo><msub id="A1.E10.m1.3.3.3.1.1.1" xref="A1.E10.m1.3.3.3.1.1.1.cmml"><mi id="A1.E10.m1.3.3.3.1.1.1.2" xref="A1.E10.m1.3.3.3.1.1.1.2.cmml">ψ</mi><mi id="A1.E10.m1.3.3.3.1.1.1.3" xref="A1.E10.m1.3.3.3.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="A1.E10.m1.3.3.3.1.1.3" xref="A1.E10.m1.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E10.m1.3b"><apply id="A1.E10.m1.3.3.cmml" xref="A1.E10.m1.3.3"><eq id="A1.E10.m1.3.3.4.cmml" xref="A1.E10.m1.3.3.4"></eq><apply id="A1.E10.m1.2.2.2.cmml" xref="A1.E10.m1.2.2.2"><times id="A1.E10.m1.2.2.2.3.cmml" xref="A1.E10.m1.2.2.2.3"></times><apply id="A1.E10.m1.2.2.2.4.cmml" xref="A1.E10.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.E10.m1.2.2.2.4.1.cmml" xref="A1.E10.m1.2.2.2.4">subscript</csymbol><ci id="A1.E10.m1.2.2.2.4.2.cmml" xref="A1.E10.m1.2.2.2.4.2">𝑑</ci><ci id="A1.E10.m1.2.2.2.4.3.cmml" xref="A1.E10.m1.2.2.2.4.3">𝗄𝗇𝗇</ci></apply><interval closure="open" id="A1.E10.m1.2.2.2.2.3.cmml" xref="A1.E10.m1.2.2.2.2.2"><apply id="A1.E10.m1.1.1.1.1.1.1.cmml" xref="A1.E10.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E10.m1.1.1.1.1.1.1.1.cmml" xref="A1.E10.m1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.E10.m1.1.1.1.1.1.1.2.cmml" xref="A1.E10.m1.1.1.1.1.1.1.2">𝜓</ci><ci id="A1.E10.m1.1.1.1.1.1.1.3.cmml" xref="A1.E10.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.E10.m1.2.2.2.2.2.2.cmml" xref="A1.E10.m1.2.2.2.2.2.2"><setdiff id="A1.E10.m1.2.2.2.2.2.2.1.cmml" xref="A1.E10.m1.2.2.2.2.2.2.1"></setdiff><ci id="A1.E10.m1.2.2.2.2.2.2.2.cmml" xref="A1.E10.m1.2.2.2.2.2.2.2">Ψ</ci><apply id="A1.E10.m1.2.2.2.2.2.2.3.cmml" xref="A1.E10.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="A1.E10.m1.2.2.2.2.2.2.3.1.cmml" xref="A1.E10.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="A1.E10.m1.2.2.2.2.2.2.3.2.cmml" xref="A1.E10.m1.2.2.2.2.2.2.3.2">𝜓</ci><ci id="A1.E10.m1.2.2.2.2.2.2.3.3.cmml" xref="A1.E10.m1.2.2.2.2.2.2.3.3">𝑖</ci></apply></apply></interval></apply><apply id="A1.E10.m1.3.3.3.cmml" xref="A1.E10.m1.3.3.3"><times id="A1.E10.m1.3.3.3.2.cmml" xref="A1.E10.m1.3.3.3.2"></times><ci id="A1.E10.m1.3.3.3.3.cmml" xref="A1.E10.m1.3.3.3.3">𝒮</ci><apply id="A1.E10.m1.3.3.3.1.1.1.cmml" xref="A1.E10.m1.3.3.3.1.1"><csymbol cd="ambiguous" id="A1.E10.m1.3.3.3.1.1.1.1.cmml" xref="A1.E10.m1.3.3.3.1.1">subscript</csymbol><ci id="A1.E10.m1.3.3.3.1.1.1.2.cmml" xref="A1.E10.m1.3.3.3.1.1.1.2">𝜓</ci><ci id="A1.E10.m1.3.3.3.1.1.1.3.cmml" xref="A1.E10.m1.3.3.3.1.1.1.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E10.m1.3c">\displaystyle d_{\mathsf{knn}}(\psi_{i},\Psi\setminus\psi_{i})=\mathcal{S}(\psi_{j})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td>
</tr></tbody>
</table>
<p id="A1.p2.15" class="ltx_p">where <math id="A1.p2.14.m1.1" class="ltx_Math" alttext="d_{\texttt{knn}}" display="inline"><semantics id="A1.p2.14.m1.1a"><msub id="A1.p2.14.m1.1.1" xref="A1.p2.14.m1.1.1.cmml"><mi id="A1.p2.14.m1.1.1.2" xref="A1.p2.14.m1.1.1.2.cmml">d</mi><mtext class="ltx_mathvariant_monospace" id="A1.p2.14.m1.1.1.3" xref="A1.p2.14.m1.1.1.3a.cmml">knn</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.p2.14.m1.1b"><apply id="A1.p2.14.m1.1.1.cmml" xref="A1.p2.14.m1.1.1"><csymbol cd="ambiguous" id="A1.p2.14.m1.1.1.1.cmml" xref="A1.p2.14.m1.1.1">subscript</csymbol><ci id="A1.p2.14.m1.1.1.2.cmml" xref="A1.p2.14.m1.1.1.2">𝑑</ci><ci id="A1.p2.14.m1.1.1.3a.cmml" xref="A1.p2.14.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="A1.p2.14.m1.1.1.3.cmml" xref="A1.p2.14.m1.1.1.3">knn</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.14.m1.1c">d_{\texttt{knn}}</annotation></semantics></math> returns the set of indices of its <math id="A1.p2.15.m2.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.p2.15.m2.1a"><mi id="A1.p2.15.m2.1.1" xref="A1.p2.15.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.p2.15.m2.1b"><ci id="A1.p2.15.m2.1.1.cmml" xref="A1.p2.15.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.15.m2.1c">k</annotation></semantics></math>-nearest neighbors. Then we measure its average intersection via</p>
<table id="A6.EGx8" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E11.m1.3" class="ltx_Math" alttext="\displaystyle m_{\texttt{NN}}(\phi_{i},\psi_{i})=\frac{1}{k}\lvert\mathcal{S}(\phi_{i})\cap\mathcal{S}(\psi_{j})\rvert" display="inline"><semantics id="A1.E11.m1.3a"><mrow id="A1.E11.m1.3.3" xref="A1.E11.m1.3.3.cmml"><mrow id="A1.E11.m1.2.2.2" xref="A1.E11.m1.2.2.2.cmml"><msub id="A1.E11.m1.2.2.2.4" xref="A1.E11.m1.2.2.2.4.cmml"><mi id="A1.E11.m1.2.2.2.4.2" xref="A1.E11.m1.2.2.2.4.2.cmml">m</mi><mtext class="ltx_mathvariant_monospace" id="A1.E11.m1.2.2.2.4.3" xref="A1.E11.m1.2.2.2.4.3a.cmml">NN</mtext></msub><mo lspace="0em" rspace="0em" id="A1.E11.m1.2.2.2.3" xref="A1.E11.m1.2.2.2.3.cmml">​</mo><mrow id="A1.E11.m1.2.2.2.2.2" xref="A1.E11.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="A1.E11.m1.2.2.2.2.2.3" xref="A1.E11.m1.2.2.2.2.3.cmml">(</mo><msub id="A1.E11.m1.1.1.1.1.1.1" xref="A1.E11.m1.1.1.1.1.1.1.cmml"><mi id="A1.E11.m1.1.1.1.1.1.1.2" xref="A1.E11.m1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.E11.m1.1.1.1.1.1.1.3" xref="A1.E11.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E11.m1.2.2.2.2.2.4" xref="A1.E11.m1.2.2.2.2.3.cmml">,</mo><msub id="A1.E11.m1.2.2.2.2.2.2" xref="A1.E11.m1.2.2.2.2.2.2.cmml"><mi id="A1.E11.m1.2.2.2.2.2.2.2" xref="A1.E11.m1.2.2.2.2.2.2.2.cmml">ψ</mi><mi id="A1.E11.m1.2.2.2.2.2.2.3" xref="A1.E11.m1.2.2.2.2.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="A1.E11.m1.2.2.2.2.2.5" xref="A1.E11.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.E11.m1.3.3.4" xref="A1.E11.m1.3.3.4.cmml">=</mo><mrow id="A1.E11.m1.3.3.3" xref="A1.E11.m1.3.3.3.cmml"><mstyle displaystyle="true" id="A1.E11.m1.3.3.3.3" xref="A1.E11.m1.3.3.3.3.cmml"><mfrac id="A1.E11.m1.3.3.3.3a" xref="A1.E11.m1.3.3.3.3.cmml"><mn id="A1.E11.m1.3.3.3.3.2" xref="A1.E11.m1.3.3.3.3.2.cmml">1</mn><mi id="A1.E11.m1.3.3.3.3.3" xref="A1.E11.m1.3.3.3.3.3.cmml">k</mi></mfrac></mstyle><mo lspace="0em" rspace="0em" id="A1.E11.m1.3.3.3.2" xref="A1.E11.m1.3.3.3.2.cmml">​</mo><mrow id="A1.E11.m1.3.3.3.1.1" 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xref="A1.E11.m1.3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.E11.m1.3.3.3.1.1.1.3" xref="A1.E11.m1.3.3.3.1.1.1.3.cmml">∩</mo><mrow id="A1.E11.m1.3.3.3.1.1.1.2" xref="A1.E11.m1.3.3.3.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.E11.m1.3.3.3.1.1.1.2.3" xref="A1.E11.m1.3.3.3.1.1.1.2.3.cmml">𝒮</mi><mo lspace="0em" rspace="0em" id="A1.E11.m1.3.3.3.1.1.1.2.2" xref="A1.E11.m1.3.3.3.1.1.1.2.2.cmml">​</mo><mrow id="A1.E11.m1.3.3.3.1.1.1.2.1.1" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.cmml"><mo stretchy="false" id="A1.E11.m1.3.3.3.1.1.1.2.1.1.2" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.cmml">(</mo><msub id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.cmml"><mi id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.2" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.2.cmml">ψ</mi><mi id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.3" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="A1.E11.m1.3.3.3.1.1.1.2.1.1.3" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo stretchy="false" id="A1.E11.m1.3.3.3.1.1.3" xref="A1.E11.m1.3.3.3.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E11.m1.3b"><apply id="A1.E11.m1.3.3.cmml" xref="A1.E11.m1.3.3"><eq id="A1.E11.m1.3.3.4.cmml" xref="A1.E11.m1.3.3.4"></eq><apply id="A1.E11.m1.2.2.2.cmml" xref="A1.E11.m1.2.2.2"><times id="A1.E11.m1.2.2.2.3.cmml" xref="A1.E11.m1.2.2.2.3"></times><apply id="A1.E11.m1.2.2.2.4.cmml" xref="A1.E11.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.E11.m1.2.2.2.4.1.cmml" xref="A1.E11.m1.2.2.2.4">subscript</csymbol><ci id="A1.E11.m1.2.2.2.4.2.cmml" xref="A1.E11.m1.2.2.2.4.2">𝑚</ci><ci id="A1.E11.m1.2.2.2.4.3a.cmml" xref="A1.E11.m1.2.2.2.4.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="A1.E11.m1.2.2.2.4.3.cmml" xref="A1.E11.m1.2.2.2.4.3">NN</mtext></ci></apply><interval closure="open" id="A1.E11.m1.2.2.2.2.3.cmml" xref="A1.E11.m1.2.2.2.2.2"><apply id="A1.E11.m1.1.1.1.1.1.1.cmml" xref="A1.E11.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E11.m1.1.1.1.1.1.1.1.cmml" xref="A1.E11.m1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.E11.m1.1.1.1.1.1.1.2.cmml" xref="A1.E11.m1.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="A1.E11.m1.1.1.1.1.1.1.3.cmml" xref="A1.E11.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.E11.m1.2.2.2.2.2.2.cmml" xref="A1.E11.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.E11.m1.2.2.2.2.2.2.1.cmml" xref="A1.E11.m1.2.2.2.2.2.2">subscript</csymbol><ci id="A1.E11.m1.2.2.2.2.2.2.2.cmml" xref="A1.E11.m1.2.2.2.2.2.2.2">𝜓</ci><ci id="A1.E11.m1.2.2.2.2.2.2.3.cmml" xref="A1.E11.m1.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply><apply id="A1.E11.m1.3.3.3.cmml" xref="A1.E11.m1.3.3.3"><times id="A1.E11.m1.3.3.3.2.cmml" xref="A1.E11.m1.3.3.3.2"></times><apply id="A1.E11.m1.3.3.3.3.cmml" xref="A1.E11.m1.3.3.3.3"><divide id="A1.E11.m1.3.3.3.3.1.cmml" xref="A1.E11.m1.3.3.3.3"></divide><cn type="integer" id="A1.E11.m1.3.3.3.3.2.cmml" 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xref="A1.E11.m1.3.3.3.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="A1.E11.m1.3.3.3.1.1.1.2.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2"><times id="A1.E11.m1.3.3.3.1.1.1.2.2.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.2"></times><ci id="A1.E11.m1.3.3.3.1.1.1.2.3.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.3">𝒮</ci><apply id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1"><csymbol cd="ambiguous" id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.1.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1">subscript</csymbol><ci id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.2.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.2">𝜓</ci><ci id="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.3.cmml" xref="A1.E11.m1.3.3.3.1.1.1.2.1.1.1.3">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E11.m1.3c">\displaystyle m_{\texttt{NN}}(\phi_{i},\psi_{i})=\frac{1}{k}\lvert\mathcal{S}(\phi_{i})\cap\mathcal{S}(\psi_{j})\rvert</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td>
</tr></tbody>
</table>
<p id="A1.p2.16" class="ltx_p">where <math id="A1.p2.16.m1.1" class="ltx_Math" alttext="\lvert{}\cdot{}\rvert" display="inline"><semantics id="A1.p2.16.m1.1a"><mrow id="A1.p2.16.m1.1.2.2" xref="A1.p2.16.m1.1.2.1.cmml"><mo stretchy="false" id="A1.p2.16.m1.1.2.2.1" xref="A1.p2.16.m1.1.2.1.1.cmml">|</mo><mo lspace="0em" rspace="0em" id="A1.p2.16.m1.1.1" xref="A1.p2.16.m1.1.1.cmml">⋅</mo><mo stretchy="false" id="A1.p2.16.m1.1.2.2.2" xref="A1.p2.16.m1.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.16.m1.1b"><apply id="A1.p2.16.m1.1.2.1.cmml" xref="A1.p2.16.m1.1.2.2"><abs id="A1.p2.16.m1.1.2.1.1.cmml" xref="A1.p2.16.m1.1.2.2.1"></abs><ci id="A1.p2.16.m1.1.1.cmml" xref="A1.p2.16.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.16.m1.1c">\lvert{}\cdot{}\rvert</annotation></semantics></math> is the size of the intersection.</p>
</div>
<section id="A1.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">The choice to use mutual nearest-neighbors</h4>

<div id="A1.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px1.p1.1" class="ltx_p">Our initial efforts to measure alignment with CKA revealed a very weak trend of alignment between models, even when comparing models within their own modality. This has also been observed by <cite class="ltx_cite ltx_citemacro_citep">(Bansal et al., <a href="#bib.bib9" title="" class="ltx_ref">2021</a>)</cite>, which had relied on alternative metrics such as model-stitching as it “reveals aspects of representations that measures such as centered kernel alignment (CKA) cannot” <cite class="ltx_cite ltx_citemacro_citep">(Bansal et al., <a href="#bib.bib9" title="" class="ltx_ref">2021</a>)</cite>.</p>
</div>
<div id="A1.SS0.SSS0.Px1.p2" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px1.p2.1" class="ltx_p">We chose to use nearest-neighbor as a metric, as methods like CKA has a very strict definition of alignment, which may not fit our current needs.
For instance, understanding the precise similarity between unrelated items, such as an orange and Bill Gates, may not be critical.</p>
</div>
</section>
<section id="A1.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Relationship between CKA and Mutual Nearest-Neighbors</h4>

<div id="A1.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px2.p1.6" class="ltx_p">Let <math id="A1.SS0.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="\phi_{i}\in\mathbb{R}^{n}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.1.m1.1a"><mrow id="A1.SS0.SSS0.Px2.p1.1.m1.1.1" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.cmml"><msub id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml"><mi id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.2" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.2.cmml">ϕ</mi><mi id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.3" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.1" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.1.cmml">∈</mo><msup id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml"><mi mathvariant="normal" id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.2" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.3" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.1.m1.1b"><apply id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1"><in id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.1"></in><apply id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.1.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.2">italic-ϕ</ci><ci id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.1.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3">superscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.2.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.2">ℝ</ci><ci id="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.3.cmml" xref="A1.SS0.SSS0.Px2.p1.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.1.m1.1c">\phi_{i}\in\mathbb{R}^{n}</annotation></semantics></math> and <math id="A1.SS0.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="\psi_{i}\in\mathbb{R}^{m}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.2.m2.1a"><mrow id="A1.SS0.SSS0.Px2.p1.2.m2.1.1" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.cmml"><msub id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml"><mi id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.2" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.2.cmml">ψ</mi><mi id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.3" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.1" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.1.cmml">∈</mo><msup id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi mathvariant="normal" id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.2" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.3" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.2.m2.1b"><apply id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1"><in id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.1"></in><apply id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.1.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.2">𝜓</ci><ci id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3">superscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.2">ℝ</ci><ci id="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" xref="A1.SS0.SSS0.Px2.p1.2.m2.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.2.m2.1c">\psi_{i}\in\mathbb{R}^{m}</annotation></semantics></math> be vectorized features of two models (<span id="A1.SS0.SSS0.Px2.p1.6.1" class="ltx_text ltx_font_italic">e.g.</span> language and vision models). Let <math id="A1.SS0.SSS0.Px2.p1.3.m3.2" class="ltx_Math" alttext="\mathbf{K}_{ij}=\kappa(\phi_{i},\phi_{j})" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.3.m3.2a"><mrow id="A1.SS0.SSS0.Px2.p1.3.m3.2.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.cmml"><msub id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.cmml"><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.2.cmml">𝐊</mi><mrow id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.cmml"><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.2.cmml">i</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.1" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.1.cmml">​</mo><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.3.cmml">j</mi></mrow></msub><mo id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.3.cmml">=</mo><mrow id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.cmml"><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.4" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.4.cmml">κ</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.3.cmml">​</mo><mrow id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml"><mo stretchy="false" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml">(</mo><msub id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.4" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml">,</mo><msub id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.cmml"><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.2" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.2.cmml">ϕ</mi><mi id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.3" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.5" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.3.m3.2b"><apply id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2"><eq id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.3"></eq><apply id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.1.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.2">𝐊</ci><apply id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3"><times id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.1.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.1"></times><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.2">𝑖</ci><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.4.3.3">𝑗</ci></apply></apply><apply id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2"><times id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.3"></times><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.4.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.4">𝜅</ci><interval closure="open" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2"><apply id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.1.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.2">italic-ϕ</ci><ci id="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.3.m3.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.3.m3.2c">\mathbf{K}_{ij}=\kappa(\phi_{i},\phi_{j})</annotation></semantics></math> and <math id="A1.SS0.SSS0.Px2.p1.4.m4.2" class="ltx_Math" alttext="\mathbf{L}_{ij}=\kappa(\psi_{i},\psi_{j})" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.4.m4.2a"><mrow id="A1.SS0.SSS0.Px2.p1.4.m4.2.2" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.cmml"><msub id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.cmml"><mi id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.2" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.2.cmml">𝐋</mi><mrow id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.cmml"><mi id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.2" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.2.cmml">i</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.1" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.1.cmml">​</mo><mi id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.3" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.4.3.3.cmml">j</mi></mrow></msub><mo id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.3" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.3.cmml">=</mo><mrow id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.cmml"><mi id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.4" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.4.cmml">κ</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.3" 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cd="ambiguous" id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.2">𝜓</ci><ci id="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="A1.SS0.SSS0.Px2.p1.4.m4.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.4.m4.2c">\mathbf{L}_{ij}=\kappa(\psi_{i},\psi_{j})</annotation></semantics></math> be the kernel matrices computed from a dataset using some kernel-function <math id="A1.SS0.SSS0.Px2.p1.5.m5.1" class="ltx_Math" alttext="\kappa" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.5.m5.1a"><mi id="A1.SS0.SSS0.Px2.p1.5.m5.1.1" xref="A1.SS0.SSS0.Px2.p1.5.m5.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.5.m5.1b"><ci id="A1.SS0.SSS0.Px2.p1.5.m5.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.5.m5.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.5.m5.1c">\kappa</annotation></semantics></math>. Using an inner-product kernel, the <math id="A1.SS0.SSS0.Px2.p1.6.m6.1" class="ltx_Math" alttext="ij" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.6.m6.1a"><mrow id="A1.SS0.SSS0.Px2.p1.6.m6.1.1" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.cmml"><mi id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.2" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml">i</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.1" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.1.cmml">​</mo><mi id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.3" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.6.m6.1b"><apply id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1"><times id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.1"></times><ci id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.2">𝑖</ci><ci id="A1.SS0.SSS0.Px2.p1.6.m6.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p1.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.6.m6.1c">ij</annotation></semantics></math>-th entry of the centered counterpart of these Kernel matrices is:</p>
<table id="A6.EGx9" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E12.m1.2" class="ltx_Math" alttext="\displaystyle\bar{\mathbf{K}}_{ij}=\langle\phi_{i},\phi_{j}\rangle-\mathbb{E}_{l}[\langle\phi_{i},\phi_{l}\rangle]\qquad\qquad\bar{\mathbf{L}}_{ij}=\langle\psi_{i},\psi_{j}\rangle-\mathbb{E}_{l}[\langle\psi_{i},\psi_{l}\rangle]" display="inline"><semantics id="A1.E12.m1.2a"><mrow id="A1.E12.m1.2.2.2" xref="A1.E12.m1.2.2.3.cmml"><mrow id="A1.E12.m1.1.1.1.1" xref="A1.E12.m1.1.1.1.1.cmml"><msub id="A1.E12.m1.1.1.1.1.5" xref="A1.E12.m1.1.1.1.1.5.cmml"><mover accent="true" id="A1.E12.m1.1.1.1.1.5.2" xref="A1.E12.m1.1.1.1.1.5.2.cmml"><mi id="A1.E12.m1.1.1.1.1.5.2.2" xref="A1.E12.m1.1.1.1.1.5.2.2.cmml">𝐊</mi><mo id="A1.E12.m1.1.1.1.1.5.2.1" xref="A1.E12.m1.1.1.1.1.5.2.1.cmml">¯</mo></mover><mrow id="A1.E12.m1.1.1.1.1.5.3" xref="A1.E12.m1.1.1.1.1.5.3.cmml"><mi id="A1.E12.m1.1.1.1.1.5.3.2" xref="A1.E12.m1.1.1.1.1.5.3.2.cmml">i</mi><mo lspace="0em" rspace="0em" id="A1.E12.m1.1.1.1.1.5.3.1" xref="A1.E12.m1.1.1.1.1.5.3.1.cmml">​</mo><mi id="A1.E12.m1.1.1.1.1.5.3.3" xref="A1.E12.m1.1.1.1.1.5.3.3.cmml">j</mi></mrow></msub><mo id="A1.E12.m1.1.1.1.1.4" xref="A1.E12.m1.1.1.1.1.4.cmml">=</mo><mrow id="A1.E12.m1.1.1.1.1.3" xref="A1.E12.m1.1.1.1.1.3.cmml"><mrow id="A1.E12.m1.1.1.1.1.2.2.2" xref="A1.E12.m1.1.1.1.1.2.2.3.cmml"><mo stretchy="false" id="A1.E12.m1.1.1.1.1.2.2.2.3" xref="A1.E12.m1.1.1.1.1.2.2.3.cmml">⟨</mo><msub id="A1.E12.m1.1.1.1.1.1.1.1.1" xref="A1.E12.m1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.E12.m1.1.1.1.1.1.1.1.1.2" xref="A1.E12.m1.1.1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.E12.m1.1.1.1.1.1.1.1.1.3" xref="A1.E12.m1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E12.m1.1.1.1.1.2.2.2.4" xref="A1.E12.m1.1.1.1.1.2.2.3.cmml">,</mo><msub id="A1.E12.m1.1.1.1.1.2.2.2.2" xref="A1.E12.m1.1.1.1.1.2.2.2.2.cmml"><mi id="A1.E12.m1.1.1.1.1.2.2.2.2.2" xref="A1.E12.m1.1.1.1.1.2.2.2.2.2.cmml">ϕ</mi><mi id="A1.E12.m1.1.1.1.1.2.2.2.2.3" 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xref="A1.E12.m1.2.2.2.2.2.2.2.2.3">𝑗</ci></apply></list><apply id="A1.E12.m1.2.2.2.2.3.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3"><times id="A1.E12.m1.2.2.2.2.3.3.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.2"></times><apply id="A1.E12.m1.2.2.2.2.3.3.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3.3"><csymbol cd="ambiguous" id="A1.E12.m1.2.2.2.2.3.3.3.1.cmml" xref="A1.E12.m1.2.2.2.2.3.3.3">subscript</csymbol><ci id="A1.E12.m1.2.2.2.2.3.3.3.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.3.2">𝔼</ci><ci id="A1.E12.m1.2.2.2.2.3.3.3.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3.3.3">𝑙</ci></apply><apply id="A1.E12.m1.2.2.2.2.3.3.1.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1"><csymbol cd="latexml" id="A1.E12.m1.2.2.2.2.3.3.1.2.1.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.2">delimited-[]</csymbol><list id="A1.E12.m1.2.2.2.2.3.3.1.1.1.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.2"><apply id="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.1.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1">subscript</csymbol><ci id="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.2">𝜓</ci><ci id="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.1.1.3">𝑖</ci></apply><apply id="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.1.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2">subscript</csymbol><ci id="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.2.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.2">𝜓</ci><ci id="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.3.cmml" xref="A1.E12.m1.2.2.2.2.3.3.1.1.1.2.2.3">𝑙</ci></apply></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E12.m1.2c">\displaystyle\bar{\mathbf{K}}_{ij}=\langle\phi_{i},\phi_{j}\rangle-\mathbb{E}_{l}[\langle\phi_{i},\phi_{l}\rangle]\qquad\qquad\bar{\mathbf{L}}_{ij}=\langle\psi_{i},\psi_{j}\rangle-\mathbb{E}_{l}[\langle\psi_{i},\psi_{l}\rangle]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td>
</tr></tbody>
</table>
<p id="A1.SS0.SSS0.Px2.p1.8" class="ltx_p">Then, the cross-covariance of <math id="A1.SS0.SSS0.Px2.p1.7.m1.1" class="ltx_Math" alttext="\mathbf{K}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.7.m1.1a"><mi id="A1.SS0.SSS0.Px2.p1.7.m1.1.1" xref="A1.SS0.SSS0.Px2.p1.7.m1.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.7.m1.1b"><ci id="A1.SS0.SSS0.Px2.p1.7.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.7.m1.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.7.m1.1c">\mathbf{K}</annotation></semantics></math> and <math id="A1.SS0.SSS0.Px2.p1.8.m2.1" class="ltx_Math" alttext="\mathbf{L}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.8.m2.1a"><mi id="A1.SS0.SSS0.Px2.p1.8.m2.1.1" xref="A1.SS0.SSS0.Px2.p1.8.m2.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.8.m2.1b"><ci id="A1.SS0.SSS0.Px2.p1.8.m2.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.8.m2.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.8.m2.1c">\mathbf{L}</annotation></semantics></math> is given by:</p>
<table id="A6.EGx10" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E13.m1.4" class="ltx_Math" alttext="\displaystyle\mathsf{HSIC}(\mathbf{K},\mathbf{L})=\frac{1}{(n-1)^{2}}\mathsf{Trace}(\bar{\mathbf{K}}\bar{\mathbf{L}})" display="inline"><semantics id="A1.E13.m1.4a"><mrow id="A1.E13.m1.4.4" xref="A1.E13.m1.4.4.cmml"><mrow id="A1.E13.m1.4.4.3" xref="A1.E13.m1.4.4.3.cmml"><mi id="A1.E13.m1.4.4.3.2" xref="A1.E13.m1.4.4.3.2.cmml">𝖧𝖲𝖨𝖢</mi><mo lspace="0em" rspace="0em" id="A1.E13.m1.4.4.3.1" xref="A1.E13.m1.4.4.3.1.cmml">​</mo><mrow id="A1.E13.m1.4.4.3.3.2" xref="A1.E13.m1.4.4.3.3.1.cmml"><mo stretchy="false" id="A1.E13.m1.4.4.3.3.2.1" xref="A1.E13.m1.4.4.3.3.1.cmml">(</mo><mi id="A1.E13.m1.2.2" xref="A1.E13.m1.2.2.cmml">𝐊</mi><mo id="A1.E13.m1.4.4.3.3.2.2" xref="A1.E13.m1.4.4.3.3.1.cmml">,</mo><mi id="A1.E13.m1.3.3" xref="A1.E13.m1.3.3.cmml">𝐋</mi><mo stretchy="false" id="A1.E13.m1.4.4.3.3.2.3" xref="A1.E13.m1.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="A1.E13.m1.4.4.2" xref="A1.E13.m1.4.4.2.cmml">=</mo><mrow id="A1.E13.m1.4.4.1" xref="A1.E13.m1.4.4.1.cmml"><mstyle displaystyle="true" id="A1.E13.m1.1.1" xref="A1.E13.m1.1.1.cmml"><mfrac id="A1.E13.m1.1.1a" xref="A1.E13.m1.1.1.cmml"><mn id="A1.E13.m1.1.1.3" xref="A1.E13.m1.1.1.3.cmml">1</mn><msup id="A1.E13.m1.1.1.1" xref="A1.E13.m1.1.1.1.cmml"><mrow id="A1.E13.m1.1.1.1.1.1" xref="A1.E13.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.E13.m1.1.1.1.1.1.2" xref="A1.E13.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.E13.m1.1.1.1.1.1.1" xref="A1.E13.m1.1.1.1.1.1.1.cmml"><mi id="A1.E13.m1.1.1.1.1.1.1.2" xref="A1.E13.m1.1.1.1.1.1.1.2.cmml">n</mi><mo id="A1.E13.m1.1.1.1.1.1.1.1" xref="A1.E13.m1.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.E13.m1.1.1.1.1.1.1.3" xref="A1.E13.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo stretchy="false" id="A1.E13.m1.1.1.1.1.1.3" xref="A1.E13.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="A1.E13.m1.1.1.1.3" xref="A1.E13.m1.1.1.1.3.cmml">2</mn></msup></mfrac></mstyle><mo lspace="0em" rspace="0em" id="A1.E13.m1.4.4.1.2" xref="A1.E13.m1.4.4.1.2.cmml">​</mo><mi id="A1.E13.m1.4.4.1.3" xref="A1.E13.m1.4.4.1.3.cmml">𝖳𝗋𝖺𝖼𝖾</mi><mo lspace="0em" rspace="0em" id="A1.E13.m1.4.4.1.2a" xref="A1.E13.m1.4.4.1.2.cmml">​</mo><mrow id="A1.E13.m1.4.4.1.1.1" xref="A1.E13.m1.4.4.1.1.1.1.cmml"><mo stretchy="false" id="A1.E13.m1.4.4.1.1.1.2" xref="A1.E13.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="A1.E13.m1.4.4.1.1.1.1" xref="A1.E13.m1.4.4.1.1.1.1.cmml"><mover accent="true" id="A1.E13.m1.4.4.1.1.1.1.2" xref="A1.E13.m1.4.4.1.1.1.1.2.cmml"><mi id="A1.E13.m1.4.4.1.1.1.1.2.2" xref="A1.E13.m1.4.4.1.1.1.1.2.2.cmml">𝐊</mi><mo id="A1.E13.m1.4.4.1.1.1.1.2.1" xref="A1.E13.m1.4.4.1.1.1.1.2.1.cmml">¯</mo></mover><mo lspace="0em" rspace="0em" id="A1.E13.m1.4.4.1.1.1.1.1" xref="A1.E13.m1.4.4.1.1.1.1.1.cmml">​</mo><mover accent="true" id="A1.E13.m1.4.4.1.1.1.1.3" xref="A1.E13.m1.4.4.1.1.1.1.3.cmml"><mi id="A1.E13.m1.4.4.1.1.1.1.3.2" xref="A1.E13.m1.4.4.1.1.1.1.3.2.cmml">𝐋</mi><mo id="A1.E13.m1.4.4.1.1.1.1.3.1" xref="A1.E13.m1.4.4.1.1.1.1.3.1.cmml">¯</mo></mover></mrow><mo stretchy="false" id="A1.E13.m1.4.4.1.1.1.3" xref="A1.E13.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E13.m1.4b"><apply id="A1.E13.m1.4.4.cmml" xref="A1.E13.m1.4.4"><eq id="A1.E13.m1.4.4.2.cmml" xref="A1.E13.m1.4.4.2"></eq><apply id="A1.E13.m1.4.4.3.cmml" xref="A1.E13.m1.4.4.3"><times id="A1.E13.m1.4.4.3.1.cmml" xref="A1.E13.m1.4.4.3.1"></times><ci id="A1.E13.m1.4.4.3.2.cmml" xref="A1.E13.m1.4.4.3.2">𝖧𝖲𝖨𝖢</ci><interval closure="open" id="A1.E13.m1.4.4.3.3.1.cmml" xref="A1.E13.m1.4.4.3.3.2"><ci id="A1.E13.m1.2.2.cmml" xref="A1.E13.m1.2.2">𝐊</ci><ci id="A1.E13.m1.3.3.cmml" xref="A1.E13.m1.3.3">𝐋</ci></interval></apply><apply id="A1.E13.m1.4.4.1.cmml" xref="A1.E13.m1.4.4.1"><times id="A1.E13.m1.4.4.1.2.cmml" xref="A1.E13.m1.4.4.1.2"></times><apply id="A1.E13.m1.1.1.cmml" xref="A1.E13.m1.1.1"><divide id="A1.E13.m1.1.1.2.cmml" xref="A1.E13.m1.1.1"></divide><cn type="integer" id="A1.E13.m1.1.1.3.cmml" xref="A1.E13.m1.1.1.3">1</cn><apply id="A1.E13.m1.1.1.1.cmml" xref="A1.E13.m1.1.1.1"><csymbol cd="ambiguous" id="A1.E13.m1.1.1.1.2.cmml" xref="A1.E13.m1.1.1.1">superscript</csymbol><apply id="A1.E13.m1.1.1.1.1.1.1.cmml" xref="A1.E13.m1.1.1.1.1.1"><minus id="A1.E13.m1.1.1.1.1.1.1.1.cmml" xref="A1.E13.m1.1.1.1.1.1.1.1"></minus><ci id="A1.E13.m1.1.1.1.1.1.1.2.cmml" xref="A1.E13.m1.1.1.1.1.1.1.2">𝑛</ci><cn type="integer" id="A1.E13.m1.1.1.1.1.1.1.3.cmml" xref="A1.E13.m1.1.1.1.1.1.1.3">1</cn></apply><cn type="integer" id="A1.E13.m1.1.1.1.3.cmml" xref="A1.E13.m1.1.1.1.3">2</cn></apply></apply><ci id="A1.E13.m1.4.4.1.3.cmml" xref="A1.E13.m1.4.4.1.3">𝖳𝗋𝖺𝖼𝖾</ci><apply id="A1.E13.m1.4.4.1.1.1.1.cmml" xref="A1.E13.m1.4.4.1.1.1"><times id="A1.E13.m1.4.4.1.1.1.1.1.cmml" xref="A1.E13.m1.4.4.1.1.1.1.1"></times><apply id="A1.E13.m1.4.4.1.1.1.1.2.cmml" xref="A1.E13.m1.4.4.1.1.1.1.2"><ci id="A1.E13.m1.4.4.1.1.1.1.2.1.cmml" xref="A1.E13.m1.4.4.1.1.1.1.2.1">¯</ci><ci id="A1.E13.m1.4.4.1.1.1.1.2.2.cmml" xref="A1.E13.m1.4.4.1.1.1.1.2.2">𝐊</ci></apply><apply id="A1.E13.m1.4.4.1.1.1.1.3.cmml" xref="A1.E13.m1.4.4.1.1.1.1.3"><ci id="A1.E13.m1.4.4.1.1.1.1.3.1.cmml" xref="A1.E13.m1.4.4.1.1.1.1.3.1">¯</ci><ci id="A1.E13.m1.4.4.1.1.1.1.3.2.cmml" xref="A1.E13.m1.4.4.1.1.1.1.3.2">𝐋</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E13.m1.4c">\displaystyle\mathsf{HSIC}(\mathbf{K},\mathbf{L})=\frac{1}{(n-1)^{2}}\mathsf{Trace}(\bar{\mathbf{K}}\bar{\mathbf{L}})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td>
</tr></tbody>
</table>
<p id="A1.SS0.SSS0.Px2.p1.12" class="ltx_p">which serves as an empirical estimator of the Hilbert-Schmidt Independence Criterion <cite class="ltx_cite ltx_citemacro_citep">(Gretton et al., <a href="#bib.bib37" title="" class="ltx_ref">2005</a>)</cite>. The Centered Kernel Alignment (CKA) <cite class="ltx_cite ltx_citemacro_citep">(Kornblith et al., <a href="#bib.bib61" title="" class="ltx_ref">2019</a>)</cite> is then its normalized counterpart:</p>
<table id="A6.EGx11" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E14.m1.8" class="ltx_Math" alttext="\displaystyle\mathsf{CKA}(\mathbf{K},\mathbf{L})=\frac{\mathsf{HSIC}(\mathbf{K},\mathbf{L})}{\sqrt{\mathsf{HSIC}(\mathbf{K},\mathbf{K})\mathsf{HSIC}(\mathbf{L},\mathbf{L})}}" display="inline"><semantics id="A1.E14.m1.8a"><mrow id="A1.E14.m1.8.9" xref="A1.E14.m1.8.9.cmml"><mrow id="A1.E14.m1.8.9.2" xref="A1.E14.m1.8.9.2.cmml"><mi id="A1.E14.m1.8.9.2.2" xref="A1.E14.m1.8.9.2.2.cmml">𝖢𝖪𝖠</mi><mo lspace="0em" rspace="0em" id="A1.E14.m1.8.9.2.1" xref="A1.E14.m1.8.9.2.1.cmml">​</mo><mrow id="A1.E14.m1.8.9.2.3.2" xref="A1.E14.m1.8.9.2.3.1.cmml"><mo stretchy="false" id="A1.E14.m1.8.9.2.3.2.1" xref="A1.E14.m1.8.9.2.3.1.cmml">(</mo><mi id="A1.E14.m1.7.7" xref="A1.E14.m1.7.7.cmml">𝐊</mi><mo id="A1.E14.m1.8.9.2.3.2.2" xref="A1.E14.m1.8.9.2.3.1.cmml">,</mo><mi id="A1.E14.m1.8.8" xref="A1.E14.m1.8.8.cmml">𝐋</mi><mo stretchy="false" id="A1.E14.m1.8.9.2.3.2.3" 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xref="A1.E14.m1.6.6.6.4.4.9.1.cmml"><mo stretchy="false" id="A1.E14.m1.6.6.6.4.4.9.2.1" xref="A1.E14.m1.6.6.6.4.4.9.1.cmml">(</mo><mi id="A1.E14.m1.5.5.5.3.3.3" xref="A1.E14.m1.5.5.5.3.3.3.cmml">𝐋</mi><mo id="A1.E14.m1.6.6.6.4.4.9.2.2" xref="A1.E14.m1.6.6.6.4.4.9.1.cmml">,</mo><mi id="A1.E14.m1.6.6.6.4.4.4" xref="A1.E14.m1.6.6.6.4.4.4.cmml">𝐋</mi><mo stretchy="false" id="A1.E14.m1.6.6.6.4.4.9.2.3" xref="A1.E14.m1.6.6.6.4.4.9.1.cmml">)</mo></mrow></mrow></msqrt></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="A1.E14.m1.8b"><apply id="A1.E14.m1.8.9.cmml" xref="A1.E14.m1.8.9"><eq id="A1.E14.m1.8.9.1.cmml" xref="A1.E14.m1.8.9.1"></eq><apply id="A1.E14.m1.8.9.2.cmml" xref="A1.E14.m1.8.9.2"><times id="A1.E14.m1.8.9.2.1.cmml" xref="A1.E14.m1.8.9.2.1"></times><ci id="A1.E14.m1.8.9.2.2.cmml" xref="A1.E14.m1.8.9.2.2">𝖢𝖪𝖠</ci><interval closure="open" id="A1.E14.m1.8.9.2.3.1.cmml" xref="A1.E14.m1.8.9.2.3.2"><ci id="A1.E14.m1.7.7.cmml" xref="A1.E14.m1.7.7">𝐊</ci><ci 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xref="A1.E14.m1.6.6.6.4.4.7.2"><ci id="A1.E14.m1.3.3.3.1.1.1.cmml" xref="A1.E14.m1.3.3.3.1.1.1">𝐊</ci><ci id="A1.E14.m1.4.4.4.2.2.2.cmml" xref="A1.E14.m1.4.4.4.2.2.2">𝐊</ci></interval><ci id="A1.E14.m1.6.6.6.4.4.8.cmml" xref="A1.E14.m1.6.6.6.4.4.8">𝖧𝖲𝖨𝖢</ci><interval closure="open" id="A1.E14.m1.6.6.6.4.4.9.1.cmml" xref="A1.E14.m1.6.6.6.4.4.9.2"><ci id="A1.E14.m1.5.5.5.3.3.3.cmml" xref="A1.E14.m1.5.5.5.3.3.3">𝐋</ci><ci id="A1.E14.m1.6.6.6.4.4.4.cmml" xref="A1.E14.m1.6.6.6.4.4.4">𝐋</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E14.m1.8c">\displaystyle\mathsf{CKA}(\mathbf{K},\mathbf{L})=\frac{\mathsf{HSIC}(\mathbf{K},\mathbf{L})}{\sqrt{\mathsf{HSIC}(\mathbf{K},\mathbf{K})\mathsf{HSIC}(\mathbf{L},\mathbf{L})}}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td>
</tr></tbody>
</table>
<p id="A1.SS0.SSS0.Px2.p1.10" class="ltx_p">CKA measures the congruence between two random variables, with a maximum alignment of <math id="A1.SS0.SSS0.Px2.p1.9.m1.1" class="ltx_Math" alttext="1" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.9.m1.1a"><mn id="A1.SS0.SSS0.Px2.p1.9.m1.1.1" xref="A1.SS0.SSS0.Px2.p1.9.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.9.m1.1b"><cn type="integer" id="A1.SS0.SSS0.Px2.p1.9.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.9.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.9.m1.1c">1</annotation></semantics></math> and a minimum of <math id="A1.SS0.SSS0.Px2.p1.10.m2.1" class="ltx_Math" alttext="0" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.10.m2.1a"><mn id="A1.SS0.SSS0.Px2.p1.10.m2.1.1" xref="A1.SS0.SSS0.Px2.p1.10.m2.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.10.m2.1b"><cn type="integer" id="A1.SS0.SSS0.Px2.p1.10.m2.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.10.m2.1.1">0</cn></annotation-xml></semantics></math>. It is invariant to isotropic scaling and offers a strict notion of alignment, measuring alignment across all samples. Hence, the CKA score reflects the global similarities of the models. This can be illustrated by expanding the trace term in HSIC:</p>
<table id="A6.EGx12" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.E15.m1.3.3.3.2.2.2.1.1.3.1.1.1.2.2.2">𝜓</ci><ci id="A1.E15.m1.3.3.3.2.2.2.1.1.3.1.1.1.2.2.3.cmml" xref="A1.E15.m1.3.3.3.2.2.2.1.1.3.1.1.1.2.2.3">𝑙</ci></apply></list></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E15.m1.3c">\displaystyle\mathsf{Trace}(\bar{\mathbf{K}}\bar{\mathbf{L}})=\sum_{i}\sum_{j}\left(\langle\phi_{i},\phi_{j}\rangle-\mathbb{E}_{l}[\langle\phi_{i},\phi_{l}\rangle]\right)\left(\langle\psi_{i},\psi_{j}\rangle-\mathbb{E}_{l}[\langle\psi_{i},\psi_{l}\rangle]\right)</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td>
</tr></tbody>
</table>
<p id="A1.SS0.SSS0.Px2.p1.11" class="ltx_p">One can modify the definition of alignment to restrict the cross-covariance measurement to samples considered to be nearest neighbors of the current sample <math id="A1.SS0.SSS0.Px2.p1.11.m1.1" class="ltx_Math" alttext="i" display="inline"><semantics id="A1.SS0.SSS0.Px2.p1.11.m1.1a"><mi id="A1.SS0.SSS0.Px2.p1.11.m1.1.1" xref="A1.SS0.SSS0.Px2.p1.11.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p1.11.m1.1b"><ci id="A1.SS0.SSS0.Px2.p1.11.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p1.11.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p1.11.m1.1c">i</annotation></semantics></math>. This emphasizes similarity over dissimilarity, biasing the measure toward local alignment:</p>
<table id="A6.EGx13" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E16.m1.2" class="ltx_Math" alttext="\displaystyle\mathsf{Align_{knn}}(\mathbf{K},\mathbf{L})" display="inline"><semantics id="A1.E16.m1.2a"><mrow id="A1.E16.m1.2.3" xref="A1.E16.m1.2.3.cmml"><msub id="A1.E16.m1.2.3.2" xref="A1.E16.m1.2.3.2.cmml"><mi id="A1.E16.m1.2.3.2.2" xref="A1.E16.m1.2.3.2.2.cmml">𝖠𝗅𝗂𝗀𝗇</mi><mi id="A1.E16.m1.2.3.2.3" xref="A1.E16.m1.2.3.2.3.cmml">𝗄𝗇𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A1.E16.m1.2.3.1" xref="A1.E16.m1.2.3.1.cmml">​</mo><mrow id="A1.E16.m1.2.3.3.2" xref="A1.E16.m1.2.3.3.1.cmml"><mo stretchy="false" id="A1.E16.m1.2.3.3.2.1" xref="A1.E16.m1.2.3.3.1.cmml">(</mo><mi id="A1.E16.m1.1.1" xref="A1.E16.m1.1.1.cmml">𝐊</mi><mo id="A1.E16.m1.2.3.3.2.2" xref="A1.E16.m1.2.3.3.1.cmml">,</mo><mi id="A1.E16.m1.2.2" xref="A1.E16.m1.2.2.cmml">𝐋</mi><mo stretchy="false" id="A1.E16.m1.2.3.3.2.3" xref="A1.E16.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E16.m1.2b"><apply id="A1.E16.m1.2.3.cmml" xref="A1.E16.m1.2.3"><times id="A1.E16.m1.2.3.1.cmml" xref="A1.E16.m1.2.3.1"></times><apply id="A1.E16.m1.2.3.2.cmml" xref="A1.E16.m1.2.3.2"><csymbol cd="ambiguous" id="A1.E16.m1.2.3.2.1.cmml" xref="A1.E16.m1.2.3.2">subscript</csymbol><ci id="A1.E16.m1.2.3.2.2.cmml" xref="A1.E16.m1.2.3.2.2">𝖠𝗅𝗂𝗀𝗇</ci><ci id="A1.E16.m1.2.3.2.3.cmml" xref="A1.E16.m1.2.3.2.3">𝗄𝗇𝗇</ci></apply><interval closure="open" id="A1.E16.m1.2.3.3.1.cmml" xref="A1.E16.m1.2.3.3.2"><ci id="A1.E16.m1.1.1.cmml" xref="A1.E16.m1.1.1">𝐊</ci><ci id="A1.E16.m1.2.2.cmml" xref="A1.E16.m1.2.2">𝐋</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E16.m1.2c">\displaystyle\mathsf{Align_{knn}}(\mathbf{K},\mathbf{L})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.E16.m2.4" class="ltx_Math" alttext="\displaystyle=\sum_{i}\sum_{j}\alpha(i,j)\cdot\left(\langle\phi_{i},\phi_{j}\rangle-\mathbb{E}_{l}[\langle\phi_{i},\phi_{l}\rangle]\right)\left(\langle\psi_{i},\psi_{j}\rangle-\mathbb{E}_{l}[\langle\psi_{i},\psi_{l}\rangle]\right)" display="inline"><semantics id="A1.E16.m2.4a"><mrow id="A1.E16.m2.4.4" xref="A1.E16.m2.4.4.cmml"><mi id="A1.E16.m2.4.4.4" xref="A1.E16.m2.4.4.4.cmml"></mi><mo id="A1.E16.m2.4.4.3" xref="A1.E16.m2.4.4.3.cmml">=</mo><mrow id="A1.E16.m2.4.4.2" xref="A1.E16.m2.4.4.2.cmml"><mstyle displaystyle="true" id="A1.E16.m2.4.4.2.3" xref="A1.E16.m2.4.4.2.3.cmml"><munder id="A1.E16.m2.4.4.2.3a" xref="A1.E16.m2.4.4.2.3.cmml"><mo movablelimits="false" id="A1.E16.m2.4.4.2.3.2" xref="A1.E16.m2.4.4.2.3.2.cmml">∑</mo><mi id="A1.E16.m2.4.4.2.3.3" xref="A1.E16.m2.4.4.2.3.3.cmml">i</mi></munder></mstyle><mrow id="A1.E16.m2.4.4.2.2" xref="A1.E16.m2.4.4.2.2.cmml"><mstyle displaystyle="true" id="A1.E16.m2.4.4.2.2.3" xref="A1.E16.m2.4.4.2.2.3.cmml"><munder id="A1.E16.m2.4.4.2.2.3a" xref="A1.E16.m2.4.4.2.2.3.cmml"><mo movablelimits="false" id="A1.E16.m2.4.4.2.2.3.2" xref="A1.E16.m2.4.4.2.2.3.2.cmml">∑</mo><mi id="A1.E16.m2.4.4.2.2.3.3" xref="A1.E16.m2.4.4.2.2.3.3.cmml">j</mi></munder></mstyle><mrow id="A1.E16.m2.4.4.2.2.2" xref="A1.E16.m2.4.4.2.2.2.cmml"><mrow id="A1.E16.m2.3.3.1.1.1.1" xref="A1.E16.m2.3.3.1.1.1.1.cmml"><mrow id="A1.E16.m2.3.3.1.1.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.3.cmml"><mi id="A1.E16.m2.3.3.1.1.1.1.3.2" xref="A1.E16.m2.3.3.1.1.1.1.3.2.cmml">α</mi><mo lspace="0em" rspace="0em" id="A1.E16.m2.3.3.1.1.1.1.3.1" xref="A1.E16.m2.3.3.1.1.1.1.3.1.cmml">​</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.3.3.2" xref="A1.E16.m2.3.3.1.1.1.1.3.3.1.cmml"><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.3.3.2.1" xref="A1.E16.m2.3.3.1.1.1.1.3.3.1.cmml">(</mo><mi id="A1.E16.m2.1.1" xref="A1.E16.m2.1.1.cmml">i</mi><mo id="A1.E16.m2.3.3.1.1.1.1.3.3.2.2" xref="A1.E16.m2.3.3.1.1.1.1.3.3.1.cmml">,</mo><mi id="A1.E16.m2.2.2" xref="A1.E16.m2.2.2.cmml">j</mi><mo rspace="0.055em" stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.3.3.2.3" xref="A1.E16.m2.3.3.1.1.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo rspace="0.222em" id="A1.E16.m2.3.3.1.1.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.2.cmml">⋅</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.cmml"><mo id="A1.E16.m2.3.3.1.1.1.1.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1.1" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.cmml"><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.3.cmml"><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.3.cmml">⟨</mo><msub id="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.4" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.3.cmml">,</mo><msub id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2.2.cmml">ϕ</mi><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.2.5" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.2.3.cmml">⟩</mo></mrow><mo id="A1.E16.m2.3.3.1.1.1.1.1.1.1.4" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.4.cmml">−</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.cmml"><msub id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3.cmml"><mi mathvariant="normal" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3.2.cmml">𝔼</mi><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.3.3.cmml">l</mi></msub><mo lspace="0em" rspace="0em" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.2.cmml">​</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.2.cmml"><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.2.1.cmml">[</mo><mrow id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.3.cmml"><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.3.cmml">⟨</mo><msub id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1.cmml"><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1.2.cmml">ϕ</mi><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.4" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.3.cmml">,</mo><msub id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2.cmml"><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2.2" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2.2.cmml">ϕ</mi><mi id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.2.3.cmml">l</mi></msub><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.2.5" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.1.3.cmml">⟩</mo></mrow><mo stretchy="false" id="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.3.1.2.1.cmml">]</mo></mrow></mrow></mrow><mo id="A1.E16.m2.3.3.1.1.1.1.1.1.3" xref="A1.E16.m2.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo lspace="0em" rspace="0em" id="A1.E16.m2.4.4.2.2.2.3" xref="A1.E16.m2.4.4.2.2.2.3.cmml">​</mo><mrow id="A1.E16.m2.4.4.2.2.2.2.1" xref="A1.E16.m2.4.4.2.2.2.2.1.1.cmml"><mo id="A1.E16.m2.4.4.2.2.2.2.1.2" xref="A1.E16.m2.4.4.2.2.2.2.1.1.cmml">(</mo><mrow id="A1.E16.m2.4.4.2.2.2.2.1.1" xref="A1.E16.m2.4.4.2.2.2.2.1.1.cmml"><mrow id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.3.cmml"><mo stretchy="false" id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.3" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.3.cmml">⟨</mo><msub id="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1" xref="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1.cmml"><mi id="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1.2" xref="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1.2.cmml">ψ</mi><mi id="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1.3" xref="A1.E16.m2.4.4.2.2.2.2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.4" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.3.cmml">,</mo><msub id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2.cmml"><mi id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2.2" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2.2.cmml">ψ</mi><mi id="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2.3" xref="A1.E16.m2.4.4.2.2.2.2.1.1.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td>
</tr></tbody>
<tbody id="A1.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.E17.m1.5" class="ltx_Math" alttext="\displaystyle\text{where}\qquad\alpha(i,j)=\mathbb{1}[\phi_{j}\in\mathsf{knn}(\phi_{i})\land\psi_{j}\in\mathsf{knn}(\psi_{i})\land i\neq j]" display="inline"><semantics id="A1.E17.m1.5a"><mrow id="A1.E17.m1.5.5" xref="A1.E17.m1.5.5.cmml"><mrow id="A1.E17.m1.4.4.1.1" xref="A1.E17.m1.4.4.1.2.cmml"><mtext id="A1.E17.m1.3.3" xref="A1.E17.m1.3.3a.cmml">where</mtext><mspace width="2em" id="A1.E17.m1.4.4.1.1.2" xref="A1.E17.m1.4.4.1.2.cmml"></mspace><mrow id="A1.E17.m1.4.4.1.1.1" xref="A1.E17.m1.4.4.1.1.1.cmml"><mi id="A1.E17.m1.4.4.1.1.1.2" xref="A1.E17.m1.4.4.1.1.1.2.cmml">α</mi><mo lspace="0em" rspace="0em" id="A1.E17.m1.4.4.1.1.1.1" xref="A1.E17.m1.4.4.1.1.1.1.cmml">​</mo><mrow id="A1.E17.m1.4.4.1.1.1.3.2" xref="A1.E17.m1.4.4.1.1.1.3.1.cmml"><mo stretchy="false" id="A1.E17.m1.4.4.1.1.1.3.2.1" xref="A1.E17.m1.4.4.1.1.1.3.1.cmml">(</mo><mi id="A1.E17.m1.1.1" xref="A1.E17.m1.1.1.cmml">i</mi><mo 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xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.cmml">(</mo><msub id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.cmml"><mi id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.2" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.2.cmml">ψ</mi><mi id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.3" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.3" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.E17.m1.5.5.2.1.1.1.2.2" xref="A1.E17.m1.5.5.2.1.1.1.2.2.cmml">∧</mo><mi id="A1.E17.m1.5.5.2.1.1.1.2.3" xref="A1.E17.m1.5.5.2.1.1.1.2.3.cmml">i</mi></mrow><mo id="A1.E17.m1.5.5.2.1.1.1.7" xref="A1.E17.m1.5.5.2.1.1.1.7.cmml">≠</mo><mi id="A1.E17.m1.5.5.2.1.1.1.8" xref="A1.E17.m1.5.5.2.1.1.1.8.cmml">j</mi></mrow><mo stretchy="false" id="A1.E17.m1.5.5.2.1.1.3" xref="A1.E17.m1.5.5.2.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E17.m1.5b"><apply id="A1.E17.m1.5.5.cmml" xref="A1.E17.m1.5.5"><eq id="A1.E17.m1.5.5.3.cmml" xref="A1.E17.m1.5.5.3"></eq><list id="A1.E17.m1.4.4.1.2.cmml" xref="A1.E17.m1.4.4.1.1"><ci id="A1.E17.m1.3.3a.cmml" xref="A1.E17.m1.3.3"><mtext id="A1.E17.m1.3.3.cmml" xref="A1.E17.m1.3.3">where</mtext></ci><apply id="A1.E17.m1.4.4.1.1.1.cmml" xref="A1.E17.m1.4.4.1.1.1"><times id="A1.E17.m1.4.4.1.1.1.1.cmml" xref="A1.E17.m1.4.4.1.1.1.1"></times><ci id="A1.E17.m1.4.4.1.1.1.2.cmml" xref="A1.E17.m1.4.4.1.1.1.2">𝛼</ci><interval closure="open" id="A1.E17.m1.4.4.1.1.1.3.1.cmml" xref="A1.E17.m1.4.4.1.1.1.3.2"><ci id="A1.E17.m1.1.1.cmml" xref="A1.E17.m1.1.1">𝑖</ci><ci id="A1.E17.m1.2.2.cmml" xref="A1.E17.m1.2.2">𝑗</ci></interval></apply></list><apply id="A1.E17.m1.5.5.2.cmml" xref="A1.E17.m1.5.5.2"><times id="A1.E17.m1.5.5.2.2.cmml" xref="A1.E17.m1.5.5.2.2"></times><ci id="A1.E17.m1.5.5.2.3.cmml" xref="A1.E17.m1.5.5.2.3">double-struck-𝟙</ci><apply id="A1.E17.m1.5.5.2.1.2.cmml" xref="A1.E17.m1.5.5.2.1.1"><csymbol cd="latexml" 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xref="A1.E17.m1.5.5.2.1.1.1"><in id="A1.E17.m1.5.5.2.1.1.1.6.cmml" xref="A1.E17.m1.5.5.2.1.1.1.6"></in><share href="#A1.E17.m1.5.5.2.1.1.1.1.cmml" id="A1.E17.m1.5.5.2.1.1.1d.cmml" xref="A1.E17.m1.5.5.2.1.1.1"></share><apply id="A1.E17.m1.5.5.2.1.1.1.2.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2"><and id="A1.E17.m1.5.5.2.1.1.1.2.2.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.2"></and><apply id="A1.E17.m1.5.5.2.1.1.1.2.1.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1"><times id="A1.E17.m1.5.5.2.1.1.1.2.1.2.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.2"></times><ci id="A1.E17.m1.5.5.2.1.1.1.2.1.3.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.3">𝗄𝗇𝗇</ci><apply id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1"><csymbol cd="ambiguous" id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.1.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1">subscript</csymbol><ci id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.2.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.2">𝜓</ci><ci id="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.3.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.1.1.1.1.3">𝑖</ci></apply></apply><ci id="A1.E17.m1.5.5.2.1.1.1.2.3.cmml" xref="A1.E17.m1.5.5.2.1.1.1.2.3">𝑖</ci></apply></apply><apply id="A1.E17.m1.5.5.2.1.1.1e.cmml" xref="A1.E17.m1.5.5.2.1.1.1"><neq id="A1.E17.m1.5.5.2.1.1.1.7.cmml" xref="A1.E17.m1.5.5.2.1.1.1.7"></neq><share href="#A1.E17.m1.5.5.2.1.1.1.2.cmml" id="A1.E17.m1.5.5.2.1.1.1f.cmml" xref="A1.E17.m1.5.5.2.1.1.1"></share><ci id="A1.E17.m1.5.5.2.1.1.1.8.cmml" xref="A1.E17.m1.5.5.2.1.1.1.8">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E17.m1.5c">\displaystyle\text{where}\qquad\alpha(i,j)=\mathbb{1}[\phi_{j}\in\mathsf{knn}(\phi_{i})\land\psi_{j}\in\mathsf{knn}(\psi_{i})\land i\neq j]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td>
</tr></tbody>
</table>
</div>
<figure id="A1.F10" class="ltx_figure"><img src="/html/2405.07987/assets/x12.png" id="A1.F10.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="452" height="135" alt="Refer to caption">
<br class="ltx_break ltx_centering">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10: </span><span id="A1.F10.12.6" class="ltx_text ltx_font_bold" style="font-size:90%;">Cross-modal alignment increases locally:<span id="A1.F10.12.6.6" class="ltx_text ltx_font_medium">
Alignment trend when varying the top-<math id="A1.F10.7.1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.F10.7.1.1.m1.1b"><mi id="A1.F10.7.1.1.m1.1.1" xref="A1.F10.7.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.F10.7.1.1.m1.1c"><ci id="A1.F10.7.1.1.m1.1.1.cmml" xref="A1.F10.7.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.7.1.1.m1.1d">k</annotation></semantics></math> nearest neighbors in the CKNNA metrics (Eqn. <a href="#A1.E18" title="Equation 18 ‣ Relationship between CKA and Mutual Nearest-Neighbors ‣ Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">18</span></a>). We center alignment score to the smallest language model and divide the total trend by the standard deviation. When <math id="A1.F10.8.2.2.m2.1" class="ltx_Math" alttext="k=1024" display="inline"><semantics id="A1.F10.8.2.2.m2.1b"><mrow id="A1.F10.8.2.2.m2.1.1" xref="A1.F10.8.2.2.m2.1.1.cmml"><mi id="A1.F10.8.2.2.m2.1.1.2" xref="A1.F10.8.2.2.m2.1.1.2.cmml">k</mi><mo id="A1.F10.8.2.2.m2.1.1.1" xref="A1.F10.8.2.2.m2.1.1.1.cmml">=</mo><mn id="A1.F10.8.2.2.m2.1.1.3" xref="A1.F10.8.2.2.m2.1.1.3.cmml">1024</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.F10.8.2.2.m2.1c"><apply id="A1.F10.8.2.2.m2.1.1.cmml" xref="A1.F10.8.2.2.m2.1.1"><eq id="A1.F10.8.2.2.m2.1.1.1.cmml" xref="A1.F10.8.2.2.m2.1.1.1"></eq><ci id="A1.F10.8.2.2.m2.1.1.2.cmml" xref="A1.F10.8.2.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A1.F10.8.2.2.m2.1.1.3.cmml" xref="A1.F10.8.2.2.m2.1.1.3">1024</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.8.2.2.m2.1d">k=1024</annotation></semantics></math>, we recover the original CKA metric, and when <math id="A1.F10.9.3.3.m3.1" class="ltx_Math" alttext="k&lt;|\mathcal{X}|" display="inline"><semantics id="A1.F10.9.3.3.m3.1b"><mrow id="A1.F10.9.3.3.m3.1.2" xref="A1.F10.9.3.3.m3.1.2.cmml"><mi id="A1.F10.9.3.3.m3.1.2.2" xref="A1.F10.9.3.3.m3.1.2.2.cmml">k</mi><mo id="A1.F10.9.3.3.m3.1.2.1" xref="A1.F10.9.3.3.m3.1.2.1.cmml">&lt;</mo><mrow id="A1.F10.9.3.3.m3.1.2.3.2" xref="A1.F10.9.3.3.m3.1.2.3.1.cmml"><mo stretchy="false" id="A1.F10.9.3.3.m3.1.2.3.2.1" xref="A1.F10.9.3.3.m3.1.2.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="A1.F10.9.3.3.m3.1.1" xref="A1.F10.9.3.3.m3.1.1.cmml">𝒳</mi><mo stretchy="false" id="A1.F10.9.3.3.m3.1.2.3.2.2" xref="A1.F10.9.3.3.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.F10.9.3.3.m3.1c"><apply id="A1.F10.9.3.3.m3.1.2.cmml" xref="A1.F10.9.3.3.m3.1.2"><lt id="A1.F10.9.3.3.m3.1.2.1.cmml" xref="A1.F10.9.3.3.m3.1.2.1"></lt><ci id="A1.F10.9.3.3.m3.1.2.2.cmml" xref="A1.F10.9.3.3.m3.1.2.2">𝑘</ci><apply id="A1.F10.9.3.3.m3.1.2.3.1.cmml" xref="A1.F10.9.3.3.m3.1.2.3.2"><abs id="A1.F10.9.3.3.m3.1.2.3.1.1.cmml" xref="A1.F10.9.3.3.m3.1.2.3.2.1"></abs><ci id="A1.F10.9.3.3.m3.1.1.cmml" xref="A1.F10.9.3.3.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.9.3.3.m3.1d">k&lt;|\mathcal{X}|</annotation></semantics></math> it closely resembles the mutual nearest-neighbor metric <math id="A1.F10.10.4.4.m4.1" class="ltx_Math" alttext="m_{\texttt{NN}}" display="inline"><semantics id="A1.F10.10.4.4.m4.1b"><msub id="A1.F10.10.4.4.m4.1.1" xref="A1.F10.10.4.4.m4.1.1.cmml"><mi id="A1.F10.10.4.4.m4.1.1.2" xref="A1.F10.10.4.4.m4.1.1.2.cmml">m</mi><mtext class="ltx_mathvariant_monospace" id="A1.F10.10.4.4.m4.1.1.3" xref="A1.F10.10.4.4.m4.1.1.3a.cmml">NN</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.F10.10.4.4.m4.1c"><apply id="A1.F10.10.4.4.m4.1.1.cmml" xref="A1.F10.10.4.4.m4.1.1"><csymbol cd="ambiguous" id="A1.F10.10.4.4.m4.1.1.1.cmml" xref="A1.F10.10.4.4.m4.1.1">subscript</csymbol><ci id="A1.F10.10.4.4.m4.1.1.2.cmml" xref="A1.F10.10.4.4.m4.1.1.2">𝑚</ci><ci id="A1.F10.10.4.4.m4.1.1.3a.cmml" xref="A1.F10.10.4.4.m4.1.1.3"><mtext class="ltx_mathvariant_monospace" mathsize="70%" id="A1.F10.10.4.4.m4.1.1.3.cmml" xref="A1.F10.10.4.4.m4.1.1.3">NN</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.10.4.4.m4.1d">m_{\texttt{NN}}</annotation></semantics></math>. Each line represents the average of all LLM models for a specific <math id="A1.F10.11.5.5.m5.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.F10.11.5.5.m5.1b"><mi id="A1.F10.11.5.5.m5.1.1" xref="A1.F10.11.5.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.F10.11.5.5.m5.1c"><ci id="A1.F10.11.5.5.m5.1.1.cmml" xref="A1.F10.11.5.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.11.5.5.m5.1d">k</annotation></semantics></math>. As we decrease <math id="A1.F10.12.6.6.m6.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.F10.12.6.6.m6.1b"><mi id="A1.F10.12.6.6.m6.1.1" xref="A1.F10.12.6.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.F10.12.6.6.m6.1c"><ci id="A1.F10.12.6.6.m6.1.1.cmml" xref="A1.F10.12.6.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.F10.12.6.6.m6.1d">k</annotation></semantics></math>, the alignment becomes more pronounced. Certain visual tasks, such as CLIP, show global alignment, while methods like ImageNet1k classification only exhibit local alignment.</span></span></figcaption>
</figure>
<div id="A1.SS0.SSS0.Px2.p2" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px2.p2.7" class="ltx_p">Where <math id="A1.SS0.SSS0.Px2.p2.1.m1.2" class="ltx_Math" alttext="\alpha(i,j)" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.1.m1.2a"><mrow id="A1.SS0.SSS0.Px2.p2.1.m1.2.3" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.cmml"><mi id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.2" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.2.cmml">α</mi><mo lspace="0em" rspace="0em" id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.1" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.1.cmml">​</mo><mrow id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.2" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.1.cmml"><mo stretchy="false" id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.2.1" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="A1.SS0.SSS0.Px2.p2.1.m1.1.1" xref="A1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml">i</mi><mo id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.2.2" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="A1.SS0.SSS0.Px2.p2.1.m1.2.2" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.2.cmml">j</mi><mo stretchy="false" id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.2.3" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.1.m1.2b"><apply id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3"><times id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.1.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.1"></times><ci id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.2.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.2">𝛼</ci><interval closure="open" id="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.1.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.3.3.2"><ci id="A1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.1.1">𝑖</ci><ci id="A1.SS0.SSS0.Px2.p2.1.m1.2.2.cmml" xref="A1.SS0.SSS0.Px2.p2.1.m1.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.1.m1.2c">\alpha(i,j)</annotation></semantics></math> is a scalar weighting that assigns <math id="A1.SS0.SSS0.Px2.p2.2.m2.1" class="ltx_Math" alttext="1" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.2.m2.1a"><mn id="A1.SS0.SSS0.Px2.p2.2.m2.1.1" xref="A1.SS0.SSS0.Px2.p2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.2.m2.1b"><cn type="integer" id="A1.SS0.SSS0.Px2.p2.2.m2.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.2.m2.1c">1</annotation></semantics></math> if <math id="A1.SS0.SSS0.Px2.p2.3.m3.1" class="ltx_Math" alttext="j" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.3.m3.1a"><mi id="A1.SS0.SSS0.Px2.p2.3.m3.1.1" xref="A1.SS0.SSS0.Px2.p2.3.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.3.m3.1b"><ci id="A1.SS0.SSS0.Px2.p2.3.m3.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.3.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.3.m3.1c">j</annotation></semantics></math> is a mutual nearest neighbors to both <math id="A1.SS0.SSS0.Px2.p2.4.m4.1" class="ltx_Math" alttext="\phi_{i}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.4.m4.1a"><msub id="A1.SS0.SSS0.Px2.p2.4.m4.1.1" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1.cmml"><mi id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.2" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1.2.cmml">ϕ</mi><mi id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.3" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.4.m4.1b"><apply id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1.2">italic-ϕ</ci><ci id="A1.SS0.SSS0.Px2.p2.4.m4.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p2.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.4.m4.1c">\phi_{i}</annotation></semantics></math> and <math id="A1.SS0.SSS0.Px2.p2.5.m5.1" class="ltx_Math" alttext="\psi_{i}" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.5.m5.1a"><msub id="A1.SS0.SSS0.Px2.p2.5.m5.1.1" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1.cmml"><mi id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.2" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1.2.cmml">ψ</mi><mi id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.3" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.5.m5.1b"><apply id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1">subscript</csymbol><ci id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.2.cmml" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1.2">𝜓</ci><ci id="A1.SS0.SSS0.Px2.p2.5.m5.1.1.3.cmml" xref="A1.SS0.SSS0.Px2.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.5.m5.1c">\psi_{i}</annotation></semantics></math>, and <math id="A1.SS0.SSS0.Px2.p2.6.m6.1" class="ltx_Math" alttext="0" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.6.m6.1a"><mn id="A1.SS0.SSS0.Px2.p2.6.m6.1.1" xref="A1.SS0.SSS0.Px2.p2.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.6.m6.1b"><cn type="integer" id="A1.SS0.SSS0.Px2.p2.6.m6.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.6.m6.1.1">0</cn></annotation-xml></semantics></math> otherwise. We refer to this metric as the Centered Kernel Nearest-Neighbor Alignment (CKNNA) metric. As the number of nearest neighbors <math id="A1.SS0.SSS0.Px2.p2.7.m7.1" class="ltx_Math" alttext="k\rightarrow\dim(\mathbf{K})" display="inline"><semantics id="A1.SS0.SSS0.Px2.p2.7.m7.1a"><mrow id="A1.SS0.SSS0.Px2.p2.7.m7.1.2" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.cmml"><mi id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.2" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.2.cmml">k</mi><mo rspace="0.1389em" stretchy="false" id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.1" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.1.cmml">→</mo><mrow id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.cmml"><mo lspace="0.1389em" rspace="0em" id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.1" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.1.cmml">dim</mo><mrow id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.2.2" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.cmml"><mo stretchy="false" id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.2.2.1" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.cmml">(</mo><mi id="A1.SS0.SSS0.Px2.p2.7.m7.1.1" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.1.cmml">𝐊</mi><mo stretchy="false" id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.2.2.2" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p2.7.m7.1b"><apply id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2"><ci id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.1.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.1">→</ci><ci id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.2.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.2">𝑘</ci><apply id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3"><csymbol cd="latexml" id="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.1.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.2.3.1">dimension</csymbol><ci id="A1.SS0.SSS0.Px2.p2.7.m7.1.1.cmml" xref="A1.SS0.SSS0.Px2.p2.7.m7.1.1">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p2.7.m7.1c">k\rightarrow\dim(\mathbf{K})</annotation></semantics></math>, we recover the original CKA metric.</p>
</div>
<div id="A1.SS0.SSS0.Px2.p3" class="ltx_para ltx_noindent">
<table id="A6.EGx14" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E18.m1.10" class="ltx_Math" alttext="\displaystyle\mathsf{CKNNA}(\mathbf{K},\mathbf{L})=\frac{\mathsf{Align_{knn}}(\mathbf{K},\mathbf{L})}{\sqrt{\mathsf{Align_{knn}}(\mathbf{K},\mathbf{K}),\mathsf{Align_{knn}}(\mathbf{L},\mathbf{L})}}" display="inline"><semantics id="A1.E18.m1.10a"><mrow id="A1.E18.m1.10.11" xref="A1.E18.m1.10.11.cmml"><mrow id="A1.E18.m1.10.11.2" xref="A1.E18.m1.10.11.2.cmml"><mi id="A1.E18.m1.10.11.2.2" xref="A1.E18.m1.10.11.2.2.cmml">𝖢𝖪𝖭𝖭𝖠</mi><mo lspace="0em" rspace="0em" id="A1.E18.m1.10.11.2.1" xref="A1.E18.m1.10.11.2.1.cmml">​</mo><mrow id="A1.E18.m1.10.11.2.3.2" xref="A1.E18.m1.10.11.2.3.1.cmml"><mo stretchy="false" id="A1.E18.m1.10.11.2.3.2.1" xref="A1.E18.m1.10.11.2.3.1.cmml">(</mo><mi id="A1.E18.m1.9.9" xref="A1.E18.m1.9.9.cmml">𝐊</mi><mo id="A1.E18.m1.10.11.2.3.2.2" xref="A1.E18.m1.10.11.2.3.1.cmml">,</mo><mi id="A1.E18.m1.10.10" xref="A1.E18.m1.10.10.cmml">𝐋</mi><mo stretchy="false" 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id="A1.E18.m1.2.2.2.5.2.3" xref="A1.E18.m1.2.2.2.5.1.cmml">)</mo></mrow></mrow><msqrt id="A1.E18.m1.8.8.8" xref="A1.E18.m1.8.8.8.cmml"><mrow id="A1.E18.m1.8.8.8.6.6.6" xref="A1.E18.m1.8.8.8.6.6.7.cmml"><mrow id="A1.E18.m1.7.7.7.5.5.5.1" xref="A1.E18.m1.7.7.7.5.5.5.1.cmml"><msub id="A1.E18.m1.7.7.7.5.5.5.1.2" xref="A1.E18.m1.7.7.7.5.5.5.1.2.cmml"><mi id="A1.E18.m1.7.7.7.5.5.5.1.2.2" xref="A1.E18.m1.7.7.7.5.5.5.1.2.2.cmml">𝖠𝗅𝗂𝗀𝗇</mi><mi id="A1.E18.m1.7.7.7.5.5.5.1.2.3" xref="A1.E18.m1.7.7.7.5.5.5.1.2.3.cmml">𝗄𝗇𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A1.E18.m1.7.7.7.5.5.5.1.1" xref="A1.E18.m1.7.7.7.5.5.5.1.1.cmml">​</mo><mrow id="A1.E18.m1.7.7.7.5.5.5.1.3.2" xref="A1.E18.m1.7.7.7.5.5.5.1.3.1.cmml"><mo stretchy="false" id="A1.E18.m1.7.7.7.5.5.5.1.3.2.1" xref="A1.E18.m1.7.7.7.5.5.5.1.3.1.cmml">(</mo><mi id="A1.E18.m1.3.3.3.1.1.1" xref="A1.E18.m1.3.3.3.1.1.1.cmml">𝐊</mi><mo id="A1.E18.m1.7.7.7.5.5.5.1.3.2.2" xref="A1.E18.m1.7.7.7.5.5.5.1.3.1.cmml">,</mo><mi id="A1.E18.m1.4.4.4.2.2.2" xref="A1.E18.m1.4.4.4.2.2.2.cmml">𝐊</mi><mo stretchy="false" id="A1.E18.m1.7.7.7.5.5.5.1.3.2.3" xref="A1.E18.m1.7.7.7.5.5.5.1.3.1.cmml">)</mo></mrow></mrow><mo id="A1.E18.m1.8.8.8.6.6.6.3" xref="A1.E18.m1.8.8.8.6.6.7.cmml">,</mo><mrow id="A1.E18.m1.8.8.8.6.6.6.2" xref="A1.E18.m1.8.8.8.6.6.6.2.cmml"><msub id="A1.E18.m1.8.8.8.6.6.6.2.2" xref="A1.E18.m1.8.8.8.6.6.6.2.2.cmml"><mi id="A1.E18.m1.8.8.8.6.6.6.2.2.2" xref="A1.E18.m1.8.8.8.6.6.6.2.2.2.cmml">𝖠𝗅𝗂𝗀𝗇</mi><mi id="A1.E18.m1.8.8.8.6.6.6.2.2.3" xref="A1.E18.m1.8.8.8.6.6.6.2.2.3.cmml">𝗄𝗇𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A1.E18.m1.8.8.8.6.6.6.2.1" xref="A1.E18.m1.8.8.8.6.6.6.2.1.cmml">​</mo><mrow id="A1.E18.m1.8.8.8.6.6.6.2.3.2" xref="A1.E18.m1.8.8.8.6.6.6.2.3.1.cmml"><mo stretchy="false" id="A1.E18.m1.8.8.8.6.6.6.2.3.2.1" xref="A1.E18.m1.8.8.8.6.6.6.2.3.1.cmml">(</mo><mi id="A1.E18.m1.5.5.5.3.3.3" xref="A1.E18.m1.5.5.5.3.3.3.cmml">𝐋</mi><mo id="A1.E18.m1.8.8.8.6.6.6.2.3.2.2" xref="A1.E18.m1.8.8.8.6.6.6.2.3.1.cmml">,</mo><mi id="A1.E18.m1.6.6.6.4.4.4" xref="A1.E18.m1.6.6.6.4.4.4.cmml">𝐋</mi><mo stretchy="false" id="A1.E18.m1.8.8.8.6.6.6.2.3.2.3" xref="A1.E18.m1.8.8.8.6.6.6.2.3.1.cmml">)</mo></mrow></mrow></mrow></msqrt></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="A1.E18.m1.10b"><apply id="A1.E18.m1.10.11.cmml" xref="A1.E18.m1.10.11"><eq id="A1.E18.m1.10.11.1.cmml" xref="A1.E18.m1.10.11.1"></eq><apply id="A1.E18.m1.10.11.2.cmml" xref="A1.E18.m1.10.11.2"><times id="A1.E18.m1.10.11.2.1.cmml" xref="A1.E18.m1.10.11.2.1"></times><ci id="A1.E18.m1.10.11.2.2.cmml" xref="A1.E18.m1.10.11.2.2">𝖢𝖪𝖭𝖭𝖠</ci><interval closure="open" id="A1.E18.m1.10.11.2.3.1.cmml" xref="A1.E18.m1.10.11.2.3.2"><ci id="A1.E18.m1.9.9.cmml" xref="A1.E18.m1.9.9">𝐊</ci><ci id="A1.E18.m1.10.10.cmml" xref="A1.E18.m1.10.10">𝐋</ci></interval></apply><apply id="A1.E18.m1.8.8.cmml" xref="A1.E18.m1.8.8"><divide id="A1.E18.m1.8.8.9.cmml" xref="A1.E18.m1.8.8"></divide><apply id="A1.E18.m1.2.2.2.cmml" 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id="A1.E18.m1.8.8.8.6.6.6.2.2.2.cmml" xref="A1.E18.m1.8.8.8.6.6.6.2.2.2">𝖠𝗅𝗂𝗀𝗇</ci><ci id="A1.E18.m1.8.8.8.6.6.6.2.2.3.cmml" xref="A1.E18.m1.8.8.8.6.6.6.2.2.3">𝗄𝗇𝗇</ci></apply><interval closure="open" id="A1.E18.m1.8.8.8.6.6.6.2.3.1.cmml" xref="A1.E18.m1.8.8.8.6.6.6.2.3.2"><ci id="A1.E18.m1.5.5.5.3.3.3.cmml" xref="A1.E18.m1.5.5.5.3.3.3">𝐋</ci><ci id="A1.E18.m1.6.6.6.4.4.4.cmml" xref="A1.E18.m1.6.6.6.4.4.4">𝐋</ci></interval></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E18.m1.10c">\displaystyle\mathsf{CKNNA}(\mathbf{K},\mathbf{L})=\frac{\mathsf{Align_{knn}}(\mathbf{K},\mathbf{L})}{\sqrt{\mathsf{Align_{knn}}(\mathbf{K},\mathbf{K}),\mathsf{Align_{knn}}(\mathbf{L},\mathbf{L})}}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td>
</tr></tbody>
</table>
</div>
<div id="A1.SS0.SSS0.Px2.p4" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px2.p4.1" class="ltx_p">We can further relax the metric to treat the cross-covariance term identically across all nearest-neighbor samples. This is equivalent to the assumption that all nearby samples have the same distance. This simplification leads us back to the mutual nearest neighbor metric:</p>
<table id="A6.EGx15" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E19.m1.4" class="ltx_Math" alttext="\displaystyle\sum_{i}\sum_{j}\alpha(i,j)\cdot 1=n\cdot k\cdot m_{\texttt{NN}}(\phi_{i},\psi_{i})" display="inline"><semantics id="A1.E19.m1.4a"><mrow id="A1.E19.m1.4.4" xref="A1.E19.m1.4.4.cmml"><mrow id="A1.E19.m1.4.4.4" xref="A1.E19.m1.4.4.4.cmml"><mstyle displaystyle="true" id="A1.E19.m1.4.4.4.1" xref="A1.E19.m1.4.4.4.1.cmml"><munder id="A1.E19.m1.4.4.4.1a" xref="A1.E19.m1.4.4.4.1.cmml"><mo movablelimits="false" id="A1.E19.m1.4.4.4.1.2" xref="A1.E19.m1.4.4.4.1.2.cmml">∑</mo><mi id="A1.E19.m1.4.4.4.1.3" xref="A1.E19.m1.4.4.4.1.3.cmml">i</mi></munder></mstyle><mrow id="A1.E19.m1.4.4.4.2" xref="A1.E19.m1.4.4.4.2.cmml"><mstyle displaystyle="true" id="A1.E19.m1.4.4.4.2.1" xref="A1.E19.m1.4.4.4.2.1.cmml"><munder id="A1.E19.m1.4.4.4.2.1a" xref="A1.E19.m1.4.4.4.2.1.cmml"><mo movablelimits="false" id="A1.E19.m1.4.4.4.2.1.2" xref="A1.E19.m1.4.4.4.2.1.2.cmml">∑</mo><mi id="A1.E19.m1.4.4.4.2.1.3" 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encoding="application/x-tex" id="A1.E19.m1.4c">\displaystyle\sum_{i}\sum_{j}\alpha(i,j)\cdot 1=n\cdot k\cdot m_{\texttt{NN}}(\phi_{i},\psi_{i})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td>
</tr></tbody>
</table>
</div>
<div id="A1.SS0.SSS0.Px2.p5" class="ltx_para ltx_noindent">
<p id="A1.SS0.SSS0.Px2.p5.4" class="ltx_p">By equating these metrics, we analyze the changes in alignment between language and vision models as we vary the number of neighbors <math id="A1.SS0.SSS0.Px2.p5.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.SS0.SSS0.Px2.p5.1.m1.1a"><mi id="A1.SS0.SSS0.Px2.p5.1.m1.1.1" xref="A1.SS0.SSS0.Px2.p5.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p5.1.m1.1b"><ci id="A1.SS0.SSS0.Px2.p5.1.m1.1.1.cmml" xref="A1.SS0.SSS0.Px2.p5.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p5.1.m1.1c">k</annotation></semantics></math> in Eqn. <a href="#A1.E18" title="Equation 18 ‣ Relationship between CKA and Mutual Nearest-Neighbors ‣ Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">18</span></a>. In Figure <a href="#A1.F10" title="Figure 10 ‣ Relationship between CKA and Mutual Nearest-Neighbors ‣ Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">10</span></a>, we compute the average alignment score across all LLM models. For each <math id="A1.SS0.SSS0.Px2.p5.2.m2.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.SS0.SSS0.Px2.p5.2.m2.1a"><mi id="A1.SS0.SSS0.Px2.p5.2.m2.1.1" xref="A1.SS0.SSS0.Px2.p5.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p5.2.m2.1b"><ci id="A1.SS0.SSS0.Px2.p5.2.m2.1.1.cmml" xref="A1.SS0.SSS0.Px2.p5.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p5.2.m2.1c">k</annotation></semantics></math>, we center the scores to the smallest vision model and divide by the standard deviation of the scores. We find that high values of <math id="A1.SS0.SSS0.Px2.p5.3.m3.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.SS0.SSS0.Px2.p5.3.m3.1a"><mi id="A1.SS0.SSS0.Px2.p5.3.m3.1.1" xref="A1.SS0.SSS0.Px2.p5.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p5.3.m3.1b"><ci id="A1.SS0.SSS0.Px2.p5.3.m3.1.1.cmml" xref="A1.SS0.SSS0.Px2.p5.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p5.3.m3.1c">k</annotation></semantics></math> show less conclusive alignment across tasks while decreasing <math id="A1.SS0.SSS0.Px2.p5.4.m4.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A1.SS0.SSS0.Px2.p5.4.m4.1a"><mi id="A1.SS0.SSS0.Px2.p5.4.m4.1.1" xref="A1.SS0.SSS0.Px2.p5.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS0.SSS0.Px2.p5.4.m4.1b"><ci id="A1.SS0.SSS0.Px2.p5.4.m4.1.1.cmml" xref="A1.SS0.SSS0.Px2.p5.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS0.SSS0.Px2.p5.4.m4.1c">k</annotation></semantics></math> shows a coherent trend across both models and tasks. We find that certain visual tasks, such as CLIP, exhibit global alignment, whereas methods like ImageNet21k classification show only local alignment. This observation suggests that cross-modal alignment occurs locally across most common visual tasks, and global alignment may require additional language grounding, as done in the CLIP objective.</p>
</div>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</section>
<section id="A2" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix B </span>Consistency across various metrics</h2>

<div id="A2.p1" class="ltx_para ltx_noindent">
<p id="A2.p1.3" class="ltx_p">We describe the metrics in Table <a href="#A2.F11" title="Figure 11 ‣ Cross-modal comparison ‣ Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a> and their corresponding properties. The <span id="A2.p1.3.1" class="ltx_text ltx_font_italic">symmetric</span> property implies that the metric is symmetric with respect to the data points <math id="A2.p1.1.m1.4" class="ltx_Math" alttext="d(x,y)=d(y,x)" display="inline"><semantics id="A2.p1.1.m1.4a"><mrow id="A2.p1.1.m1.4.5" xref="A2.p1.1.m1.4.5.cmml"><mrow id="A2.p1.1.m1.4.5.2" xref="A2.p1.1.m1.4.5.2.cmml"><mi id="A2.p1.1.m1.4.5.2.2" xref="A2.p1.1.m1.4.5.2.2.cmml">d</mi><mo lspace="0em" rspace="0em" id="A2.p1.1.m1.4.5.2.1" xref="A2.p1.1.m1.4.5.2.1.cmml">​</mo><mrow id="A2.p1.1.m1.4.5.2.3.2" xref="A2.p1.1.m1.4.5.2.3.1.cmml"><mo stretchy="false" id="A2.p1.1.m1.4.5.2.3.2.1" xref="A2.p1.1.m1.4.5.2.3.1.cmml">(</mo><mi id="A2.p1.1.m1.1.1" xref="A2.p1.1.m1.1.1.cmml">x</mi><mo id="A2.p1.1.m1.4.5.2.3.2.2" xref="A2.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="A2.p1.1.m1.2.2" xref="A2.p1.1.m1.2.2.cmml">y</mi><mo stretchy="false" id="A2.p1.1.m1.4.5.2.3.2.3" xref="A2.p1.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.p1.1.m1.4.5.1" xref="A2.p1.1.m1.4.5.1.cmml">=</mo><mrow id="A2.p1.1.m1.4.5.3" xref="A2.p1.1.m1.4.5.3.cmml"><mi id="A2.p1.1.m1.4.5.3.2" xref="A2.p1.1.m1.4.5.3.2.cmml">d</mi><mo lspace="0em" rspace="0em" id="A2.p1.1.m1.4.5.3.1" xref="A2.p1.1.m1.4.5.3.1.cmml">​</mo><mrow id="A2.p1.1.m1.4.5.3.3.2" xref="A2.p1.1.m1.4.5.3.3.1.cmml"><mo stretchy="false" id="A2.p1.1.m1.4.5.3.3.2.1" xref="A2.p1.1.m1.4.5.3.3.1.cmml">(</mo><mi id="A2.p1.1.m1.3.3" xref="A2.p1.1.m1.3.3.cmml">y</mi><mo id="A2.p1.1.m1.4.5.3.3.2.2" xref="A2.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="A2.p1.1.m1.4.4" xref="A2.p1.1.m1.4.4.cmml">x</mi><mo stretchy="false" id="A2.p1.1.m1.4.5.3.3.2.3" xref="A2.p1.1.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.4b"><apply id="A2.p1.1.m1.4.5.cmml" xref="A2.p1.1.m1.4.5"><eq id="A2.p1.1.m1.4.5.1.cmml" xref="A2.p1.1.m1.4.5.1"></eq><apply id="A2.p1.1.m1.4.5.2.cmml" xref="A2.p1.1.m1.4.5.2"><times id="A2.p1.1.m1.4.5.2.1.cmml" xref="A2.p1.1.m1.4.5.2.1"></times><ci id="A2.p1.1.m1.4.5.2.2.cmml" xref="A2.p1.1.m1.4.5.2.2">𝑑</ci><interval closure="open" id="A2.p1.1.m1.4.5.2.3.1.cmml" xref="A2.p1.1.m1.4.5.2.3.2"><ci id="A2.p1.1.m1.1.1.cmml" xref="A2.p1.1.m1.1.1">𝑥</ci><ci id="A2.p1.1.m1.2.2.cmml" xref="A2.p1.1.m1.2.2">𝑦</ci></interval></apply><apply id="A2.p1.1.m1.4.5.3.cmml" xref="A2.p1.1.m1.4.5.3"><times id="A2.p1.1.m1.4.5.3.1.cmml" xref="A2.p1.1.m1.4.5.3.1"></times><ci id="A2.p1.1.m1.4.5.3.2.cmml" xref="A2.p1.1.m1.4.5.3.2">𝑑</ci><interval closure="open" id="A2.p1.1.m1.4.5.3.3.1.cmml" xref="A2.p1.1.m1.4.5.3.3.2"><ci id="A2.p1.1.m1.3.3.cmml" xref="A2.p1.1.m1.3.3">𝑦</ci><ci id="A2.p1.1.m1.4.4.cmml" xref="A2.p1.1.m1.4.4">𝑥</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.4c">d(x,y)=d(y,x)</annotation></semantics></math>. The <span id="A2.p1.3.2" class="ltx_text ltx_font_italic">global</span> property means all samples are used to compute the distance with respect to every sample. The <span id="A2.p1.3.3" class="ltx_text ltx_font_italic">ordinal</span> property is when the ordering of the distance is taken into consideration. For example, mutual nearest neighbor is not ordinal since the nearest neighbors <math id="A2.p1.2.m2.3" class="ltx_Math" alttext="\{a,b,c\}" display="inline"><semantics id="A2.p1.2.m2.3a"><mrow id="A2.p1.2.m2.3.4.2" xref="A2.p1.2.m2.3.4.1.cmml"><mo stretchy="false" id="A2.p1.2.m2.3.4.2.1" xref="A2.p1.2.m2.3.4.1.cmml">{</mo><mi id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml">a</mi><mo id="A2.p1.2.m2.3.4.2.2" xref="A2.p1.2.m2.3.4.1.cmml">,</mo><mi id="A2.p1.2.m2.2.2" xref="A2.p1.2.m2.2.2.cmml">b</mi><mo id="A2.p1.2.m2.3.4.2.3" xref="A2.p1.2.m2.3.4.1.cmml">,</mo><mi id="A2.p1.2.m2.3.3" xref="A2.p1.2.m2.3.3.cmml">c</mi><mo stretchy="false" id="A2.p1.2.m2.3.4.2.4" xref="A2.p1.2.m2.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.3b"><set id="A2.p1.2.m2.3.4.1.cmml" xref="A2.p1.2.m2.3.4.2"><ci id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1">𝑎</ci><ci id="A2.p1.2.m2.2.2.cmml" xref="A2.p1.2.m2.2.2">𝑏</ci><ci id="A2.p1.2.m2.3.3.cmml" xref="A2.p1.2.m2.3.3">𝑐</ci></set></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.3c">\{a,b,c\}</annotation></semantics></math> and <math id="A2.p1.3.m3.3" class="ltx_Math" alttext="\{c,a,b\}" display="inline"><semantics id="A2.p1.3.m3.3a"><mrow id="A2.p1.3.m3.3.4.2" xref="A2.p1.3.m3.3.4.1.cmml"><mo stretchy="false" id="A2.p1.3.m3.3.4.2.1" xref="A2.p1.3.m3.3.4.1.cmml">{</mo><mi id="A2.p1.3.m3.1.1" xref="A2.p1.3.m3.1.1.cmml">c</mi><mo id="A2.p1.3.m3.3.4.2.2" xref="A2.p1.3.m3.3.4.1.cmml">,</mo><mi id="A2.p1.3.m3.2.2" xref="A2.p1.3.m3.2.2.cmml">a</mi><mo id="A2.p1.3.m3.3.4.2.3" xref="A2.p1.3.m3.3.4.1.cmml">,</mo><mi id="A2.p1.3.m3.3.3" xref="A2.p1.3.m3.3.3.cmml">b</mi><mo stretchy="false" id="A2.p1.3.m3.3.4.2.4" xref="A2.p1.3.m3.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.3.m3.3b"><set id="A2.p1.3.m3.3.4.1.cmml" xref="A2.p1.3.m3.3.4.2"><ci id="A2.p1.3.m3.1.1.cmml" xref="A2.p1.3.m3.1.1">𝑐</ci><ci id="A2.p1.3.m3.2.2.cmml" xref="A2.p1.3.m3.2.2">𝑎</ci><ci id="A2.p1.3.m3.3.3.cmml" xref="A2.p1.3.m3.3.3">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.3.m3.3c">\{c,a,b\}</annotation></semantics></math> are treated equally. The <span id="A2.p1.3.4" class="ltx_text ltx_font_italic">batchable</span> property is a computational property that makes it feasible to compute in a reasonable time frame.</p>
</div>
<section id="A2.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Vision-vision comparison</h4>

<div id="A2.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="A2.SS0.SSS0.Px1.p1.1" class="ltx_p">In <a href="#A2.F12" title="In Cross-modal comparison ‣ Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">12</span></a>, we evaluate Spearman’s rank correlation among different metrics and hyperparameters over <math id="A2.SS0.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="78" display="inline"><semantics id="A2.SS0.SSS0.Px1.p1.1.m1.1a"><mn id="A2.SS0.SSS0.Px1.p1.1.m1.1.1" xref="A2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">78</mn><annotation-xml encoding="MathML-Content" id="A2.SS0.SSS0.Px1.p1.1.m1.1b"><cn type="integer" id="A2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A2.SS0.SSS0.Px1.p1.1.m1.1.1">78</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.SS0.SSS0.Px1.p1.1.m1.1c">78</annotation></semantics></math> vision models (details in <a href="#A3.SS1" title="C.1 Vision-Vision Alignment and Representation Quality ‣ Appendix C Experiments on Evaluating Alignment and Convergence ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">C.1</span></a>). We find most metrics highly correlated with each other.</p>
</div>
</section>
<section id="A2.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Cross-modal comparison</h4>

<div id="A2.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="A2.SS0.SSS0.Px2.p1.1" class="ltx_p">We measure vision-language alignment using a range of alternative metrics. We visualize the corresponding alignment results in Figure <a href="#A2.F13" title="Figure 13 ‣ Cross-modal comparison ‣ Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">13</span></a> and Figure <a href="#A2.F14" title="Figure 14 ‣ Cross-modal comparison ‣ Appendix B Consistency across various metrics ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">14</span></a>. Our findings indicate that alignment sensitivity not only depends on the metric used to compute it but also varies according to the specific tasks on which the vision models are trained.</p>
</div>
<figure id="A2.F11" class="ltx_figure">
<table id="A2.F11.7" class="ltx_tabular ltx_centering ltx_align_middle">
<tbody class="ltx_tbody">
<tr id="A2.F11.7.8.1" class="ltx_tr">
<td id="A2.F11.7.8.1.1" class="ltx_td ltx_align_right ltx_border_tt" rowspan="2"><span id="A2.F11.7.8.1.1.1" class="ltx_text ltx_font_bold">Metric</span></td>
<td id="A2.F11.7.8.1.2" class="ltx_td ltx_align_center ltx_border_tt" colspan="4"><span id="A2.F11.7.8.1.2.1" class="ltx_text ltx_font_bold">Property</span></td>
<th id="A2.F11.7.8.1.3" class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_th_row ltx_border_tt">
<span id="A2.F11.7.8.1.3.1" class="ltx_inline-block ltx_align_top">
<span id="A2.F11.7.8.1.3.1.1" class="ltx_p" style="width:233.3pt;"><span id="A2.F11.7.8.1.3.1.1.1" class="ltx_text"></span> <span id="A2.F11.7.8.1.3.1.1.2" class="ltx_text">
<span id="A2.F11.7.8.1.3.1.1.2.1" class="ltx_tabular ltx_align_middle">
<span id="A2.F11.7.8.1.3.1.1.2.1.1" class="ltx_tr">
<span id="A2.F11.7.8.1.3.1.1.2.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center ltx_rowspan ltx_rowspan_2"><span id="A2.F11.7.8.1.3.1.1.2.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:80%;">Description</span></span></span>
</span></span><span id="A2.F11.7.8.1.3.1.1.3" class="ltx_text"></span></span>
</span>
</th>
</tr>
<tr id="A2.F11.7.9.2" class="ltx_tr">
<td id="A2.F11.7.9.2.1" class="ltx_td ltx_align_center ltx_border_t"><span id="A2.F11.7.9.2.1.1" class="ltx_text" style="font-size:90%;">symmetric</span></td>
<td id="A2.F11.7.9.2.2" class="ltx_td ltx_align_center ltx_border_t"><span id="A2.F11.7.9.2.2.1" class="ltx_text" style="font-size:90%;">global</span></td>
<td id="A2.F11.7.9.2.3" class="ltx_td ltx_align_center ltx_border_t"><span id="A2.F11.7.9.2.3.1" class="ltx_text" style="font-size:90%;">ordinal</span></td>
<td id="A2.F11.7.9.2.4" class="ltx_td ltx_align_center ltx_border_t"><span id="A2.F11.7.9.2.4.1" class="ltx_text" style="font-size:90%;">batchable</span></td>
<td id="A2.F11.7.9.2.5" class="ltx_td ltx_align_top ltx_border_t"></td>
</tr>
<tr id="A2.F11.7.10.3" class="ltx_tr" style="background-color:#E6E6E6;">
<td id="A2.F11.7.10.3.1" class="ltx_td ltx_align_right ltx_border_t" style="padding-bottom:31.0001pt;"><span id="A2.F11.7.10.3.1.1" class="ltx_text" style="background-color:#E6E6E6;">CKA</span></td>
<td id="A2.F11.7.10.3.2" class="ltx_td ltx_align_center ltx_border_t" style="padding-bottom:31.0001pt;"><span id="A2.F11.7.10.3.2.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.10.3.3" class="ltx_td ltx_align_center ltx_border_t" style="padding-bottom:31.0001pt;"><span id="A2.F11.7.10.3.3.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.10.3.4" class="ltx_td ltx_align_center ltx_border_t" style="padding-bottom:31.0001pt;"><span id="A2.F11.7.10.3.4.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.10.3.5" class="ltx_td ltx_align_center ltx_border_t" style="padding-bottom:31.0001pt;"><span id="A2.F11.7.10.3.5.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.10.3.6" class="ltx_td ltx_align_justify ltx_align_top ltx_border_t" style="padding-bottom:31.0001pt;">
<span id="A2.F11.7.10.3.6.1" class="ltx_inline-block ltx_align_top" style="background-color:#E6E6E6;">
<span id="A2.F11.7.10.3.6.1.1" class="ltx_p" style="width:233.3pt;">Centered Kernel Alignment (CKA; <cite class="ltx_cite ltx_citemacro_citet">Kornblith et al. (<a href="#bib.bib61" title="" class="ltx_ref">2019</a>)</cite>) measures the similarity of neural networks by comparing the alignment of their kernel induced by their feature spaces.</span>
</span>
</td>
</tr>
<tr id="A2.F11.7.11.4" class="ltx_tr">
<td id="A2.F11.7.11.4.1" class="ltx_td ltx_align_right" style="padding-bottom:18.00005pt;"><span id="A2.F11.7.11.4.1.1" class="ltx_text">Unbiased CKA</span></td>
<td id="A2.F11.7.11.4.2" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.7.11.4.2.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.11.4.3" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.7.11.4.3.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.11.4.4" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.7.11.4.4.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.11.4.5" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.7.11.4.5.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.11.4.6" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:18.00005pt;">
<span id="A2.F11.7.11.4.6.1" class="ltx_inline-block ltx_align_top">
<span id="A2.F11.7.11.4.6.1.1" class="ltx_p" style="width:233.3pt;">Unbiased estimator of CKA that corrects for sample bias in HSIC <cite class="ltx_cite ltx_citemacro_citep">(Song et al., <a href="#bib.bib110" title="" class="ltx_ref">2012</a>)</cite>.</span>
</span>
</td>
</tr>
<tr id="A2.F11.7.12.5" class="ltx_tr" style="background-color:#E6E6E6;">
<td id="A2.F11.7.12.5.1" class="ltx_td ltx_align_right" style="padding-bottom:42.00003pt;"><span id="A2.F11.7.12.5.1.1" class="ltx_text" style="background-color:#E6E6E6;">SVCCA</span></td>
<td id="A2.F11.7.12.5.2" class="ltx_td ltx_align_center" style="padding-bottom:42.00003pt;"><span id="A2.F11.7.12.5.2.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.12.5.3" class="ltx_td ltx_align_center" style="padding-bottom:42.00003pt;" rowspan="2"><span id="A2.F11.7.12.5.3.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.12.5.4" class="ltx_td ltx_align_center" style="padding-bottom:42.00003pt;" rowspan="2"><span id="A2.F11.7.12.5.4.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.12.5.5" class="ltx_td ltx_align_center" style="padding-bottom:42.00003pt;"><span id="A2.F11.7.12.5.5.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.7.12.5.6" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:42.00003pt;">
<span id="A2.F11.7.12.5.6.1" class="ltx_inline-block ltx_align_top" style="background-color:#E6E6E6;">
<span id="A2.F11.7.12.5.6.1.1" class="ltx_p" style="width:233.3pt;">Singular Value Canonical Correlation Analysis (SVCCA; <cite class="ltx_cite ltx_citemacro_citet">Raghu et al. (<a href="#bib.bib100" title="" class="ltx_ref">2017</a>)</cite>) compares neural networks by decomposing their activities into singular vectors and measuring correlation.</span>
</span>
</td>
</tr>
<tr id="A2.F11.1.1" class="ltx_tr">
<td id="A2.F11.1.1.1" class="ltx_td ltx_align_right" style="padding-bottom:18.00005pt;"><span id="A2.F11.1.1.1.1" class="ltx_text">Mutual <math id="A2.F11.1.1.1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F11.1.1.1.1.m1.1a"><mi id="A2.F11.1.1.1.1.m1.1.1" xref="A2.F11.1.1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F11.1.1.1.1.m1.1b"><ci id="A2.F11.1.1.1.1.m1.1.1.cmml" xref="A2.F11.1.1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.1.1.1.1.m1.1c">k</annotation></semantics></math>-NN</span></td>
<td id="A2.F11.1.1.2" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.1.1.2.1" class="ltx_text">✓</span></td>
<td id="A2.F11.1.1.3" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.1.1.3.1" class="ltx_text">✓</span></td>
<td id="A2.F11.1.1.4" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:18.00005pt;">
<span id="A2.F11.1.1.4.1" class="ltx_inline-block ltx_align_top">
<span id="A2.F11.1.1.4.1.1" class="ltx_p" style="width:233.3pt;">Measures the intersection over union (IoU) of nearest neighbors between two models.</span>
</span>
</td>
</tr>
<tr id="A2.F11.2.2" class="ltx_tr" style="background-color:#E6E6E6;">
<td id="A2.F11.2.2.2" class="ltx_td ltx_align_right" style="padding-bottom:18.00005pt;"><span id="A2.F11.2.2.2.1" class="ltx_text" style="background-color:#E6E6E6;">CKNNA</span></td>
<td id="A2.F11.2.2.3" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;" rowspan="2"><span id="A2.F11.2.2.3.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.2.2.1" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;" rowspan="2"><span id="A2.F11.2.2.1.1" class="ltx_text" style="background-color:#E6E6E6;">✓<math id="A2.F11.2.2.1.1.m1.1" class="ltx_Math" alttext="\ast" display="inline"><semantics id="A2.F11.2.2.1.1.m1.1a"><mo mathbackground="#E6E6E6" id="A2.F11.2.2.1.1.m1.1.1" xref="A2.F11.2.2.1.1.m1.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="A2.F11.2.2.1.1.m1.1b"><ci id="A2.F11.2.2.1.1.m1.1.1.cmml" xref="A2.F11.2.2.1.1.m1.1.1">∗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.2.2.1.1.m1.1c">\ast</annotation></semantics></math></span></td>
<td id="A2.F11.2.2.4" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;" rowspan="2"><span id="A2.F11.2.2.4.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.2.2.5" class="ltx_td ltx_align_center" style="padding-bottom:18.00005pt;"><span id="A2.F11.2.2.5.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.2.2.6" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:18.00005pt;">
<span id="A2.F11.2.2.6.1" class="ltx_inline-block ltx_align_top" style="background-color:#E6E6E6;">
<span id="A2.F11.2.2.6.1.1" class="ltx_p" style="width:233.3pt;">Modified CKA measure that computes the kernel alignment only for its nearest neighbors. See Appendix <a href="#A1" title="Appendix A Mutual 𝑘-Nearest Neighbor Alignment Metric ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">A</span></a>.</span>
</span>
</td>
</tr>
<tr id="A2.F11.3.3" class="ltx_tr">
<td id="A2.F11.3.3.1" class="ltx_td ltx_align_right" style="padding-bottom:31.0001pt;"><span id="A2.F11.3.3.1.1" class="ltx_text">Cycle <math id="A2.F11.3.3.1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F11.3.3.1.1.m1.1a"><mi id="A2.F11.3.3.1.1.m1.1.1" xref="A2.F11.3.3.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F11.3.3.1.1.m1.1b"><ci id="A2.F11.3.3.1.1.m1.1.1.cmml" xref="A2.F11.3.3.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.3.3.1.1.m1.1c">k</annotation></semantics></math>-NN</span></td>
<td id="A2.F11.3.3.2" class="ltx_td ltx_align_center" style="padding-bottom:31.0001pt;" rowspan="3"><span id="A2.F11.3.3.2.1" class="ltx_text">✓</span></td>
<td id="A2.F11.3.3.3" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:31.0001pt;">
<span id="A2.F11.3.3.3.1" class="ltx_inline-block ltx_align_top">
<span id="A2.F11.3.3.3.1.1" class="ltx_p" style="width:233.3pt;">Measures whether the nearest neighbor in one domain also considers the original sample as its nearest neighbor in the other domain.</span>
</span>
</td>
</tr>
<tr id="A2.F11.5.5" class="ltx_tr" style="background-color:#E6E6E6;">
<td id="A2.F11.4.4.1" class="ltx_td ltx_align_right" style="padding-bottom:31.0001pt;"><span id="A2.F11.4.4.1.1" class="ltx_text" style="background-color:#E6E6E6;">Edit <math id="A2.F11.4.4.1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F11.4.4.1.1.m1.1a"><mi mathbackground="#E6E6E6" id="A2.F11.4.4.1.1.m1.1.1" xref="A2.F11.4.4.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F11.4.4.1.1.m1.1b"><ci id="A2.F11.4.4.1.1.m1.1.1.cmml" xref="A2.F11.4.4.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.4.4.1.1.m1.1c">k</annotation></semantics></math>-NN</span></td>
<td id="A2.F11.5.5.3" class="ltx_td ltx_align_center" style="padding-bottom:31.0001pt;"><span id="A2.F11.5.5.3.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.5.5.2" class="ltx_td ltx_align_center" style="padding-bottom:31.0001pt;"><span id="A2.F11.5.5.2.1" class="ltx_text" style="background-color:#E6E6E6;">✓<math id="A2.F11.5.5.2.1.m1.1" class="ltx_Math" alttext="\ast" display="inline"><semantics id="A2.F11.5.5.2.1.m1.1a"><mo mathbackground="#E6E6E6" id="A2.F11.5.5.2.1.m1.1.1" xref="A2.F11.5.5.2.1.m1.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="A2.F11.5.5.2.1.m1.1b"><ci id="A2.F11.5.5.2.1.m1.1.1.cmml" xref="A2.F11.5.5.2.1.m1.1.1">∗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.5.5.2.1.m1.1c">\ast</annotation></semantics></math></span></td>
<td id="A2.F11.5.5.4" class="ltx_td ltx_align_center" style="padding-bottom:31.0001pt;"><span id="A2.F11.5.5.4.1" class="ltx_text" style="background-color:#E6E6E6;">✓</span></td>
<td id="A2.F11.5.5.5" class="ltx_td ltx_align_justify ltx_align_top" style="padding-bottom:31.0001pt;">
<span id="A2.F11.5.5.5.1" class="ltx_inline-block ltx_align_top" style="background-color:#E6E6E6;">
<span id="A2.F11.5.5.5.1.1" class="ltx_p" style="width:233.3pt;">Computes the edit distance required to match the nearest neighbors between two datasets. The score is normalized by the maximum edit distance.</span>
</span>
</td>
</tr>
<tr id="A2.F11.7.7" class="ltx_tr">
<td id="A2.F11.6.6.1" class="ltx_td ltx_align_right ltx_border_bb" style="padding-bottom:16.99998pt;" rowspan="2"><span id="A2.F11.6.6.1.1" class="ltx_text">LCS <math id="A2.F11.6.6.1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F11.6.6.1.1.m1.1a"><mi id="A2.F11.6.6.1.1.m1.1.1" xref="A2.F11.6.6.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F11.6.6.1.1.m1.1b"><ci id="A2.F11.6.6.1.1.m1.1.1.cmml" xref="A2.F11.6.6.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.6.6.1.1.m1.1c">k</annotation></semantics></math>-NN</span></td>
<td id="A2.F11.7.7.3" class="ltx_td ltx_align_center ltx_border_bb" style="padding-bottom:16.99998pt;" rowspan="2"><span id="A2.F11.7.7.3.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.7.2" class="ltx_td ltx_align_center ltx_border_bb" style="padding-bottom:16.99998pt;" rowspan="2"><span id="A2.F11.7.7.2.1" class="ltx_text">✓<math id="A2.F11.7.7.2.1.m1.1" class="ltx_Math" alttext="\ast" display="inline"><semantics id="A2.F11.7.7.2.1.m1.1a"><mo id="A2.F11.7.7.2.1.m1.1.1" xref="A2.F11.7.7.2.1.m1.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="A2.F11.7.7.2.1.m1.1b"><ci id="A2.F11.7.7.2.1.m1.1.1.cmml" xref="A2.F11.7.7.2.1.m1.1.1">∗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.7.7.2.1.m1.1c">\ast</annotation></semantics></math></span></td>
<td id="A2.F11.7.7.4" class="ltx_td ltx_align_center ltx_border_bb" style="padding-bottom:16.99998pt;" rowspan="2"><span id="A2.F11.7.7.4.1" class="ltx_text">✓</span></td>
<td id="A2.F11.7.7.5" class="ltx_td ltx_align_justify ltx_align_top ltx_border_bb" style="padding-bottom:16.99998pt;">
<span id="A2.F11.7.7.5.1" class="ltx_inline-block ltx_align_top">
<span id="A2.F11.7.7.5.1.1" class="ltx_p" style="width:233.3pt;">Calculates the longest common subsequence of nearest neighbors and is normalized by the sequence length.</span>
</span>
</td>
</tr>
</tbody>
</table>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 11: </span>Comparative analysis of neural network similarity metrics. ✓<math id="A2.F11.11.m1.1" class="ltx_Math" alttext="\ast" display="inline"><semantics id="A2.F11.11.m1.1b"><mo id="A2.F11.11.m1.1.1" xref="A2.F11.11.m1.1.1.cmml">∗</mo><annotation-xml encoding="MathML-Content" id="A2.F11.11.m1.1c"><ci id="A2.F11.11.m1.1.1.cmml" xref="A2.F11.11.m1.1.1">∗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.11.m1.1d">\ast</annotation></semantics></math> indicates the metric is global and still meaningful when the nearest neighbor <math id="A2.F11.12.m2.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F11.12.m2.1b"><mi id="A2.F11.12.m2.1.1" xref="A2.F11.12.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F11.12.m2.1c"><ci id="A2.F11.12.m2.1.1.cmml" xref="A2.F11.12.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.12.m2.1d">k</annotation></semantics></math> is set to maximum batch-size <math id="A2.F11.13.m3.1" class="ltx_Math" alttext="k=|\mathcal{X}|" display="inline"><semantics id="A2.F11.13.m3.1b"><mrow id="A2.F11.13.m3.1.2" xref="A2.F11.13.m3.1.2.cmml"><mi id="A2.F11.13.m3.1.2.2" xref="A2.F11.13.m3.1.2.2.cmml">k</mi><mo id="A2.F11.13.m3.1.2.1" xref="A2.F11.13.m3.1.2.1.cmml">=</mo><mrow id="A2.F11.13.m3.1.2.3.2" xref="A2.F11.13.m3.1.2.3.1.cmml"><mo stretchy="false" id="A2.F11.13.m3.1.2.3.2.1" xref="A2.F11.13.m3.1.2.3.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="A2.F11.13.m3.1.1" xref="A2.F11.13.m3.1.1.cmml">𝒳</mi><mo stretchy="false" id="A2.F11.13.m3.1.2.3.2.2" xref="A2.F11.13.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.F11.13.m3.1c"><apply id="A2.F11.13.m3.1.2.cmml" xref="A2.F11.13.m3.1.2"><eq id="A2.F11.13.m3.1.2.1.cmml" xref="A2.F11.13.m3.1.2.1"></eq><ci id="A2.F11.13.m3.1.2.2.cmml" xref="A2.F11.13.m3.1.2.2">𝑘</ci><apply id="A2.F11.13.m3.1.2.3.1.cmml" xref="A2.F11.13.m3.1.2.3.2"><abs id="A2.F11.13.m3.1.2.3.1.1.cmml" xref="A2.F11.13.m3.1.2.3.2.1"></abs><ci id="A2.F11.13.m3.1.1.cmml" xref="A2.F11.13.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F11.13.m3.1d">k=|\mathcal{X}|</annotation></semantics></math>.</figcaption>
</figure>
<figure id="A2.F12" class="ltx_figure"><img src="/html/2405.07987/assets/x13.png" id="A2.F12.g1" class="ltx_graphics ltx_centering ltx_img_square" width="461" height="390" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 12: </span><span id="A2.F12.16.1" class="ltx_text ltx_font_bold">Vision-vision alignment measured with various metrics.</span> Spearman’s rank correlation among different metrics and batch sizes (<math id="A2.F12.8.m1.1" class="ltx_Math" alttext="\mathsf{bsz}" display="inline"><semantics id="A2.F12.8.m1.1b"><mi id="A2.F12.8.m1.1.1" xref="A2.F12.8.m1.1.1.cmml">𝖻𝗌𝗓</mi><annotation-xml encoding="MathML-Content" id="A2.F12.8.m1.1c"><ci id="A2.F12.8.m1.1.1.cmml" xref="A2.F12.8.m1.1.1">𝖻𝗌𝗓</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.8.m1.1d">\mathsf{bsz}</annotation></semantics></math>) when used to measure alignment among <math id="A2.F12.9.m2.1" class="ltx_Math" alttext="78" display="inline"><semantics id="A2.F12.9.m2.1b"><mn id="A2.F12.9.m2.1.1" xref="A2.F12.9.m2.1.1.cmml">78</mn><annotation-xml encoding="MathML-Content" id="A2.F12.9.m2.1c"><cn type="integer" id="A2.F12.9.m2.1.1.cmml" xref="A2.F12.9.m2.1.1">78</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.9.m2.1d">78</annotation></semantics></math> vision models (see <a href="#A3.SS1" title="C.1 Vision-Vision Alignment and Representation Quality ‣ Appendix C Experiments on Evaluating Alignment and Convergence ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">C.1</span></a> for details of these models). All <math id="A2.F12.10.m3.1" class="ltx_Math" alttext="p" display="inline"><semantics id="A2.F12.10.m3.1b"><mi id="A2.F12.10.m3.1.1" xref="A2.F12.10.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A2.F12.10.m3.1c"><ci id="A2.F12.10.m3.1.1.cmml" xref="A2.F12.10.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.10.m3.1d">p</annotation></semantics></math>-values are below <math id="A2.F12.11.m4.1" class="ltx_Math" alttext="2.24\times 10^{-105}" display="inline"><semantics id="A2.F12.11.m4.1b"><mrow id="A2.F12.11.m4.1.1" xref="A2.F12.11.m4.1.1.cmml"><mn id="A2.F12.11.m4.1.1.2" xref="A2.F12.11.m4.1.1.2.cmml">2.24</mn><mo lspace="0.222em" rspace="0.222em" id="A2.F12.11.m4.1.1.1" xref="A2.F12.11.m4.1.1.1.cmml">×</mo><msup id="A2.F12.11.m4.1.1.3" xref="A2.F12.11.m4.1.1.3.cmml"><mn id="A2.F12.11.m4.1.1.3.2" xref="A2.F12.11.m4.1.1.3.2.cmml">10</mn><mrow id="A2.F12.11.m4.1.1.3.3" xref="A2.F12.11.m4.1.1.3.3.cmml"><mo id="A2.F12.11.m4.1.1.3.3b" xref="A2.F12.11.m4.1.1.3.3.cmml">−</mo><mn id="A2.F12.11.m4.1.1.3.3.2" xref="A2.F12.11.m4.1.1.3.3.2.cmml">105</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.F12.11.m4.1c"><apply id="A2.F12.11.m4.1.1.cmml" xref="A2.F12.11.m4.1.1"><times id="A2.F12.11.m4.1.1.1.cmml" xref="A2.F12.11.m4.1.1.1"></times><cn type="float" id="A2.F12.11.m4.1.1.2.cmml" xref="A2.F12.11.m4.1.1.2">2.24</cn><apply id="A2.F12.11.m4.1.1.3.cmml" xref="A2.F12.11.m4.1.1.3"><csymbol cd="ambiguous" id="A2.F12.11.m4.1.1.3.1.cmml" xref="A2.F12.11.m4.1.1.3">superscript</csymbol><cn type="integer" id="A2.F12.11.m4.1.1.3.2.cmml" xref="A2.F12.11.m4.1.1.3.2">10</cn><apply id="A2.F12.11.m4.1.1.3.3.cmml" xref="A2.F12.11.m4.1.1.3.3"><minus id="A2.F12.11.m4.1.1.3.3.1.cmml" xref="A2.F12.11.m4.1.1.3.3"></minus><cn type="integer" id="A2.F12.11.m4.1.1.3.3.2.cmml" xref="A2.F12.11.m4.1.1.3.3.2">105</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.11.m4.1d">2.24\times 10^{-105}</annotation></semantics></math>. Our vision-vision analysis in <a href="#S2.F2.6" title="In Preliminaries ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">2</span></a> is based on the first metric (Mutual <math id="A2.F12.12.m5.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F12.12.m5.1b"><mi id="A2.F12.12.m5.1.1" xref="A2.F12.12.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F12.12.m5.1c"><ci id="A2.F12.12.m5.1.1.cmml" xref="A2.F12.12.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.12.m5.1d">k</annotation></semantics></math>-NN with <math id="A2.F12.13.m6.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F12.13.m6.1b"><mrow id="A2.F12.13.m6.1.1" xref="A2.F12.13.m6.1.1.cmml"><mi id="A2.F12.13.m6.1.1.2" xref="A2.F12.13.m6.1.1.2.cmml">k</mi><mo id="A2.F12.13.m6.1.1.1" xref="A2.F12.13.m6.1.1.1.cmml">=</mo><mn id="A2.F12.13.m6.1.1.3" xref="A2.F12.13.m6.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F12.13.m6.1c"><apply id="A2.F12.13.m6.1.1.cmml" xref="A2.F12.13.m6.1.1"><eq id="A2.F12.13.m6.1.1.1.cmml" xref="A2.F12.13.m6.1.1.1"></eq><ci id="A2.F12.13.m6.1.1.2.cmml" xref="A2.F12.13.m6.1.1.2">𝑘</ci><cn type="integer" id="A2.F12.13.m6.1.1.3.cmml" xref="A2.F12.13.m6.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.13.m6.1d">k=10</annotation></semantics></math> and <math id="A2.F12.14.m7.1" class="ltx_Math" alttext="\mathsf{bsz}=1000" display="inline"><semantics id="A2.F12.14.m7.1b"><mrow id="A2.F12.14.m7.1.1" xref="A2.F12.14.m7.1.1.cmml"><mi id="A2.F12.14.m7.1.1.2" xref="A2.F12.14.m7.1.1.2.cmml">𝖻𝗌𝗓</mi><mo id="A2.F12.14.m7.1.1.1" xref="A2.F12.14.m7.1.1.1.cmml">=</mo><mn id="A2.F12.14.m7.1.1.3" xref="A2.F12.14.m7.1.1.3.cmml">1000</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F12.14.m7.1c"><apply id="A2.F12.14.m7.1.1.cmml" xref="A2.F12.14.m7.1.1"><eq id="A2.F12.14.m7.1.1.1.cmml" xref="A2.F12.14.m7.1.1.1"></eq><ci id="A2.F12.14.m7.1.1.2.cmml" xref="A2.F12.14.m7.1.1.2">𝖻𝗌𝗓</ci><cn type="integer" id="A2.F12.14.m7.1.1.3.cmml" xref="A2.F12.14.m7.1.1.3">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F12.14.m7.1d">\mathsf{bsz}=1000</annotation></semantics></math>).</figcaption>
</figure>
<figure id="A2.F13" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F13.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x14.png" id="A2.F13.sf1.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F13.sf1.2.1.1" class="ltx_text" style="font-size:80%;">(a)</span> </span><span id="A2.F13.sf1.3.2" class="ltx_text" style="font-size:80%;">CKA</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F13.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x15.png" id="A2.F13.sf2.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F13.sf2.2.1.1" class="ltx_text" style="font-size:80%;">(b)</span> </span><span id="A2.F13.sf2.3.2" class="ltx_text" style="font-size:80%;">Unbiased CKA</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F13.sf3" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x16.png" id="A2.F13.sf3.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F13.sf3.2.1.1" class="ltx_text" style="font-size:80%;">(c)</span> </span><span id="A2.F13.sf3.3.2" class="ltx_text" style="font-size:80%;">SVCCA</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F13.sf4" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x17.png" id="A2.F13.sf4.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F13.sf4.6.3.1" class="ltx_text" style="font-size:80%;">(d)</span> </span><span id="A2.F13.sf4.4.2" class="ltx_text" style="font-size:80%;">Mutual <math id="A2.F13.sf4.3.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F13.sf4.3.1.m1.1b"><mi id="A2.F13.sf4.3.1.m1.1.1" xref="A2.F13.sf4.3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F13.sf4.3.1.m1.1c"><ci id="A2.F13.sf4.3.1.m1.1.1.cmml" xref="A2.F13.sf4.3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F13.sf4.3.1.m1.1d">k</annotation></semantics></math>-NN (<math id="A2.F13.sf4.4.2.m2.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F13.sf4.4.2.m2.1b"><mrow id="A2.F13.sf4.4.2.m2.1.1" xref="A2.F13.sf4.4.2.m2.1.1.cmml"><mi id="A2.F13.sf4.4.2.m2.1.1.2" xref="A2.F13.sf4.4.2.m2.1.1.2.cmml">k</mi><mo id="A2.F13.sf4.4.2.m2.1.1.1" xref="A2.F13.sf4.4.2.m2.1.1.1.cmml">=</mo><mn id="A2.F13.sf4.4.2.m2.1.1.3" xref="A2.F13.sf4.4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F13.sf4.4.2.m2.1c"><apply id="A2.F13.sf4.4.2.m2.1.1.cmml" xref="A2.F13.sf4.4.2.m2.1.1"><eq id="A2.F13.sf4.4.2.m2.1.1.1.cmml" xref="A2.F13.sf4.4.2.m2.1.1.1"></eq><ci id="A2.F13.sf4.4.2.m2.1.1.2.cmml" xref="A2.F13.sf4.4.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A2.F13.sf4.4.2.m2.1.1.3.cmml" xref="A2.F13.sf4.4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F13.sf4.4.2.m2.1d">k=10</annotation></semantics></math>)</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 13: </span><span id="A2.F13.2.1" class="ltx_text ltx_font_bold">Cross-modal alignment for various metrics</span></figcaption>
</figure>
<figure id="A2.F14" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F14.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x18.png" id="A2.F14.sf1.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F14.sf1.4.2.1" class="ltx_text" style="font-size:80%;">(a)</span> </span><span id="A2.F14.sf1.2.1" class="ltx_text" style="font-size:80%;">CKNNA (<math id="A2.F14.sf1.2.1.m1.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F14.sf1.2.1.m1.1b"><mrow id="A2.F14.sf1.2.1.m1.1.1" xref="A2.F14.sf1.2.1.m1.1.1.cmml"><mi id="A2.F14.sf1.2.1.m1.1.1.2" xref="A2.F14.sf1.2.1.m1.1.1.2.cmml">k</mi><mo id="A2.F14.sf1.2.1.m1.1.1.1" xref="A2.F14.sf1.2.1.m1.1.1.1.cmml">=</mo><mn id="A2.F14.sf1.2.1.m1.1.1.3" xref="A2.F14.sf1.2.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F14.sf1.2.1.m1.1c"><apply id="A2.F14.sf1.2.1.m1.1.1.cmml" xref="A2.F14.sf1.2.1.m1.1.1"><eq id="A2.F14.sf1.2.1.m1.1.1.1.cmml" xref="A2.F14.sf1.2.1.m1.1.1.1"></eq><ci id="A2.F14.sf1.2.1.m1.1.1.2.cmml" xref="A2.F14.sf1.2.1.m1.1.1.2">𝑘</ci><cn type="integer" id="A2.F14.sf1.2.1.m1.1.1.3.cmml" xref="A2.F14.sf1.2.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf1.2.1.m1.1d">k=10</annotation></semantics></math>)</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F14.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x19.png" id="A2.F14.sf2.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F14.sf2.6.3.1" class="ltx_text" style="font-size:80%;">(b)</span> </span><span id="A2.F14.sf2.4.2" class="ltx_text" style="font-size:80%;">Cycle <math id="A2.F14.sf2.3.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F14.sf2.3.1.m1.1b"><mi id="A2.F14.sf2.3.1.m1.1.1" xref="A2.F14.sf2.3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F14.sf2.3.1.m1.1c"><ci id="A2.F14.sf2.3.1.m1.1.1.cmml" xref="A2.F14.sf2.3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf2.3.1.m1.1d">k</annotation></semantics></math>-NN (<math id="A2.F14.sf2.4.2.m2.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F14.sf2.4.2.m2.1b"><mrow id="A2.F14.sf2.4.2.m2.1.1" xref="A2.F14.sf2.4.2.m2.1.1.cmml"><mi id="A2.F14.sf2.4.2.m2.1.1.2" xref="A2.F14.sf2.4.2.m2.1.1.2.cmml">k</mi><mo id="A2.F14.sf2.4.2.m2.1.1.1" xref="A2.F14.sf2.4.2.m2.1.1.1.cmml">=</mo><mn id="A2.F14.sf2.4.2.m2.1.1.3" xref="A2.F14.sf2.4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F14.sf2.4.2.m2.1c"><apply id="A2.F14.sf2.4.2.m2.1.1.cmml" xref="A2.F14.sf2.4.2.m2.1.1"><eq id="A2.F14.sf2.4.2.m2.1.1.1.cmml" xref="A2.F14.sf2.4.2.m2.1.1.1"></eq><ci id="A2.F14.sf2.4.2.m2.1.1.2.cmml" xref="A2.F14.sf2.4.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A2.F14.sf2.4.2.m2.1.1.3.cmml" xref="A2.F14.sf2.4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf2.4.2.m2.1d">k=10</annotation></semantics></math>)</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F14.sf3" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x20.png" id="A2.F14.sf3.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F14.sf3.6.3.1" class="ltx_text" style="font-size:80%;">(c)</span> </span><span id="A2.F14.sf3.4.2" class="ltx_text" style="font-size:80%;">Edit-distance <math id="A2.F14.sf3.3.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F14.sf3.3.1.m1.1b"><mi id="A2.F14.sf3.3.1.m1.1.1" xref="A2.F14.sf3.3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F14.sf3.3.1.m1.1c"><ci id="A2.F14.sf3.3.1.m1.1.1.cmml" xref="A2.F14.sf3.3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf3.3.1.m1.1d">k</annotation></semantics></math>-NN (<math id="A2.F14.sf3.4.2.m2.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F14.sf3.4.2.m2.1b"><mrow id="A2.F14.sf3.4.2.m2.1.1" xref="A2.F14.sf3.4.2.m2.1.1.cmml"><mi id="A2.F14.sf3.4.2.m2.1.1.2" xref="A2.F14.sf3.4.2.m2.1.1.2.cmml">k</mi><mo id="A2.F14.sf3.4.2.m2.1.1.1" xref="A2.F14.sf3.4.2.m2.1.1.1.cmml">=</mo><mn id="A2.F14.sf3.4.2.m2.1.1.3" xref="A2.F14.sf3.4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F14.sf3.4.2.m2.1c"><apply id="A2.F14.sf3.4.2.m2.1.1.cmml" xref="A2.F14.sf3.4.2.m2.1.1"><eq id="A2.F14.sf3.4.2.m2.1.1.1.cmml" xref="A2.F14.sf3.4.2.m2.1.1.1"></eq><ci id="A2.F14.sf3.4.2.m2.1.1.2.cmml" xref="A2.F14.sf3.4.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A2.F14.sf3.4.2.m2.1.1.3.cmml" xref="A2.F14.sf3.4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf3.4.2.m2.1d">k=10</annotation></semantics></math>)</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A2.F14.sf4" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2405.07987/assets/x21.png" id="A2.F14.sf4.g1" class="ltx_graphics ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="A2.F14.sf4.6.3.1" class="ltx_text" style="font-size:80%;">(d)</span> </span><span id="A2.F14.sf4.4.2" class="ltx_text" style="font-size:80%;">Longest-Common-Subsequence <math id="A2.F14.sf4.3.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F14.sf4.3.1.m1.1b"><mi id="A2.F14.sf4.3.1.m1.1.1" xref="A2.F14.sf4.3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F14.sf4.3.1.m1.1c"><ci id="A2.F14.sf4.3.1.m1.1.1.cmml" xref="A2.F14.sf4.3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf4.3.1.m1.1d">k</annotation></semantics></math>-NN (<math id="A2.F14.sf4.4.2.m2.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A2.F14.sf4.4.2.m2.1b"><mrow id="A2.F14.sf4.4.2.m2.1.1" xref="A2.F14.sf4.4.2.m2.1.1.cmml"><mi id="A2.F14.sf4.4.2.m2.1.1.2" xref="A2.F14.sf4.4.2.m2.1.1.2.cmml">k</mi><mo id="A2.F14.sf4.4.2.m2.1.1.1" xref="A2.F14.sf4.4.2.m2.1.1.1.cmml">=</mo><mn id="A2.F14.sf4.4.2.m2.1.1.3" xref="A2.F14.sf4.4.2.m2.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.F14.sf4.4.2.m2.1c"><apply id="A2.F14.sf4.4.2.m2.1.1.cmml" xref="A2.F14.sf4.4.2.m2.1.1"><eq id="A2.F14.sf4.4.2.m2.1.1.1.cmml" xref="A2.F14.sf4.4.2.m2.1.1.1"></eq><ci id="A2.F14.sf4.4.2.m2.1.1.2.cmml" xref="A2.F14.sf4.4.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A2.F14.sf4.4.2.m2.1.1.3.cmml" xref="A2.F14.sf4.4.2.m2.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F14.sf4.4.2.m2.1d">k=10</annotation></semantics></math>)</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 14: </span><span id="A2.F14.2.1" class="ltx_text ltx_font_bold">Cross-modal alignment measured with various metrics</span></figcaption>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</section>
<section id="A3" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix C </span>Experiments on Evaluating Alignment and Convergence</h2>

<div id="A3.p1" class="ltx_para ltx_noindent">
<p id="A3.p1.1" class="ltx_p">To demonstrate representational convergence, we take off-the-shelf models at multiple scales and multiple modalities and measure their representational alignment.</p>
</div>
<section id="A3.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">C.1 </span>Vision-Vision Alignment and Representation Quality</h3>

<div id="A3.SS1.p1" class="ltx_para ltx_noindent">
<p id="A3.SS1.p1.1" class="ltx_p">We consider 78 vision models in total:</p>
<ul id="A3.I1" class="ltx_itemize">
<li id="A3.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i1.p1" class="ltx_para ltx_noindent">
<p id="A3.I1.i1.p1.1" class="ltx_p"><math id="A3.I1.i1.p1.1.m1.1" class="ltx_Math" alttext="17" display="inline"><semantics id="A3.I1.i1.p1.1.m1.1a"><mn id="A3.I1.i1.p1.1.m1.1.1" xref="A3.I1.i1.p1.1.m1.1.1.cmml">17</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i1.p1.1.m1.1b"><cn type="integer" id="A3.I1.i1.p1.1.m1.1.1.cmml" xref="A3.I1.i1.p1.1.m1.1.1">17</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i1.p1.1.m1.1c">17</annotation></semantics></math> ViT models ranging from ViT-tiny to ViT-giant, trained on tasks including ImageNet-21k <cite class="ltx_cite ltx_citemacro_citep">(Dosovitskiy et al., <a href="#bib.bib28" title="" class="ltx_ref">2020</a>)</cite> classification, Masked Autoencoders <cite class="ltx_cite ltx_citemacro_citep">(He et al., <a href="#bib.bib45" title="" class="ltx_ref">2021</a>)</cite>, DINO <cite class="ltx_cite ltx_citemacro_citep">(Caron et al., <a href="#bib.bib18" title="" class="ltx_ref">2021</a>)</cite>, and CLIP <cite class="ltx_cite ltx_citemacro_citep">(Radford et al., <a href="#bib.bib99" title="" class="ltx_ref">2021</a>)</cite>, including some finetuned on ImageNet-12k.</p>
</div>
</li>
<li id="A3.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i2.p1" class="ltx_para">
<p id="A3.I1.i2.p1.1" class="ltx_p"><math id="A3.I1.i2.p1.1.m1.1" class="ltx_Math" alttext="1" display="inline"><semantics id="A3.I1.i2.p1.1.m1.1a"><mn id="A3.I1.i2.p1.1.m1.1.1" xref="A3.I1.i2.p1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i2.p1.1.m1.1b"><cn type="integer" id="A3.I1.i2.p1.1.m1.1.1.cmml" xref="A3.I1.i2.p1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i2.p1.1.m1.1c">1</annotation></semantics></math> randomly initialized ResNet-50.</p>
</div>
</li>
<li id="A3.I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i3.p1" class="ltx_para">
<p id="A3.I1.i3.p1.2" class="ltx_p"><math id="A3.I1.i3.p1.1.m1.1" class="ltx_Math" alttext="11" display="inline"><semantics id="A3.I1.i3.p1.1.m1.1a"><mn id="A3.I1.i3.p1.1.m1.1.1" xref="A3.I1.i3.p1.1.m1.1.1.cmml">11</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i3.p1.1.m1.1b"><cn type="integer" id="A3.I1.i3.p1.1.m1.1.1.cmml" xref="A3.I1.i3.p1.1.m1.1.1">11</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i3.p1.1.m1.1c">11</annotation></semantics></math> ResNet-50 models trained with contrastive learning on ImageNet-1k, Places-365 <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a href="#bib.bib137" title="" class="ltx_ref">2017</a>; López-Cifuentes et al., <a href="#bib.bib74" title="" class="ltx_ref">2020</a>)</cite>, and <math id="A3.I1.i3.p1.2.m2.1" class="ltx_Math" alttext="9" display="inline"><semantics id="A3.I1.i3.p1.2.m2.1a"><mn id="A3.I1.i3.p1.2.m2.1.1" xref="A3.I1.i3.p1.2.m2.1.1.cmml">9</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i3.p1.2.m2.1b"><cn type="integer" id="A3.I1.i3.p1.2.m2.1.1.cmml" xref="A3.I1.i3.p1.2.m2.1.1">9</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i3.p1.2.m2.1c">9</annotation></semantics></math> synthetic image datasets used in <cite class="ltx_cite ltx_citemacro_citet">Baradad et al. (<a href="#bib.bib11" title="" class="ltx_ref">2022</a>)</cite>.</p>
</div>
</li>
<li id="A3.I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A3.I1.i4.p1" class="ltx_para ltx_noindent">
<p id="A3.I1.i4.p1.2" class="ltx_p"><math id="A3.I1.i4.p1.1.m1.1" class="ltx_Math" alttext="49" display="inline"><semantics id="A3.I1.i4.p1.1.m1.1a"><mn id="A3.I1.i4.p1.1.m1.1.1" xref="A3.I1.i4.p1.1.m1.1.1.cmml">49</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i4.p1.1.m1.1b"><cn type="integer" id="A3.I1.i4.p1.1.m1.1.1.cmml" xref="A3.I1.i4.p1.1.m1.1.1">49</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i4.p1.1.m1.1c">49</annotation></semantics></math> ResNet-18 models trained with Alignment and Uniformity contrastive loss <cite class="ltx_cite ltx_citemacro_citep">(Wang &amp; Isola, <a href="#bib.bib126" title="" class="ltx_ref">2020</a>)</cite> on ImageNet-100, Places-365, and <math id="A3.I1.i4.p1.2.m2.1" class="ltx_Math" alttext="47" display="inline"><semantics id="A3.I1.i4.p1.2.m2.1a"><mn id="A3.I1.i4.p1.2.m2.1.1" xref="A3.I1.i4.p1.2.m2.1.1.cmml">47</mn><annotation-xml encoding="MathML-Content" id="A3.I1.i4.p1.2.m2.1b"><cn type="integer" id="A3.I1.i4.p1.2.m2.1.1.cmml" xref="A3.I1.i4.p1.2.m2.1.1">47</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.I1.i4.p1.2.m2.1c">47</annotation></semantics></math> realistic and synthetic image datasets from <cite class="ltx_cite ltx_citemacro_citet">Baradad et al. (<a href="#bib.bib10" title="" class="ltx_ref">2021</a>)</cite>.</p>
</div>
</li>
</ul>
</div>
<div id="A3.SS1.p2" class="ltx_para ltx_noindent">
<p id="A3.SS1.p2.1" class="ltx_p">To test representation quality, we evaluate linear probing performance on all 19 VTAB classification tasks <cite class="ltx_cite ltx_citemacro_citep">(Zhai et al., <a href="#bib.bib135" title="" class="ltx_ref">2019</a>)</cite>, which is a standard multi-task transfer learning benchmark containing structured, specialized, and natural datasets covering diverse domains. To reduce compute requirements, we subsample training and validation datasets to have at most 10,000 samples. We consider a representation solves a task if its performance is <math id="A3.SS1.p2.1.m1.1" class="ltx_Math" alttext="\geq 80\%" display="inline"><semantics id="A3.SS1.p2.1.m1.1a"><mrow id="A3.SS1.p2.1.m1.1.1" xref="A3.SS1.p2.1.m1.1.1.cmml"><mi id="A3.SS1.p2.1.m1.1.1.2" xref="A3.SS1.p2.1.m1.1.1.2.cmml"></mi><mo id="A3.SS1.p2.1.m1.1.1.1" xref="A3.SS1.p2.1.m1.1.1.1.cmml">≥</mo><mrow id="A3.SS1.p2.1.m1.1.1.3" xref="A3.SS1.p2.1.m1.1.1.3.cmml"><mn id="A3.SS1.p2.1.m1.1.1.3.2" xref="A3.SS1.p2.1.m1.1.1.3.2.cmml">80</mn><mo id="A3.SS1.p2.1.m1.1.1.3.1" xref="A3.SS1.p2.1.m1.1.1.3.1.cmml">%</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.1.m1.1b"><apply id="A3.SS1.p2.1.m1.1.1.cmml" xref="A3.SS1.p2.1.m1.1.1"><geq id="A3.SS1.p2.1.m1.1.1.1.cmml" xref="A3.SS1.p2.1.m1.1.1.1"></geq><csymbol cd="latexml" id="A3.SS1.p2.1.m1.1.1.2.cmml" xref="A3.SS1.p2.1.m1.1.1.2">absent</csymbol><apply id="A3.SS1.p2.1.m1.1.1.3.cmml" xref="A3.SS1.p2.1.m1.1.1.3"><csymbol cd="latexml" id="A3.SS1.p2.1.m1.1.1.3.1.cmml" xref="A3.SS1.p2.1.m1.1.1.3.1">percent</csymbol><cn type="integer" id="A3.SS1.p2.1.m1.1.1.3.2.cmml" xref="A3.SS1.p2.1.m1.1.1.3.2">80</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.1.m1.1c">\geq 80\%</annotation></semantics></math> of the best performance on that task across all 78 models.</p>
</div>
<div id="A3.SS1.p3" class="ltx_para ltx_noindent">
<p id="A3.SS1.p3.2" class="ltx_p">To compute the alignment metric, we use <math id="A3.SS1.p3.1.m1.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A3.SS1.p3.1.m1.1a"><mrow id="A3.SS1.p3.1.m1.1.1" xref="A3.SS1.p3.1.m1.1.1.cmml"><mi id="A3.SS1.p3.1.m1.1.1.2" xref="A3.SS1.p3.1.m1.1.1.2.cmml">k</mi><mo id="A3.SS1.p3.1.m1.1.1.1" xref="A3.SS1.p3.1.m1.1.1.1.cmml">=</mo><mn id="A3.SS1.p3.1.m1.1.1.3" xref="A3.SS1.p3.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p3.1.m1.1b"><apply id="A3.SS1.p3.1.m1.1.1.cmml" xref="A3.SS1.p3.1.m1.1.1"><eq id="A3.SS1.p3.1.m1.1.1.1.cmml" xref="A3.SS1.p3.1.m1.1.1.1"></eq><ci id="A3.SS1.p3.1.m1.1.1.2.cmml" xref="A3.SS1.p3.1.m1.1.1.2">𝑘</ci><cn type="integer" id="A3.SS1.p3.1.m1.1.1.3.cmml" xref="A3.SS1.p3.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p3.1.m1.1c">k=10</annotation></semantics></math> nearest neighbors over <math id="A3.SS1.p3.2.m2.1" class="ltx_Math" alttext="1000" display="inline"><semantics id="A3.SS1.p3.2.m2.1a"><mn id="A3.SS1.p3.2.m2.1.1" xref="A3.SS1.p3.2.m2.1.1.cmml">1000</mn><annotation-xml encoding="MathML-Content" id="A3.SS1.p3.2.m2.1b"><cn type="integer" id="A3.SS1.p3.2.m2.1.1.cmml" xref="A3.SS1.p3.2.m2.1.1">1000</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p3.2.m2.1c">1000</annotation></semantics></math> image representations computed on Places-365’s validation dataset <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a href="#bib.bib137" title="" class="ltx_ref">2017</a>)</cite>. This dataset is disjoint from VTAB datasets, although both contain natural images.</p>
</div>
</section>
<section id="A3.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">C.2 </span>Cross-Modal Alignment</h3>

<div id="A3.SS2.p1" class="ltx_para ltx_noindent">
<p id="A3.SS2.p1.1" class="ltx_p">We compare the representation of an image in a vision model to the representation of a caption describing that image in a language model. The language model families we consider are BLOOM <cite class="ltx_cite ltx_citemacro_citep">(BigScience et al., <a href="#bib.bib14" title="" class="ltx_ref">2022</a>)</cite>, OpenLlama <cite class="ltx_cite ltx_citemacro_citep">(Geng &amp; Liu, <a href="#bib.bib34" title="" class="ltx_ref">2023</a>)</cite>, and LLama <cite class="ltx_cite ltx_citemacro_citep">(Touvron et al., <a href="#bib.bib121" title="" class="ltx_ref">2023</a>)</cite>.
For Figure <a href="#S2.F4" title="Figure 4 ‣ 2.2 Alignment increases with scale and performance ‣ 2 Representations are converging ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a>, we included more recent model families such as OLMo <cite class="ltx_cite ltx_citemacro_citep">(Groeneveld et al., <a href="#bib.bib38" title="" class="ltx_ref">2024</a>)</cite>, LLama3 <cite class="ltx_cite ltx_citemacro_citep">(Meta, <a href="#bib.bib80" title="" class="ltx_ref">2024</a>)</cite>, Gemma <cite class="ltx_cite ltx_citemacro_citep">(Team et al., <a href="#bib.bib116" title="" class="ltx_ref">2024</a>)</cite>, and Mistral/Mixtral <cite class="ltx_cite ltx_citemacro_citep">(Jiang et al., <a href="#bib.bib53" title="" class="ltx_ref">2023</a>, <a href="#bib.bib54" title="" class="ltx_ref">2024</a>)</cite>. These models were downloaded from Huggingface <cite class="ltx_cite ltx_citemacro_citep">(Wolf et al., <a href="#bib.bib129" title="" class="ltx_ref">2019</a>)</cite>.</p>
</div>
<div id="A3.SS2.p2" class="ltx_para ltx_noindent">
<p id="A3.SS2.p2.1" class="ltx_p">For vision models, we consider ViT models <cite class="ltx_cite ltx_citemacro_citep">(Dosovitskiy et al., <a href="#bib.bib28" title="" class="ltx_ref">2020</a>)</cite> of various sizes trained on various data and objectives. We mainly consider the popular vision models: classification on ImageNet21K <cite class="ltx_cite ltx_citemacro_citep">(Russakovsky et al., <a href="#bib.bib102" title="" class="ltx_ref">2015</a>)</cite>, MAE <cite class="ltx_cite ltx_citemacro_citep">(He et al., <a href="#bib.bib45" title="" class="ltx_ref">2021</a>)</cite>, DINOv2 <cite class="ltx_cite ltx_citemacro_citep">(Oquab et al., <a href="#bib.bib92" title="" class="ltx_ref">2023</a>)</cite>, CLIP <cite class="ltx_cite ltx_citemacro_citep">(Radford et al., <a href="#bib.bib99" title="" class="ltx_ref">2021</a>)</cite>, and CLIP finetuned on ImageNet12K. These models were downloaded from PyTorch Image Models (TIMM) <cite class="ltx_cite ltx_citemacro_citep">(Wightman, <a href="#bib.bib128" title="" class="ltx_ref">2021</a>)</cite>.</p>
</div>
<div id="A3.SS2.p3" class="ltx_para ltx_noindent">
<p id="A3.SS2.p3.3" class="ltx_p">To compute the alignment metric, we use <math id="A3.SS2.p3.1.m1.1" class="ltx_Math" alttext="k=10" display="inline"><semantics id="A3.SS2.p3.1.m1.1a"><mrow id="A3.SS2.p3.1.m1.1.1" xref="A3.SS2.p3.1.m1.1.1.cmml"><mi id="A3.SS2.p3.1.m1.1.1.2" xref="A3.SS2.p3.1.m1.1.1.2.cmml">k</mi><mo id="A3.SS2.p3.1.m1.1.1.1" xref="A3.SS2.p3.1.m1.1.1.1.cmml">=</mo><mn id="A3.SS2.p3.1.m1.1.1.3" xref="A3.SS2.p3.1.m1.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p3.1.m1.1b"><apply id="A3.SS2.p3.1.m1.1.1.cmml" xref="A3.SS2.p3.1.m1.1.1"><eq id="A3.SS2.p3.1.m1.1.1.1.cmml" xref="A3.SS2.p3.1.m1.1.1.1"></eq><ci id="A3.SS2.p3.1.m1.1.1.2.cmml" xref="A3.SS2.p3.1.m1.1.1.2">𝑘</ci><cn type="integer" id="A3.SS2.p3.1.m1.1.1.3.cmml" xref="A3.SS2.p3.1.m1.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p3.1.m1.1c">k=10</annotation></semantics></math> nearest neighbors over 1024 samples from WIT (Wikipedia-based Image Text)<cite class="ltx_cite ltx_citemacro_citep">(Srinivasan et al., <a href="#bib.bib111" title="" class="ltx_ref">2021</a>)</cite>. For the vision model, we use class token of each layer, and for the language model, we average pool each layer to a single token. Since it is not trivial to determine where the alignment might occur, we draw inspiration from BrainScore<cite class="ltx_cite ltx_citemacro_citep">(Schrimpf et al., <a href="#bib.bib104" title="" class="ltx_ref">2018</a>)</cite> and compute pairwise alignment scores, then take the maximum. One of these pairwise comparisons also includes concatenated features. We apply <math id="A3.SS2.p3.2.m2.1" class="ltx_Math" alttext="l_{2}" display="inline"><semantics id="A3.SS2.p3.2.m2.1a"><msub id="A3.SS2.p3.2.m2.1.1" xref="A3.SS2.p3.2.m2.1.1.cmml"><mi id="A3.SS2.p3.2.m2.1.1.2" xref="A3.SS2.p3.2.m2.1.1.2.cmml">l</mi><mn id="A3.SS2.p3.2.m2.1.1.3" xref="A3.SS2.p3.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.p3.2.m2.1b"><apply id="A3.SS2.p3.2.m2.1.1.cmml" xref="A3.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS2.p3.2.m2.1.1.1.cmml" xref="A3.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="A3.SS2.p3.2.m2.1.1.2.cmml" xref="A3.SS2.p3.2.m2.1.1.2">𝑙</ci><cn type="integer" id="A3.SS2.p3.2.m2.1.1.3.cmml" xref="A3.SS2.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p3.2.m2.1c">l_{2}</annotation></semantics></math> normalization to the features before measuring the distance. As transformer architectures have “emergent outliers” <cite class="ltx_cite ltx_citemacro_citep">(Dettmers et al., <a href="#bib.bib23" title="" class="ltx_ref">2022</a>)</cite>, we truncate the elements in the features that are above the <math id="A3.SS2.p3.3.m3.1" class="ltx_Math" alttext="95" display="inline"><semantics id="A3.SS2.p3.3.m3.1a"><mn id="A3.SS2.p3.3.m3.1.1" xref="A3.SS2.p3.3.m3.1.1.cmml">95</mn><annotation-xml encoding="MathML-Content" id="A3.SS2.p3.3.m3.1b"><cn type="integer" id="A3.SS2.p3.3.m3.1.1.cmml" xref="A3.SS2.p3.3.m3.1.1">95</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p3.3.m3.1c">95</annotation></semantics></math>-th percentile.</p>
</div>
<div id="A3.SS2.p4" class="ltx_para ltx_noindent">
<p id="A3.SS2.p4.1" class="ltx_p">Simply taking the last token did not show any strong alignment signal. We also experimented with prompting the language model and taking the last token representation. The prompt we used was</p>
<table id="A6.EGx16" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A3.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><span id="A3.Ex5.3.2.1.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">An image with the caption ‘&lt;caption&gt;’. This is an image of a &lt;fill&gt;</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A3.SS2.p4.2" class="ltx_p">Using prompting showed similar trends to average pooling but had slightly lower alignment scores.</p>
</div>
</section>
</section>
<section id="A4" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix D </span>Color Cooccurrence Experiment</h2>

<div id="A4.p1" class="ltx_para ltx_noindent">
<p id="A4.p1.1" class="ltx_p">Here we describe the details of how we created the four color representations visualized in <a href="#S4.F8" title="In 4.1 An idealized world ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">8</span></a>, from left to right.</p>
</div>
<section id="A4.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Perceptual representation from CIELAB color space</h4>

<div id="A4.SS0.SSS0.Px1.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px1.p1.1" class="ltx_p">We embed pixels taken from the CIFAR-10 image dataset <cite class="ltx_cite ltx_citemacro_citep">(Krizhevsky et al., <a href="#bib.bib62" title="" class="ltx_ref">2009</a>; Torralba et al., <a href="#bib.bib120" title="" class="ltx_ref">2008</a>)</cite> based on the CIELAB color space, which is designed as a <em id="A4.SS0.SSS0.Px1.p1.1.1" class="ltx_emph ltx_font_italic">perceptually uniform</em> space that changes numerical values correspond to similar perceived changes in color.</p>
</div>
</section>
<section id="A4.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Three representations from cooccurrence in VISION and LANGUAGE</h4>

<div id="A4.SS0.SSS0.Px2.p1" class="ltx_para ltx_noindent">
<p id="A4.SS0.SSS0.Px2.p1.7" class="ltx_p">For these three representations, we first obtain a dissimilarity matrix over colors (in different ways detailed below), then use multidimensional scaling <cite class="ltx_cite ltx_citemacro_citep">(Shepard, <a href="#bib.bib106" title="" class="ltx_ref">1980</a>)</cite> to find a 3-dimensional embedding in which Euclidean distance between the embeddings for <math id="A4.SS0.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="A" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.1.m1.1a"><mi id="A4.SS0.SSS0.Px2.p1.1.m1.1.1" xref="A4.SS0.SSS0.Px2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.1.m1.1b"><ci id="A4.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.1.m1.1c">A</annotation></semantics></math> and <math id="A4.SS0.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="B" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.2.m2.1a"><mi id="A4.SS0.SSS0.Px2.p1.2.m2.1.1" xref="A4.SS0.SSS0.Px2.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.2.m2.1b"><ci id="A4.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.2.m2.1c">B</annotation></semantics></math>, <math id="A4.SS0.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="{z}_{A}" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.3.m3.1a"><msub id="A4.SS0.SSS0.Px2.p1.3.m3.1.1" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.2" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1.2.cmml">z</mi><mi id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.3" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.3.m3.1b"><apply id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1.2">𝑧</ci><ci id="A4.SS0.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="A4.SS0.SSS0.Px2.p1.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.3.m3.1c">{z}_{A}</annotation></semantics></math> and <math id="A4.SS0.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="{z}_{B}" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.4.m4.1a"><msub id="A4.SS0.SSS0.Px2.p1.4.m4.1.1" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1.cmml"><mi id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.2" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml">z</mi><mi id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.3" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.4.m4.1b"><apply id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1">subscript</csymbol><ci id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1.2">𝑧</ci><ci id="A4.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml" xref="A4.SS0.SSS0.Px2.p1.4.m4.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.4.m4.1c">{z}_{B}</annotation></semantics></math>, best matches this dissimilarity matrix. We use <math id="A4.SS0.SSS0.Px2.p1.5.m5.2" class="ltx_Math" alttext="1{,}000" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.5.m5.2a"><mrow id="A4.SS0.SSS0.Px2.p1.5.m5.2.3.2" xref="A4.SS0.SSS0.Px2.p1.5.m5.2.3.1.cmml"><mn id="A4.SS0.SSS0.Px2.p1.5.m5.1.1" xref="A4.SS0.SSS0.Px2.p1.5.m5.1.1.cmml">1</mn><mo id="A4.SS0.SSS0.Px2.p1.5.m5.2.3.2.1" xref="A4.SS0.SSS0.Px2.p1.5.m5.2.3.1.cmml">,</mo><mn id="A4.SS0.SSS0.Px2.p1.5.m5.2.2" xref="A4.SS0.SSS0.Px2.p1.5.m5.2.2.cmml">000</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.5.m5.2b"><list id="A4.SS0.SSS0.Px2.p1.5.m5.2.3.1.cmml" xref="A4.SS0.SSS0.Px2.p1.5.m5.2.3.2"><cn type="integer" id="A4.SS0.SSS0.Px2.p1.5.m5.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.5.m5.1.1">1</cn><cn type="integer" id="A4.SS0.SSS0.Px2.p1.5.m5.2.2.cmml" xref="A4.SS0.SSS0.Px2.p1.5.m5.2.2">000</cn></list></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.5.m5.2c">1{,}000</annotation></semantics></math> fits and take the best match. Afterward, we visually align it with the CIELAB space by finding the best rotation, translation, scaling, and flipping, by running the Kabsch-Umeyama algorithm <cite class="ltx_cite ltx_citemacro_citep">(Kabsch, <a href="#bib.bib56" title="" class="ltx_ref">1976</a>, <a href="#bib.bib57" title="" class="ltx_ref">1978</a>; Umeyama, <a href="#bib.bib123" title="" class="ltx_ref">1991</a>)</cite> twice, once on <math id="A4.SS0.SSS0.Px2.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{z}" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.6.m6.1a"><mi id="A4.SS0.SSS0.Px2.p1.6.m6.1.1" xref="A4.SS0.SSS0.Px2.p1.6.m6.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.6.m6.1b"><ci id="A4.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.6.m6.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.6.m6.1c">\mathbf{z}</annotation></semantics></math> and once on <math id="A4.SS0.SSS0.Px2.p1.7.m7.1" class="ltx_Math" alttext="-\mathbf{z}" display="inline"><semantics id="A4.SS0.SSS0.Px2.p1.7.m7.1a"><mrow id="A4.SS0.SSS0.Px2.p1.7.m7.1.1" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1.cmml"><mo id="A4.SS0.SSS0.Px2.p1.7.m7.1.1a" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1.cmml">−</mo><mi id="A4.SS0.SSS0.Px2.p1.7.m7.1.1.2" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml">𝐳</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS0.SSS0.Px2.p1.7.m7.1b"><apply id="A4.SS0.SSS0.Px2.p1.7.m7.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1"><minus id="A4.SS0.SSS0.Px2.p1.7.m7.1.1.1.cmml" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1"></minus><ci id="A4.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml" xref="A4.SS0.SSS0.Px2.p1.7.m7.1.1.2">𝐳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS0.SSS0.Px2.p1.7.m7.1c">-\mathbf{z}</annotation></semantics></math>, to account for flipping. The dissimilarity matrix we used in each case is described as following:</p>
<ul id="A4.I1" class="ltx_itemize">
<li id="A4.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A4.I1.i1.p1" class="ltx_para">
<p id="A4.I1.i1.p1.9" class="ltx_p"><span id="A4.I1.i1.p1.9.1" class="ltx_text ltx_font_bold">VISION: Pixel cooccurrence.</span>
We collect color cooccurrence statistics from the CIFAR-10 dataset, and estimate a joint distribution <math id="A4.I1.i1.p1.1.m1.2" class="ltx_Math" alttext="p(A,B)" display="inline"><semantics id="A4.I1.i1.p1.1.m1.2a"><mrow id="A4.I1.i1.p1.1.m1.2.3" xref="A4.I1.i1.p1.1.m1.2.3.cmml"><mi id="A4.I1.i1.p1.1.m1.2.3.2" xref="A4.I1.i1.p1.1.m1.2.3.2.cmml">p</mi><mo lspace="0em" rspace="0em" id="A4.I1.i1.p1.1.m1.2.3.1" xref="A4.I1.i1.p1.1.m1.2.3.1.cmml">​</mo><mrow id="A4.I1.i1.p1.1.m1.2.3.3.2" xref="A4.I1.i1.p1.1.m1.2.3.3.1.cmml"><mo stretchy="false" id="A4.I1.i1.p1.1.m1.2.3.3.2.1" xref="A4.I1.i1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="A4.I1.i1.p1.1.m1.1.1" xref="A4.I1.i1.p1.1.m1.1.1.cmml">A</mi><mo id="A4.I1.i1.p1.1.m1.2.3.3.2.2" xref="A4.I1.i1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="A4.I1.i1.p1.1.m1.2.2" xref="A4.I1.i1.p1.1.m1.2.2.cmml">B</mi><mo stretchy="false" id="A4.I1.i1.p1.1.m1.2.3.3.2.3" xref="A4.I1.i1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.1.m1.2b"><apply id="A4.I1.i1.p1.1.m1.2.3.cmml" xref="A4.I1.i1.p1.1.m1.2.3"><times id="A4.I1.i1.p1.1.m1.2.3.1.cmml" xref="A4.I1.i1.p1.1.m1.2.3.1"></times><ci id="A4.I1.i1.p1.1.m1.2.3.2.cmml" xref="A4.I1.i1.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="A4.I1.i1.p1.1.m1.2.3.3.1.cmml" xref="A4.I1.i1.p1.1.m1.2.3.3.2"><ci id="A4.I1.i1.p1.1.m1.1.1.cmml" xref="A4.I1.i1.p1.1.m1.1.1">𝐴</ci><ci id="A4.I1.i1.p1.1.m1.2.2.cmml" xref="A4.I1.i1.p1.1.m1.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.1.m1.2c">p(A,B)</annotation></semantics></math> over <math id="A4.I1.i1.p1.2.m2.2" class="ltx_Math" alttext="300{,}000" display="inline"><semantics id="A4.I1.i1.p1.2.m2.2a"><mrow id="A4.I1.i1.p1.2.m2.2.3.2" xref="A4.I1.i1.p1.2.m2.2.3.1.cmml"><mn id="A4.I1.i1.p1.2.m2.1.1" xref="A4.I1.i1.p1.2.m2.1.1.cmml">300</mn><mo id="A4.I1.i1.p1.2.m2.2.3.2.1" xref="A4.I1.i1.p1.2.m2.2.3.1.cmml">,</mo><mn id="A4.I1.i1.p1.2.m2.2.2" xref="A4.I1.i1.p1.2.m2.2.2.cmml">000</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.2.m2.2b"><list id="A4.I1.i1.p1.2.m2.2.3.1.cmml" xref="A4.I1.i1.p1.2.m2.2.3.2"><cn type="integer" id="A4.I1.i1.p1.2.m2.1.1.cmml" xref="A4.I1.i1.p1.2.m2.1.1">300</cn><cn type="integer" id="A4.I1.i1.p1.2.m2.2.2.cmml" xref="A4.I1.i1.p1.2.m2.2.2">000</cn></list></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.2.m2.2c">300{,}000</annotation></semantics></math> randomly sampled pixel colors <math id="A4.I1.i1.p1.3.m3.1" class="ltx_Math" alttext="A" display="inline"><semantics id="A4.I1.i1.p1.3.m3.1a"><mi id="A4.I1.i1.p1.3.m3.1.1" xref="A4.I1.i1.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.3.m3.1b"><ci id="A4.I1.i1.p1.3.m3.1.1.cmml" xref="A4.I1.i1.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.3.m3.1c">A</annotation></semantics></math> and <math id="A4.I1.i1.p1.4.m4.1" class="ltx_Math" alttext="B" display="inline"><semantics id="A4.I1.i1.p1.4.m4.1a"><mi id="A4.I1.i1.p1.4.m4.1.1" xref="A4.I1.i1.p1.4.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.4.m4.1b"><ci id="A4.I1.i1.p1.4.m4.1.1.cmml" xref="A4.I1.i1.p1.4.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.4.m4.1c">B</annotation></semantics></math> that occur within a radius of at most 4 pixels of one another. Colors are quantized on a grid in RGB space and represented as discrete variables, and <math id="A4.I1.i1.p1.5.m5.2" class="ltx_Math" alttext="p(A,B)" display="inline"><semantics id="A4.I1.i1.p1.5.m5.2a"><mrow id="A4.I1.i1.p1.5.m5.2.3" xref="A4.I1.i1.p1.5.m5.2.3.cmml"><mi id="A4.I1.i1.p1.5.m5.2.3.2" xref="A4.I1.i1.p1.5.m5.2.3.2.cmml">p</mi><mo lspace="0em" rspace="0em" id="A4.I1.i1.p1.5.m5.2.3.1" xref="A4.I1.i1.p1.5.m5.2.3.1.cmml">​</mo><mrow id="A4.I1.i1.p1.5.m5.2.3.3.2" xref="A4.I1.i1.p1.5.m5.2.3.3.1.cmml"><mo stretchy="false" id="A4.I1.i1.p1.5.m5.2.3.3.2.1" xref="A4.I1.i1.p1.5.m5.2.3.3.1.cmml">(</mo><mi id="A4.I1.i1.p1.5.m5.1.1" xref="A4.I1.i1.p1.5.m5.1.1.cmml">A</mi><mo id="A4.I1.i1.p1.5.m5.2.3.3.2.2" xref="A4.I1.i1.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="A4.I1.i1.p1.5.m5.2.2" xref="A4.I1.i1.p1.5.m5.2.2.cmml">B</mi><mo stretchy="false" id="A4.I1.i1.p1.5.m5.2.3.3.2.3" xref="A4.I1.i1.p1.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.5.m5.2b"><apply id="A4.I1.i1.p1.5.m5.2.3.cmml" xref="A4.I1.i1.p1.5.m5.2.3"><times id="A4.I1.i1.p1.5.m5.2.3.1.cmml" xref="A4.I1.i1.p1.5.m5.2.3.1"></times><ci id="A4.I1.i1.p1.5.m5.2.3.2.cmml" xref="A4.I1.i1.p1.5.m5.2.3.2">𝑝</ci><interval closure="open" id="A4.I1.i1.p1.5.m5.2.3.3.1.cmml" xref="A4.I1.i1.p1.5.m5.2.3.3.2"><ci id="A4.I1.i1.p1.5.m5.1.1.cmml" xref="A4.I1.i1.p1.5.m5.1.1">𝐴</ci><ci id="A4.I1.i1.p1.5.m5.2.2.cmml" xref="A4.I1.i1.p1.5.m5.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.5.m5.2c">p(A,B)</annotation></semantics></math> is modeled as a table of normalized counts, from which we compute the empirical pointwise mutual information matrix <math id="A4.I1.i1.p1.6.m6.2" class="ltx_Math" alttext="K_{\mathsf{PMI}}(A,B)" display="inline"><semantics id="A4.I1.i1.p1.6.m6.2a"><mrow id="A4.I1.i1.p1.6.m6.2.3" xref="A4.I1.i1.p1.6.m6.2.3.cmml"><msub id="A4.I1.i1.p1.6.m6.2.3.2" xref="A4.I1.i1.p1.6.m6.2.3.2.cmml"><mi id="A4.I1.i1.p1.6.m6.2.3.2.2" xref="A4.I1.i1.p1.6.m6.2.3.2.2.cmml">K</mi><mi id="A4.I1.i1.p1.6.m6.2.3.2.3" xref="A4.I1.i1.p1.6.m6.2.3.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A4.I1.i1.p1.6.m6.2.3.1" xref="A4.I1.i1.p1.6.m6.2.3.1.cmml">​</mo><mrow id="A4.I1.i1.p1.6.m6.2.3.3.2" xref="A4.I1.i1.p1.6.m6.2.3.3.1.cmml"><mo stretchy="false" id="A4.I1.i1.p1.6.m6.2.3.3.2.1" xref="A4.I1.i1.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="A4.I1.i1.p1.6.m6.1.1" xref="A4.I1.i1.p1.6.m6.1.1.cmml">A</mi><mo id="A4.I1.i1.p1.6.m6.2.3.3.2.2" xref="A4.I1.i1.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="A4.I1.i1.p1.6.m6.2.2" xref="A4.I1.i1.p1.6.m6.2.2.cmml">B</mi><mo stretchy="false" id="A4.I1.i1.p1.6.m6.2.3.3.2.3" xref="A4.I1.i1.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.6.m6.2b"><apply id="A4.I1.i1.p1.6.m6.2.3.cmml" xref="A4.I1.i1.p1.6.m6.2.3"><times id="A4.I1.i1.p1.6.m6.2.3.1.cmml" xref="A4.I1.i1.p1.6.m6.2.3.1"></times><apply id="A4.I1.i1.p1.6.m6.2.3.2.cmml" xref="A4.I1.i1.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="A4.I1.i1.p1.6.m6.2.3.2.1.cmml" xref="A4.I1.i1.p1.6.m6.2.3.2">subscript</csymbol><ci id="A4.I1.i1.p1.6.m6.2.3.2.2.cmml" xref="A4.I1.i1.p1.6.m6.2.3.2.2">𝐾</ci><ci id="A4.I1.i1.p1.6.m6.2.3.2.3.cmml" xref="A4.I1.i1.p1.6.m6.2.3.2.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="A4.I1.i1.p1.6.m6.2.3.3.1.cmml" xref="A4.I1.i1.p1.6.m6.2.3.3.2"><ci id="A4.I1.i1.p1.6.m6.1.1.cmml" xref="A4.I1.i1.p1.6.m6.1.1">𝐴</ci><ci id="A4.I1.i1.p1.6.m6.2.2.cmml" xref="A4.I1.i1.p1.6.m6.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.6.m6.2c">K_{\mathsf{PMI}}(A,B)</annotation></semantics></math>. Quantization ensures that there is no bias from how color distances are represented in RGB space. Dissimilarity matrix is defined as <math id="A4.I1.i1.p1.7.m7.2" class="ltx_Math" alttext="-K_{\mathsf{PMI}}(A,B)+c" display="inline"><semantics id="A4.I1.i1.p1.7.m7.2a"><mrow id="A4.I1.i1.p1.7.m7.2.3" xref="A4.I1.i1.p1.7.m7.2.3.cmml"><mrow id="A4.I1.i1.p1.7.m7.2.3.2" xref="A4.I1.i1.p1.7.m7.2.3.2.cmml"><mo id="A4.I1.i1.p1.7.m7.2.3.2a" xref="A4.I1.i1.p1.7.m7.2.3.2.cmml">−</mo><mrow id="A4.I1.i1.p1.7.m7.2.3.2.2" xref="A4.I1.i1.p1.7.m7.2.3.2.2.cmml"><msub id="A4.I1.i1.p1.7.m7.2.3.2.2.2" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2.cmml"><mi id="A4.I1.i1.p1.7.m7.2.3.2.2.2.2" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2.2.cmml">K</mi><mi id="A4.I1.i1.p1.7.m7.2.3.2.2.2.3" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A4.I1.i1.p1.7.m7.2.3.2.2.1" xref="A4.I1.i1.p1.7.m7.2.3.2.2.1.cmml">​</mo><mrow id="A4.I1.i1.p1.7.m7.2.3.2.2.3.2" xref="A4.I1.i1.p1.7.m7.2.3.2.2.3.1.cmml"><mo stretchy="false" id="A4.I1.i1.p1.7.m7.2.3.2.2.3.2.1" xref="A4.I1.i1.p1.7.m7.2.3.2.2.3.1.cmml">(</mo><mi id="A4.I1.i1.p1.7.m7.1.1" xref="A4.I1.i1.p1.7.m7.1.1.cmml">A</mi><mo id="A4.I1.i1.p1.7.m7.2.3.2.2.3.2.2" xref="A4.I1.i1.p1.7.m7.2.3.2.2.3.1.cmml">,</mo><mi id="A4.I1.i1.p1.7.m7.2.2" xref="A4.I1.i1.p1.7.m7.2.2.cmml">B</mi><mo stretchy="false" id="A4.I1.i1.p1.7.m7.2.3.2.2.3.2.3" xref="A4.I1.i1.p1.7.m7.2.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.I1.i1.p1.7.m7.2.3.1" xref="A4.I1.i1.p1.7.m7.2.3.1.cmml">+</mo><mi id="A4.I1.i1.p1.7.m7.2.3.3" xref="A4.I1.i1.p1.7.m7.2.3.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.7.m7.2b"><apply id="A4.I1.i1.p1.7.m7.2.3.cmml" xref="A4.I1.i1.p1.7.m7.2.3"><plus id="A4.I1.i1.p1.7.m7.2.3.1.cmml" xref="A4.I1.i1.p1.7.m7.2.3.1"></plus><apply id="A4.I1.i1.p1.7.m7.2.3.2.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2"><minus id="A4.I1.i1.p1.7.m7.2.3.2.1.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2"></minus><apply id="A4.I1.i1.p1.7.m7.2.3.2.2.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2"><times id="A4.I1.i1.p1.7.m7.2.3.2.2.1.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.1"></times><apply id="A4.I1.i1.p1.7.m7.2.3.2.2.2.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2"><csymbol cd="ambiguous" id="A4.I1.i1.p1.7.m7.2.3.2.2.2.1.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2">subscript</csymbol><ci id="A4.I1.i1.p1.7.m7.2.3.2.2.2.2.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2.2">𝐾</ci><ci id="A4.I1.i1.p1.7.m7.2.3.2.2.2.3.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.2.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="A4.I1.i1.p1.7.m7.2.3.2.2.3.1.cmml" xref="A4.I1.i1.p1.7.m7.2.3.2.2.3.2"><ci id="A4.I1.i1.p1.7.m7.1.1.cmml" xref="A4.I1.i1.p1.7.m7.1.1">𝐴</ci><ci id="A4.I1.i1.p1.7.m7.2.2.cmml" xref="A4.I1.i1.p1.7.m7.2.2">𝐵</ci></interval></apply></apply><ci id="A4.I1.i1.p1.7.m7.2.3.3.cmml" xref="A4.I1.i1.p1.7.m7.2.3.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.7.m7.2c">-K_{\mathsf{PMI}}(A,B)+c</annotation></semantics></math>, where <math id="A4.I1.i1.p1.8.m8.4" class="ltx_Math" alttext="c=\max_{A,B}K_{\mathsf{PMI}}(A,B)" display="inline"><semantics id="A4.I1.i1.p1.8.m8.4a"><mrow id="A4.I1.i1.p1.8.m8.4.5" xref="A4.I1.i1.p1.8.m8.4.5.cmml"><mi id="A4.I1.i1.p1.8.m8.4.5.2" xref="A4.I1.i1.p1.8.m8.4.5.2.cmml">c</mi><mo id="A4.I1.i1.p1.8.m8.4.5.1" xref="A4.I1.i1.p1.8.m8.4.5.1.cmml">=</mo><mrow id="A4.I1.i1.p1.8.m8.4.5.3" xref="A4.I1.i1.p1.8.m8.4.5.3.cmml"><mrow id="A4.I1.i1.p1.8.m8.4.5.3.2" xref="A4.I1.i1.p1.8.m8.4.5.3.2.cmml"><msub id="A4.I1.i1.p1.8.m8.4.5.3.2.1" xref="A4.I1.i1.p1.8.m8.4.5.3.2.1.cmml"><mi id="A4.I1.i1.p1.8.m8.4.5.3.2.1.2" xref="A4.I1.i1.p1.8.m8.4.5.3.2.1.2.cmml">max</mi><mrow id="A4.I1.i1.p1.8.m8.2.2.2.4" xref="A4.I1.i1.p1.8.m8.2.2.2.3.cmml"><mi id="A4.I1.i1.p1.8.m8.1.1.1.1" xref="A4.I1.i1.p1.8.m8.1.1.1.1.cmml">A</mi><mo id="A4.I1.i1.p1.8.m8.2.2.2.4.1" xref="A4.I1.i1.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="A4.I1.i1.p1.8.m8.2.2.2.2" xref="A4.I1.i1.p1.8.m8.2.2.2.2.cmml">B</mi></mrow></msub><mo lspace="0.167em" id="A4.I1.i1.p1.8.m8.4.5.3.2a" xref="A4.I1.i1.p1.8.m8.4.5.3.2.cmml">⁡</mo><msub id="A4.I1.i1.p1.8.m8.4.5.3.2.2" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2.cmml"><mi id="A4.I1.i1.p1.8.m8.4.5.3.2.2.2" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2.2.cmml">K</mi><mi id="A4.I1.i1.p1.8.m8.4.5.3.2.2.3" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2.3.cmml">𝖯𝖬𝖨</mi></msub></mrow><mo lspace="0em" rspace="0em" id="A4.I1.i1.p1.8.m8.4.5.3.1" xref="A4.I1.i1.p1.8.m8.4.5.3.1.cmml">​</mo><mrow id="A4.I1.i1.p1.8.m8.4.5.3.3.2" xref="A4.I1.i1.p1.8.m8.4.5.3.3.1.cmml"><mo stretchy="false" id="A4.I1.i1.p1.8.m8.4.5.3.3.2.1" xref="A4.I1.i1.p1.8.m8.4.5.3.3.1.cmml">(</mo><mi id="A4.I1.i1.p1.8.m8.3.3" xref="A4.I1.i1.p1.8.m8.3.3.cmml">A</mi><mo id="A4.I1.i1.p1.8.m8.4.5.3.3.2.2" xref="A4.I1.i1.p1.8.m8.4.5.3.3.1.cmml">,</mo><mi id="A4.I1.i1.p1.8.m8.4.4" xref="A4.I1.i1.p1.8.m8.4.4.cmml">B</mi><mo stretchy="false" id="A4.I1.i1.p1.8.m8.4.5.3.3.2.3" xref="A4.I1.i1.p1.8.m8.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.8.m8.4b"><apply id="A4.I1.i1.p1.8.m8.4.5.cmml" xref="A4.I1.i1.p1.8.m8.4.5"><eq id="A4.I1.i1.p1.8.m8.4.5.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.1"></eq><ci id="A4.I1.i1.p1.8.m8.4.5.2.cmml" xref="A4.I1.i1.p1.8.m8.4.5.2">𝑐</ci><apply id="A4.I1.i1.p1.8.m8.4.5.3.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3"><times id="A4.I1.i1.p1.8.m8.4.5.3.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.1"></times><apply id="A4.I1.i1.p1.8.m8.4.5.3.2.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2"><apply id="A4.I1.i1.p1.8.m8.4.5.3.2.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.1"><csymbol cd="ambiguous" id="A4.I1.i1.p1.8.m8.4.5.3.2.1.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.1">subscript</csymbol><max id="A4.I1.i1.p1.8.m8.4.5.3.2.1.2.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.1.2"></max><list id="A4.I1.i1.p1.8.m8.2.2.2.3.cmml" xref="A4.I1.i1.p1.8.m8.2.2.2.4"><ci id="A4.I1.i1.p1.8.m8.1.1.1.1.cmml" xref="A4.I1.i1.p1.8.m8.1.1.1.1">𝐴</ci><ci id="A4.I1.i1.p1.8.m8.2.2.2.2.cmml" xref="A4.I1.i1.p1.8.m8.2.2.2.2">𝐵</ci></list></apply><apply id="A4.I1.i1.p1.8.m8.4.5.3.2.2.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2"><csymbol cd="ambiguous" id="A4.I1.i1.p1.8.m8.4.5.3.2.2.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2">subscript</csymbol><ci id="A4.I1.i1.p1.8.m8.4.5.3.2.2.2.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2.2">𝐾</ci><ci id="A4.I1.i1.p1.8.m8.4.5.3.2.2.3.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.2.2.3">𝖯𝖬𝖨</ci></apply></apply><interval closure="open" id="A4.I1.i1.p1.8.m8.4.5.3.3.1.cmml" xref="A4.I1.i1.p1.8.m8.4.5.3.3.2"><ci id="A4.I1.i1.p1.8.m8.3.3.cmml" xref="A4.I1.i1.p1.8.m8.3.3">𝐴</ci><ci id="A4.I1.i1.p1.8.m8.4.4.cmml" xref="A4.I1.i1.p1.8.m8.4.4">𝐵</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.8.m8.4c">c=\max_{A,B}K_{\mathsf{PMI}}(A,B)</annotation></semantics></math> is an offset to ensure non-negativity (similar to the constant in <a href="#S4.SS2" title="4.2 A family of contrastive learners converge to a representation of ℙ⁢(𝐙) ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a> and <a href="#A6.Thmtheorem1" title="Proposition F.1. ‣ F.2 Contrastive learners can represent 𝐾_𝖯𝖬𝖨 exactly under smoothness conditions ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">F.1</span></a> that ensures neural networks can express <math id="A4.I1.i1.p1.9.m9.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A4.I1.i1.p1.9.m9.1a"><msub id="A4.I1.i1.p1.9.m9.1.1" xref="A4.I1.i1.p1.9.m9.1.1.cmml"><mi id="A4.I1.i1.p1.9.m9.1.1.2" xref="A4.I1.i1.p1.9.m9.1.1.2.cmml">K</mi><mi id="A4.I1.i1.p1.9.m9.1.1.3" xref="A4.I1.i1.p1.9.m9.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A4.I1.i1.p1.9.m9.1b"><apply id="A4.I1.i1.p1.9.m9.1.1.cmml" xref="A4.I1.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A4.I1.i1.p1.9.m9.1.1.1.cmml" xref="A4.I1.i1.p1.9.m9.1.1">subscript</csymbol><ci id="A4.I1.i1.p1.9.m9.1.1.2.cmml" xref="A4.I1.i1.p1.9.m9.1.1.2">𝐾</ci><ci id="A4.I1.i1.p1.9.m9.1.1.3.cmml" xref="A4.I1.i1.p1.9.m9.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i1.p1.9.m9.1c">K_{\mathsf{PMI}}</annotation></semantics></math>).</p>
</div>
</li>
<li id="A4.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A4.I1.i2.p1" class="ltx_para ltx_noindent">
<p id="A4.I1.i2.p1.1" class="ltx_p"><span id="A4.I1.i2.p1.1.1" class="ltx_text ltx_font_bold">LANGUAGE.</span> We used an approach similar to <cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite>.</p>
<ul id="A4.I1.i2.I1" class="ltx_itemize">
<li id="A4.I1.i2.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item"><span id="A4.I1.i2.I1.i1.1.1.1" class="ltx_text ltx_font_bold">–</span></span> 
<div id="A4.I1.i2.I1.i1.p1" class="ltx_para">
<p id="A4.I1.i2.I1.i1.p1.6" class="ltx_p">We take <math id="A4.I1.i2.I1.i1.p1.1.m1.1" class="ltx_Math" alttext="20" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.1.m1.1a"><mn id="A4.I1.i2.I1.i1.p1.1.m1.1.1" xref="A4.I1.i2.I1.i1.p1.1.m1.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.1.m1.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.1.m1.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.1.m1.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.1.m1.1c">20</annotation></semantics></math> pairs of (color, word) appeared in the dataset collected by <cite class="ltx_cite ltx_citemacro_citet">Lindsey &amp; Brown (<a href="#bib.bib69" title="" class="ltx_ref">2014</a>)</cite>, where <math id="A4.I1.i2.I1.i1.p1.2.m2.1" class="ltx_Math" alttext="51" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.2.m2.1a"><mn id="A4.I1.i2.I1.i1.p1.2.m2.1.1" xref="A4.I1.i2.I1.i1.p1.2.m2.1.1.cmml">51</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.2.m2.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.2.m2.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.2.m2.1.1">51</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.2.m2.1c">51</annotation></semantics></math> participants were asked to free name each of the <math id="A4.I1.i2.I1.i1.p1.3.m3.1" class="ltx_Math" alttext="330" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.3.m3.1a"><mn id="A4.I1.i2.I1.i1.p1.3.m3.1.1" xref="A4.I1.i2.I1.i1.p1.3.m3.1.1.cmml">330</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.3.m3.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.3.m3.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.3.m3.1.1">330</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.3.m3.1c">330</annotation></semantics></math> colors from the Munsell Color Chart. We filtered words that appeared less than <math id="A4.I1.i2.I1.i1.p1.4.m4.1" class="ltx_Math" alttext="100" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.4.m4.1a"><mn id="A4.I1.i2.I1.i1.p1.4.m4.1.1" xref="A4.I1.i2.I1.i1.p1.4.m4.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.4.m4.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.4.m4.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.4.m4.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.4.m4.1c">100</annotation></semantics></math> times, and computed each word’s associate color by taking the centroid in CIELAB space. Our filtering process followed <cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite> exactly, but resulted in <math id="A4.I1.i2.I1.i1.p1.5.m5.1" class="ltx_Math" alttext="20" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.5.m5.1a"><mn id="A4.I1.i2.I1.i1.p1.5.m5.1.1" xref="A4.I1.i2.I1.i1.p1.5.m5.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.5.m5.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.5.m5.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.5.m5.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.5.m5.1c">20</annotation></semantics></math> colors, a slightly different set than the <math id="A4.I1.i2.I1.i1.p1.6.m6.1" class="ltx_Math" alttext="18" display="inline"><semantics id="A4.I1.i2.I1.i1.p1.6.m6.1a"><mn id="A4.I1.i2.I1.i1.p1.6.m6.1.1" xref="A4.I1.i2.I1.i1.p1.6.m6.1.1.cmml">18</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i1.p1.6.m6.1b"><cn type="integer" id="A4.I1.i2.I1.i1.p1.6.m6.1.1.cmml" xref="A4.I1.i2.I1.i1.p1.6.m6.1.1">18</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i1.p1.6.m6.1c">18</annotation></semantics></math> colors they claimed.</p>
</div>
</li>
<li id="A4.I1.i2.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item"><span id="A4.I1.i2.I1.i2.1.1.1" class="ltx_text ltx_font_bold">–</span></span> 
<div id="A4.I1.i2.I1.i2.p1" class="ltx_para">
<p id="A4.I1.i2.I1.i2.p1.2" class="ltx_p">For each of the <math id="A4.I1.i2.I1.i2.p1.1.m1.1" class="ltx_Math" alttext="20" display="inline"><semantics id="A4.I1.i2.I1.i2.p1.1.m1.1a"><mn id="A4.I1.i2.I1.i2.p1.1.m1.1.1" xref="A4.I1.i2.I1.i2.p1.1.m1.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i2.p1.1.m1.1b"><cn type="integer" id="A4.I1.i2.I1.i2.p1.1.m1.1.1.cmml" xref="A4.I1.i2.I1.i2.p1.1.m1.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i2.p1.1.m1.1c">20</annotation></semantics></math> color words <span id="A4.I1.i2.I1.i2.p1.2.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">&lt;col&gt;</span>, we construct three sentences:</p>
<table id="A6.EGx17" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A4.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><span id="A4.Ex6.3.2.1.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">The color &lt;col&gt;.</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A4.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><span id="A4.Ex7.3.2.1.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">This color is &lt;col&gt;.</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A4.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><span id="A4.Ex8.3.2.1.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">The color of this thing is &lt;col&gt;.</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A4.I1.i2.I1.i2.p1.3" class="ltx_p">and obtain the average sentence embedding from the language encoder, as the embedding for <span id="A4.I1.i2.I1.i2.p1.3.1" class="ltx_text ltx_markedasmath ltx_font_typewriter">&lt;col&gt;</span> (details below). We find this approach more effective than <cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite>, which uses object names that potentially have color biases, even though the objects may appear in multiple colors.</p>
</div>
</li>
<li id="A4.I1.i2.I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item"><span id="A4.I1.i2.I1.i3.1.1.1" class="ltx_text ltx_font_bold">–</span></span> 
<div id="A4.I1.i2.I1.i3.p1" class="ltx_para">
<p id="A4.I1.i2.I1.i3.p1.1" class="ltx_p">Unlike <cite class="ltx_cite ltx_citemacro_citet">Abdou et al. (<a href="#bib.bib1" title="" class="ltx_ref">2021</a>)</cite>, we did not perform linear regression from language embedding to CIELAB space, which distorts distances and easily overfits with only <math id="A4.I1.i2.I1.i3.p1.1.m1.1" class="ltx_Math" alttext="20" display="inline"><semantics id="A4.I1.i2.I1.i3.p1.1.m1.1a"><mn id="A4.I1.i2.I1.i3.p1.1.m1.1.1" xref="A4.I1.i2.I1.i3.p1.1.m1.1.1.cmml">20</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i3.p1.1.m1.1b"><cn type="integer" id="A4.I1.i2.I1.i3.p1.1.m1.1.1.cmml" xref="A4.I1.i2.I1.i3.p1.1.m1.1.1">20</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i3.p1.1.m1.1c">20</annotation></semantics></math> samples. Instead, we used multidimensional scaling to best preserve distances, as described above.</p>
</div>
</li>
<li id="A4.I1.i2.I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item"><span id="A4.I1.i2.I1.i4.1.1.1" class="ltx_text ltx_font_bold">–</span></span> 
<div id="A4.I1.i2.I1.i4.p1" class="ltx_para">
<p id="A4.I1.i2.I1.i4.p1.2" class="ltx_p"><span id="A4.I1.i2.I1.i4.p1.2.1" class="ltx_text ltx_font_bold">Masked language contrastive learning (SimCSE) embedding: </span>
We used sentence embedding from the unsupervised SimCSE RoBERTa-L <cite class="ltx_cite ltx_citemacro_citep">(Gao et al., <a href="#bib.bib31" title="" class="ltx_ref">2021</a>)</cite> to encode the above sentences into <math id="A4.I1.i2.I1.i4.p1.1.m1.1" class="ltx_Math" alttext="1024" display="inline"><semantics id="A4.I1.i2.I1.i4.p1.1.m1.1a"><mn id="A4.I1.i2.I1.i4.p1.1.m1.1.1" xref="A4.I1.i2.I1.i4.p1.1.m1.1.1.cmml">1024</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i4.p1.1.m1.1b"><cn type="integer" id="A4.I1.i2.I1.i4.p1.1.m1.1.1.cmml" xref="A4.I1.i2.I1.i4.p1.1.m1.1.1">1024</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i4.p1.1.m1.1c">1024</annotation></semantics></math>-dimensional embeddings, and used the pairwise Euclidean distances among <span id="A4.I1.i2.I1.i4.p1.2.2" class="ltx_text ltx_markedasmath ltx_font_typewriter">&lt;col&gt;</span> embeddings as the dissimilarity matrix.</p>
</div>
</li>
<li id="A4.I1.i2.I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item"><span id="A4.I1.i2.I1.i5.1.1.1" class="ltx_text ltx_font_bold">–</span></span> 
<div id="A4.I1.i2.I1.i5.p1" class="ltx_para ltx_noindent">
<p id="A4.I1.i2.I1.i5.p1.2" class="ltx_p"><span id="A4.I1.i2.I1.i5.p1.2.1" class="ltx_text ltx_font_bold">Masked language predictive learning (RoBERTa) embedding: </span>
We concatenated hidden states of the last four layers of RoBERTa-L <cite class="ltx_cite ltx_citemacro_citep">(Liu et al., <a href="#bib.bib72" title="" class="ltx_ref">2019</a>)</cite>, following <cite class="ltx_cite ltx_citemacro_citep">(Devlin et al., <a href="#bib.bib24" title="" class="ltx_ref">2018</a>)</cite>. We averaged across token dimensions, and obtained a <math id="A4.I1.i2.I1.i5.p1.1.m1.1" class="ltx_Math" alttext="4096" display="inline"><semantics id="A4.I1.i2.I1.i5.p1.1.m1.1a"><mn id="A4.I1.i2.I1.i5.p1.1.m1.1.1" xref="A4.I1.i2.I1.i5.p1.1.m1.1.1.cmml">4096</mn><annotation-xml encoding="MathML-Content" id="A4.I1.i2.I1.i5.p1.1.m1.1b"><cn type="integer" id="A4.I1.i2.I1.i5.p1.1.m1.1.1.cmml" xref="A4.I1.i2.I1.i5.p1.1.m1.1.1">4096</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.I1.i2.I1.i5.p1.1.m1.1c">4096</annotation></semantics></math>-dimensional embedding for each of the above sentences, and used the pairwise Euclidean distances among <span id="A4.I1.i2.I1.i5.p1.2.2" class="ltx_text ltx_markedasmath ltx_font_typewriter">&lt;col&gt;</span> embeddings as the dissimilarity matrix.</p>
</div>
</li>
</ul>
</div>
</li>
</ul>
</div>
</section>
</section>
<section id="A5" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix E </span>Caption Density Experiments</h2>

<div id="A5.p1" class="ltx_para ltx_noindent">
<p id="A5.p1.1" class="ltx_p">We use LLAMA3-8B-Instruct <cite class="ltx_cite ltx_citemacro_citep">(Meta, <a href="#bib.bib80" title="" class="ltx_ref">2024</a>)</cite> to generate summary captions at various densities for images in the Densely Captioned Images dataset <cite class="ltx_cite ltx_citemacro_citep">(Urbanek et al., <a href="#bib.bib124" title="" class="ltx_ref">2023</a>)</cite> from the train split. Following  <cite class="ltx_cite ltx_citemacro_citet">Urbanek et al. (<a href="#bib.bib124" title="" class="ltx_ref">2023</a>)</cite>, we prompt the language model with the following instructions to generate captions at differing granularity:</p>
</div>
<div id="A5.p2" class="ltx_para ltx_noindent">
<p id="A5.p2.1" class="ltx_p"><span id="A5.p2.1.1" class="ltx_text ltx_font_typewriter">system: You are given a full-text description of an image. You should summarize it into about &lt;num_words&gt; words, being sure to include as much salient visual information as possible given the &lt;num_words&gt; word constraint, especially information from the start of the original description. The new description should apply for the original image. Respond with only the summary, in one line.</span></p>
</div>
<div id="A5.p3" class="ltx_para ltx_noindent">
<p id="A5.p3.1" class="ltx_p"><span id="A5.p3.1.1" class="ltx_text ltx_font_typewriter">user: &lt;original_caption&gt;</span></p>
</div>
</section>
<section id="A6" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix F </span>Analysis of Contrastive Learners</h2>

<section id="A6.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">F.1 </span>Contrastive objectives learn pointwise mutual information</h3>

<div id="A6.SS1.p1" class="ltx_para ltx_noindent">
<p id="A6.SS1.p1.3" class="ltx_p">There are two widely used forms of contrastive objectives. We now discuss each form in details and show how they both are minimized by the pointwise mutual information (PMI) as stated in <a href="#S4.E5" title="In 4.2 A family of contrastive learners converge to a representation of ℙ⁢(𝐙) ‣ 4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">5</span></a>. 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id="A6.SS1.p1.3.m3.2.2.2.2.3.1.cmml" xref="A6.SS1.p1.3.m3.2.2.2.2.3">subscript</csymbol><ci id="A6.SS1.p1.3.m3.2.2.2.2.3.2.cmml" xref="A6.SS1.p1.3.m3.2.2.2.2.3.2">ℱ</ci><ci id="A6.SS1.p1.3.m3.2.2.2.2.3.3.cmml" xref="A6.SS1.p1.3.m3.2.2.2.2.3.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p1.3.m3.2c">\{g=\langle f_{X},f_{X}\rangle\colon f_{X}\in\mathcal{F}_{X}\}</annotation></semantics></math>.</p>
</div>
<div id="A6.SS1.p2" class="ltx_para ltx_noindent">
<p id="A6.SS1.p2.3" class="ltx_p">Recall that our positive pairs are sampled from <math id="A6.SS1.p2.1.m1.2" class="ltx_Math" alttext="(x,x_{+})\sim{P}_{\mathsf{coor}}" display="inline"><semantics id="A6.SS1.p2.1.m1.2a"><mrow id="A6.SS1.p2.1.m1.2.2" xref="A6.SS1.p2.1.m1.2.2.cmml"><mrow id="A6.SS1.p2.1.m1.2.2.1.1" xref="A6.SS1.p2.1.m1.2.2.1.2.cmml"><mo stretchy="false" id="A6.SS1.p2.1.m1.2.2.1.1.2" xref="A6.SS1.p2.1.m1.2.2.1.2.cmml">(</mo><mi id="A6.SS1.p2.1.m1.1.1" xref="A6.SS1.p2.1.m1.1.1.cmml">x</mi><mo id="A6.SS1.p2.1.m1.2.2.1.1.3" xref="A6.SS1.p2.1.m1.2.2.1.2.cmml">,</mo><msub id="A6.SS1.p2.1.m1.2.2.1.1.1" xref="A6.SS1.p2.1.m1.2.2.1.1.1.cmml"><mi id="A6.SS1.p2.1.m1.2.2.1.1.1.2" xref="A6.SS1.p2.1.m1.2.2.1.1.1.2.cmml">x</mi><mo id="A6.SS1.p2.1.m1.2.2.1.1.1.3" xref="A6.SS1.p2.1.m1.2.2.1.1.1.3.cmml">+</mo></msub><mo stretchy="false" id="A6.SS1.p2.1.m1.2.2.1.1.4" xref="A6.SS1.p2.1.m1.2.2.1.2.cmml">)</mo></mrow><mo id="A6.SS1.p2.1.m1.2.2.2" xref="A6.SS1.p2.1.m1.2.2.2.cmml">∼</mo><msub id="A6.SS1.p2.1.m1.2.2.3" xref="A6.SS1.p2.1.m1.2.2.3.cmml"><mi id="A6.SS1.p2.1.m1.2.2.3.2" xref="A6.SS1.p2.1.m1.2.2.3.2.cmml">P</mi><mi id="A6.SS1.p2.1.m1.2.2.3.3" xref="A6.SS1.p2.1.m1.2.2.3.3.cmml">𝖼𝗈𝗈𝗋</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.p2.1.m1.2b"><apply id="A6.SS1.p2.1.m1.2.2.cmml" xref="A6.SS1.p2.1.m1.2.2"><csymbol cd="latexml" id="A6.SS1.p2.1.m1.2.2.2.cmml" xref="A6.SS1.p2.1.m1.2.2.2">similar-to</csymbol><interval closure="open" id="A6.SS1.p2.1.m1.2.2.1.2.cmml" xref="A6.SS1.p2.1.m1.2.2.1.1"><ci id="A6.SS1.p2.1.m1.1.1.cmml" xref="A6.SS1.p2.1.m1.1.1">𝑥</ci><apply id="A6.SS1.p2.1.m1.2.2.1.1.1.cmml" xref="A6.SS1.p2.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="A6.SS1.p2.1.m1.2.2.1.1.1.1.cmml" xref="A6.SS1.p2.1.m1.2.2.1.1.1">subscript</csymbol><ci id="A6.SS1.p2.1.m1.2.2.1.1.1.2.cmml" xref="A6.SS1.p2.1.m1.2.2.1.1.1.2">𝑥</ci><plus id="A6.SS1.p2.1.m1.2.2.1.1.1.3.cmml" xref="A6.SS1.p2.1.m1.2.2.1.1.1.3"></plus></apply></interval><apply id="A6.SS1.p2.1.m1.2.2.3.cmml" xref="A6.SS1.p2.1.m1.2.2.3"><csymbol cd="ambiguous" id="A6.SS1.p2.1.m1.2.2.3.1.cmml" xref="A6.SS1.p2.1.m1.2.2.3">subscript</csymbol><ci id="A6.SS1.p2.1.m1.2.2.3.2.cmml" xref="A6.SS1.p2.1.m1.2.2.3.2">𝑃</ci><ci id="A6.SS1.p2.1.m1.2.2.3.3.cmml" xref="A6.SS1.p2.1.m1.2.2.3.3">𝖼𝗈𝗈𝗋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p2.1.m1.2c">(x,x_{+})\sim{P}_{\mathsf{coor}}</annotation></semantics></math>, and that the negative pairs are sampled independently from its marginals which we denote as <math id="A6.SS1.p2.2.m2.2" class="ltx_Math" alttext="(x,x_{-})\overset{\text{i.i.d.}}{\sim}P" display="inline"><semantics id="A6.SS1.p2.2.m2.2a"><mrow id="A6.SS1.p2.2.m2.2.2" xref="A6.SS1.p2.2.m2.2.2.cmml"><mrow id="A6.SS1.p2.2.m2.2.2.1.1" xref="A6.SS1.p2.2.m2.2.2.1.2.cmml"><mo stretchy="false" id="A6.SS1.p2.2.m2.2.2.1.1.2" xref="A6.SS1.p2.2.m2.2.2.1.2.cmml">(</mo><mi id="A6.SS1.p2.2.m2.1.1" xref="A6.SS1.p2.2.m2.1.1.cmml">x</mi><mo id="A6.SS1.p2.2.m2.2.2.1.1.3" xref="A6.SS1.p2.2.m2.2.2.1.2.cmml">,</mo><msub id="A6.SS1.p2.2.m2.2.2.1.1.1" xref="A6.SS1.p2.2.m2.2.2.1.1.1.cmml"><mi id="A6.SS1.p2.2.m2.2.2.1.1.1.2" xref="A6.SS1.p2.2.m2.2.2.1.1.1.2.cmml">x</mi><mo id="A6.SS1.p2.2.m2.2.2.1.1.1.3" xref="A6.SS1.p2.2.m2.2.2.1.1.1.3.cmml">−</mo></msub><mo stretchy="false" id="A6.SS1.p2.2.m2.2.2.1.1.4" xref="A6.SS1.p2.2.m2.2.2.1.2.cmml">)</mo></mrow><mo lspace="0em" rspace="0em" id="A6.SS1.p2.2.m2.2.2.2" xref="A6.SS1.p2.2.m2.2.2.2.cmml">​</mo><mover accent="true" id="A6.SS1.p2.2.m2.2.2.3" xref="A6.SS1.p2.2.m2.2.2.3.cmml"><mo id="A6.SS1.p2.2.m2.2.2.3.2" xref="A6.SS1.p2.2.m2.2.2.3.2.cmml">∼</mo><mtext id="A6.SS1.p2.2.m2.2.2.3.1" xref="A6.SS1.p2.2.m2.2.2.3.1a.cmml">i.i.d.</mtext></mover><mo lspace="0em" rspace="0em" id="A6.SS1.p2.2.m2.2.2.2a" xref="A6.SS1.p2.2.m2.2.2.2.cmml">​</mo><mi id="A6.SS1.p2.2.m2.2.2.4" xref="A6.SS1.p2.2.m2.2.2.4.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.p2.2.m2.2b"><apply id="A6.SS1.p2.2.m2.2.2.cmml" xref="A6.SS1.p2.2.m2.2.2"><times id="A6.SS1.p2.2.m2.2.2.2.cmml" xref="A6.SS1.p2.2.m2.2.2.2"></times><interval closure="open" id="A6.SS1.p2.2.m2.2.2.1.2.cmml" xref="A6.SS1.p2.2.m2.2.2.1.1"><ci id="A6.SS1.p2.2.m2.1.1.cmml" xref="A6.SS1.p2.2.m2.1.1">𝑥</ci><apply id="A6.SS1.p2.2.m2.2.2.1.1.1.cmml" xref="A6.SS1.p2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A6.SS1.p2.2.m2.2.2.1.1.1.1.cmml" xref="A6.SS1.p2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="A6.SS1.p2.2.m2.2.2.1.1.1.2.cmml" xref="A6.SS1.p2.2.m2.2.2.1.1.1.2">𝑥</ci><minus id="A6.SS1.p2.2.m2.2.2.1.1.1.3.cmml" xref="A6.SS1.p2.2.m2.2.2.1.1.1.3"></minus></apply></interval><apply id="A6.SS1.p2.2.m2.2.2.3.cmml" xref="A6.SS1.p2.2.m2.2.2.3"><ci id="A6.SS1.p2.2.m2.2.2.3.1a.cmml" xref="A6.SS1.p2.2.m2.2.2.3.1"><mtext id="A6.SS1.p2.2.m2.2.2.3.1.cmml" xref="A6.SS1.p2.2.m2.2.2.3.1">i.i.d.</mtext></ci><csymbol cd="latexml" id="A6.SS1.p2.2.m2.2.2.3.2.cmml" xref="A6.SS1.p2.2.m2.2.2.3.2">similar-to</csymbol></apply><ci id="A6.SS1.p2.2.m2.2.2.4.cmml" xref="A6.SS1.p2.2.m2.2.2.4">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p2.2.m2.2c">(x,x_{-})\overset{\text{i.i.d.}}{\sim}P</annotation></semantics></math> where <math id="A6.SS1.p2.3.m3.3" class="ltx_Math" alttext="P(x)=\sum_{x_{+}}{P}_{\mathsf{coor}}(x,x_{+})" display="inline"><semantics id="A6.SS1.p2.3.m3.3a"><mrow id="A6.SS1.p2.3.m3.3.3" xref="A6.SS1.p2.3.m3.3.3.cmml"><mrow id="A6.SS1.p2.3.m3.3.3.3" xref="A6.SS1.p2.3.m3.3.3.3.cmml"><mi id="A6.SS1.p2.3.m3.3.3.3.2" xref="A6.SS1.p2.3.m3.3.3.3.2.cmml">P</mi><mo lspace="0em" rspace="0em" id="A6.SS1.p2.3.m3.3.3.3.1" xref="A6.SS1.p2.3.m3.3.3.3.1.cmml">​</mo><mrow id="A6.SS1.p2.3.m3.3.3.3.3.2" xref="A6.SS1.p2.3.m3.3.3.3.cmml"><mo stretchy="false" id="A6.SS1.p2.3.m3.3.3.3.3.2.1" xref="A6.SS1.p2.3.m3.3.3.3.cmml">(</mo><mi id="A6.SS1.p2.3.m3.1.1" xref="A6.SS1.p2.3.m3.1.1.cmml">x</mi><mo stretchy="false" 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id="A6.SS1.p2.3.m3.3.3.1.1.1.1.1.3.cmml" xref="A6.SS1.p2.3.m3.3.3.1.1.1.1.1.3"></plus></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p2.3.m3.3c">P(x)=\sum_{x_{+}}{P}_{\mathsf{coor}}(x,x_{+})</annotation></semantics></math>.</p>
<ol id="A6.I1" class="ltx_enumerate">
<li id="A6.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">1.</span> 
<div id="A6.I1.i1.p1" class="ltx_para ltx_noindent">
<p id="A6.I1.i1.p1.1" class="ltx_p"><span id="A6.I1.i1.p1.1.1" class="ltx_text ltx_font_bold">The binary NCE loss <cite class="ltx_cite ltx_citemacro_citep">(Gutmann &amp; Hyvärinen, <a href="#bib.bib40" title="" class="ltx_ref">2010</a>)</cite></span> is defined with a certain prior over sampling positive vs. negative pairs. Let <math id="A6.I1.i1.p1.1.m1.1" class="ltx_Math" alttext="p_{\mathsf{pos}}" display="inline"><semantics id="A6.I1.i1.p1.1.m1.1a"><msub id="A6.I1.i1.p1.1.m1.1.1" xref="A6.I1.i1.p1.1.m1.1.1.cmml"><mi id="A6.I1.i1.p1.1.m1.1.1.2" xref="A6.I1.i1.p1.1.m1.1.1.2.cmml">p</mi><mi id="A6.I1.i1.p1.1.m1.1.1.3" xref="A6.I1.i1.p1.1.m1.1.1.3.cmml">𝗉𝗈𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="A6.I1.i1.p1.1.m1.1b"><apply id="A6.I1.i1.p1.1.m1.1.1.cmml" xref="A6.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.I1.i1.p1.1.m1.1.1.1.cmml" xref="A6.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="A6.I1.i1.p1.1.m1.1.1.2.cmml" xref="A6.I1.i1.p1.1.m1.1.1.2">𝑝</ci><ci id="A6.I1.i1.p1.1.m1.1.1.3.cmml" xref="A6.I1.i1.p1.1.m1.1.1.3">𝗉𝗈𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i1.p1.1.m1.1c">p_{\mathsf{pos}}</annotation></semantics></math> be the probability of sampling a positive pair. Then the loss is given by</p>
<table id="A6.E20" class="ltx_equation ltx_eqn_table">

<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 id="A6.E20.m1.8" class="ltx_Math" alttext="\mathcal{L}_{\mathsf{binary\mbox{-}NCE}}(g)\triangleq p_{\mathsf{pos}}\cdot\mathbb{E}_{(x,x_{+})\sim{P}_{\mathsf{coor}}}\left[-\log\sigma(g(x,x_{+}))\right]+(1-p_{\mathsf{pos}})\cdot\mathbb{E}_{(x,x_{-})\overset{\text{i.i.d.}}{\sim}P}\left[-\log\sigma(-g(x,x_{-}))\right]." display="block"><semantics id="A6.E20.m1.8a"><mrow id="A6.E20.m1.8.8.1" xref="A6.E20.m1.8.8.1.1.cmml"><mrow id="A6.E20.m1.8.8.1.1" xref="A6.E20.m1.8.8.1.1.cmml"><mrow id="A6.E20.m1.8.8.1.1.5" xref="A6.E20.m1.8.8.1.1.5.cmml"><msub id="A6.E20.m1.8.8.1.1.5.2" xref="A6.E20.m1.8.8.1.1.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.E20.m1.8.8.1.1.5.2.2" xref="A6.E20.m1.8.8.1.1.5.2.2.cmml">ℒ</mi><mrow id="A6.E20.m1.8.8.1.1.5.2.3" xref="A6.E20.m1.8.8.1.1.5.2.3.cmml"><mi id="A6.E20.m1.8.8.1.1.5.2.3.2" xref="A6.E20.m1.8.8.1.1.5.2.3.2.cmml">𝖻𝗂𝗇𝖺𝗋𝗒</mi><mo lspace="0em" rspace="0em" id="A6.E20.m1.8.8.1.1.5.2.3.1" 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p_{\mathsf{pos}}\cdot\mathbb{E}_{(x,x_{+})\sim{P}_{\mathsf{coor}}}\left[-\log\sigma(g(x,x_{+}))\right]+(1-p_{\mathsf{pos}})\cdot\mathbb{E}_{(x,x_{-})\overset{\text{i.i.d.}}{\sim}P}\left[-\log\sigma(-g(x,x_{-}))\right].</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.I1.i1.p2" class="ltx_para">
<p id="A6.I1.i1.p2.1" class="ltx_p">The Bayes optimal solution is given by</p>
<table id="A6.EGx18" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A6.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A6.E21.m1.2" class="ltx_Math" alttext="\displaystyle g(x_{a},x_{b})" display="inline"><semantics id="A6.E21.m1.2a"><mrow id="A6.E21.m1.2.2" xref="A6.E21.m1.2.2.cmml"><mi id="A6.E21.m1.2.2.4" xref="A6.E21.m1.2.2.4.cmml">g</mi><mo lspace="0em" rspace="0em" id="A6.E21.m1.2.2.3" xref="A6.E21.m1.2.2.3.cmml">​</mo><mrow id="A6.E21.m1.2.2.2.2" xref="A6.E21.m1.2.2.2.3.cmml"><mo stretchy="false" id="A6.E21.m1.2.2.2.2.3" xref="A6.E21.m1.2.2.2.3.cmml">(</mo><msub id="A6.E21.m1.1.1.1.1.1" xref="A6.E21.m1.1.1.1.1.1.cmml"><mi id="A6.E21.m1.1.1.1.1.1.2" xref="A6.E21.m1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.E21.m1.1.1.1.1.1.3" xref="A6.E21.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A6.E21.m1.2.2.2.2.4" xref="A6.E21.m1.2.2.2.3.cmml">,</mo><msub id="A6.E21.m1.2.2.2.2.2" xref="A6.E21.m1.2.2.2.2.2.cmml"><mi id="A6.E21.m1.2.2.2.2.2.2" xref="A6.E21.m1.2.2.2.2.2.2.cmml">x</mi><mi id="A6.E21.m1.2.2.2.2.2.3" xref="A6.E21.m1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A6.E21.m1.2.2.2.2.5" xref="A6.E21.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.E21.m1.2b"><apply id="A6.E21.m1.2.2.cmml" xref="A6.E21.m1.2.2"><times id="A6.E21.m1.2.2.3.cmml" xref="A6.E21.m1.2.2.3"></times><ci id="A6.E21.m1.2.2.4.cmml" xref="A6.E21.m1.2.2.4">𝑔</ci><interval closure="open" id="A6.E21.m1.2.2.2.3.cmml" xref="A6.E21.m1.2.2.2.2"><apply id="A6.E21.m1.1.1.1.1.1.cmml" xref="A6.E21.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E21.m1.1.1.1.1.1.1.cmml" xref="A6.E21.m1.1.1.1.1.1">subscript</csymbol><ci id="A6.E21.m1.1.1.1.1.1.2.cmml" xref="A6.E21.m1.1.1.1.1.1.2">𝑥</ci><ci id="A6.E21.m1.1.1.1.1.1.3.cmml" xref="A6.E21.m1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E21.m1.2.2.2.2.2.cmml" xref="A6.E21.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.E21.m1.2.2.2.2.2.1.cmml" xref="A6.E21.m1.2.2.2.2.2">subscript</csymbol><ci id="A6.E21.m1.2.2.2.2.2.2.cmml" xref="A6.E21.m1.2.2.2.2.2.2">𝑥</ci><ci id="A6.E21.m1.2.2.2.2.2.3.cmml" xref="A6.E21.m1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E21.m1.2c">\displaystyle g(x_{a},x_{b})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A6.E21.m2.2" class="ltx_Math" alttext="\displaystyle=\log\frac{P(\texttt{pos}\mathrel{|}x_{a},x_{b})}{1-P(\texttt{pos}\mathrel{|}x_{a},x_{b})}" display="inline"><semantics id="A6.E21.m2.2a"><mrow id="A6.E21.m2.2.3" xref="A6.E21.m2.2.3.cmml"><mi id="A6.E21.m2.2.3.2" xref="A6.E21.m2.2.3.2.cmml"></mi><mo id="A6.E21.m2.2.3.1" xref="A6.E21.m2.2.3.1.cmml">=</mo><mrow id="A6.E21.m2.2.3.3" xref="A6.E21.m2.2.3.3.cmml"><mi id="A6.E21.m2.2.3.3.1" xref="A6.E21.m2.2.3.3.1.cmml">log</mi><mo lspace="0.167em" id="A6.E21.m2.2.3.3a" xref="A6.E21.m2.2.3.3.cmml">⁡</mo><mstyle displaystyle="true" id="A6.E21.m2.2.2" xref="A6.E21.m2.2.2.cmml"><mfrac id="A6.E21.m2.2.2a" xref="A6.E21.m2.2.2.cmml"><mrow id="A6.E21.m2.1.1.1" xref="A6.E21.m2.1.1.1.cmml"><mi id="A6.E21.m2.1.1.1.3" xref="A6.E21.m2.1.1.1.3.cmml">P</mi><mo lspace="0em" rspace="0em" id="A6.E21.m2.1.1.1.2" xref="A6.E21.m2.1.1.1.2.cmml">​</mo><mrow id="A6.E21.m2.1.1.1.1.1" xref="A6.E21.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="A6.E21.m2.1.1.1.1.1.2" xref="A6.E21.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="A6.E21.m2.1.1.1.1.1.1" xref="A6.E21.m2.1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A6.E21.m2.1.1.1.1.1.1.4" xref="A6.E21.m2.1.1.1.1.1.1.4a.cmml">pos</mtext><mo fence="false" id="A6.E21.m2.1.1.1.1.1.1.3" xref="A6.E21.m2.1.1.1.1.1.1.3.cmml">|</mo><mrow id="A6.E21.m2.1.1.1.1.1.1.2.2" xref="A6.E21.m2.1.1.1.1.1.1.2.3.cmml"><msub id="A6.E21.m2.1.1.1.1.1.1.1.1.1" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="A6.E21.m2.1.1.1.1.1.1.1.1.1.2" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.E21.m2.1.1.1.1.1.1.1.1.1.3" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A6.E21.m2.1.1.1.1.1.1.2.2.3" xref="A6.E21.m2.1.1.1.1.1.1.2.3.cmml">,</mo><msub id="A6.E21.m2.1.1.1.1.1.1.2.2.2" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2.cmml"><mi id="A6.E21.m2.1.1.1.1.1.1.2.2.2.2" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2.2.cmml">x</mi><mi id="A6.E21.m2.1.1.1.1.1.1.2.2.2.3" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2.3.cmml">b</mi></msub></mrow></mrow><mo stretchy="false" id="A6.E21.m2.1.1.1.1.1.3" xref="A6.E21.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="A6.E21.m2.2.2.2" xref="A6.E21.m2.2.2.2.cmml"><mn id="A6.E21.m2.2.2.2.3" xref="A6.E21.m2.2.2.2.3.cmml">1</mn><mo id="A6.E21.m2.2.2.2.2" xref="A6.E21.m2.2.2.2.2.cmml">−</mo><mrow id="A6.E21.m2.2.2.2.1" xref="A6.E21.m2.2.2.2.1.cmml"><mi id="A6.E21.m2.2.2.2.1.3" xref="A6.E21.m2.2.2.2.1.3.cmml">P</mi><mo lspace="0em" rspace="0em" id="A6.E21.m2.2.2.2.1.2" xref="A6.E21.m2.2.2.2.1.2.cmml">​</mo><mrow id="A6.E21.m2.2.2.2.1.1.1" xref="A6.E21.m2.2.2.2.1.1.1.1.cmml"><mo stretchy="false" id="A6.E21.m2.2.2.2.1.1.1.2" xref="A6.E21.m2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="A6.E21.m2.2.2.2.1.1.1.1" xref="A6.E21.m2.2.2.2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A6.E21.m2.2.2.2.1.1.1.1.4" xref="A6.E21.m2.2.2.2.1.1.1.1.4a.cmml">pos</mtext><mo fence="false" id="A6.E21.m2.2.2.2.1.1.1.1.3" xref="A6.E21.m2.2.2.2.1.1.1.1.3.cmml">|</mo><mrow id="A6.E21.m2.2.2.2.1.1.1.1.2.2" xref="A6.E21.m2.2.2.2.1.1.1.1.2.3.cmml"><msub id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.2" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.3" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A6.E21.m2.2.2.2.1.1.1.1.2.2.3" xref="A6.E21.m2.2.2.2.1.1.1.1.2.3.cmml">,</mo><msub id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.cmml"><mi id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.2" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.2.cmml">x</mi><mi id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.3" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.3.cmml">b</mi></msub></mrow></mrow><mo stretchy="false" id="A6.E21.m2.2.2.2.1.1.1.3" xref="A6.E21.m2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.E21.m2.2b"><apply id="A6.E21.m2.2.3.cmml" xref="A6.E21.m2.2.3"><eq id="A6.E21.m2.2.3.1.cmml" xref="A6.E21.m2.2.3.1"></eq><csymbol cd="latexml" id="A6.E21.m2.2.3.2.cmml" xref="A6.E21.m2.2.3.2">absent</csymbol><apply id="A6.E21.m2.2.3.3.cmml" xref="A6.E21.m2.2.3.3"><log id="A6.E21.m2.2.3.3.1.cmml" xref="A6.E21.m2.2.3.3.1"></log><apply id="A6.E21.m2.2.2.cmml" xref="A6.E21.m2.2.2"><divide id="A6.E21.m2.2.2.3.cmml" xref="A6.E21.m2.2.2"></divide><apply id="A6.E21.m2.1.1.1.cmml" xref="A6.E21.m2.1.1.1"><times id="A6.E21.m2.1.1.1.2.cmml" xref="A6.E21.m2.1.1.1.2"></times><ci id="A6.E21.m2.1.1.1.3.cmml" xref="A6.E21.m2.1.1.1.3">𝑃</ci><apply id="A6.E21.m2.1.1.1.1.1.1.cmml" xref="A6.E21.m2.1.1.1.1.1"><ci id="A6.E21.m2.1.1.1.1.1.1.3.cmml" xref="A6.E21.m2.1.1.1.1.1.1.3">|</ci><ci id="A6.E21.m2.1.1.1.1.1.1.4a.cmml" xref="A6.E21.m2.1.1.1.1.1.1.4"><mtext class="ltx_mathvariant_monospace" id="A6.E21.m2.1.1.1.1.1.1.4.cmml" xref="A6.E21.m2.1.1.1.1.1.1.4">pos</mtext></ci><list id="A6.E21.m2.1.1.1.1.1.1.2.3.cmml" xref="A6.E21.m2.1.1.1.1.1.1.2.2"><apply id="A6.E21.m2.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E21.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A6.E21.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A6.E21.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="A6.E21.m2.1.1.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E21.m2.1.1.1.1.1.1.2.2.2.cmml" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A6.E21.m2.1.1.1.1.1.1.2.2.2.1.cmml" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A6.E21.m2.1.1.1.1.1.1.2.2.2.2.cmml" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2.2">𝑥</ci><ci id="A6.E21.m2.1.1.1.1.1.1.2.2.2.3.cmml" xref="A6.E21.m2.1.1.1.1.1.1.2.2.2.3">𝑏</ci></apply></list></apply></apply><apply id="A6.E21.m2.2.2.2.cmml" xref="A6.E21.m2.2.2.2"><minus id="A6.E21.m2.2.2.2.2.cmml" xref="A6.E21.m2.2.2.2.2"></minus><cn type="integer" id="A6.E21.m2.2.2.2.3.cmml" xref="A6.E21.m2.2.2.2.3">1</cn><apply id="A6.E21.m2.2.2.2.1.cmml" xref="A6.E21.m2.2.2.2.1"><times id="A6.E21.m2.2.2.2.1.2.cmml" xref="A6.E21.m2.2.2.2.1.2"></times><ci id="A6.E21.m2.2.2.2.1.3.cmml" xref="A6.E21.m2.2.2.2.1.3">𝑃</ci><apply id="A6.E21.m2.2.2.2.1.1.1.1.cmml" xref="A6.E21.m2.2.2.2.1.1.1"><ci id="A6.E21.m2.2.2.2.1.1.1.1.3.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.3">|</ci><ci id="A6.E21.m2.2.2.2.1.1.1.1.4a.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.4"><mtext class="ltx_mathvariant_monospace" id="A6.E21.m2.2.2.2.1.1.1.1.4.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.4">pos</mtext></ci><list id="A6.E21.m2.2.2.2.1.1.1.1.2.3.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2"><apply id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.1.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2">subscript</csymbol><ci id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.2.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.2">𝑥</ci><ci id="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.3.cmml" xref="A6.E21.m2.2.2.2.1.1.1.1.2.2.2.3">𝑏</ci></apply></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E21.m2.2c">\displaystyle=\log\frac{P(\texttt{pos}\mathrel{|}x_{a},x_{b})}{1-P(\texttt{pos}\mathrel{|}x_{a},x_{b})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td>
</tr></tbody>
<tbody id="A6.E22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A6.E22.m1.6" class="ltx_Math" alttext="\displaystyle=\log\frac{P(\texttt{pos},x_{a},x_{b})}{P(\texttt{neg},x_{a},x_{b})}" display="inline"><semantics id="A6.E22.m1.6a"><mrow id="A6.E22.m1.6.7" xref="A6.E22.m1.6.7.cmml"><mi id="A6.E22.m1.6.7.2" xref="A6.E22.m1.6.7.2.cmml"></mi><mo id="A6.E22.m1.6.7.1" xref="A6.E22.m1.6.7.1.cmml">=</mo><mrow id="A6.E22.m1.6.7.3" xref="A6.E22.m1.6.7.3.cmml"><mi id="A6.E22.m1.6.7.3.1" xref="A6.E22.m1.6.7.3.1.cmml">log</mi><mo lspace="0.167em" id="A6.E22.m1.6.7.3a" xref="A6.E22.m1.6.7.3.cmml">⁡</mo><mstyle displaystyle="true" id="A6.E22.m1.6.6" xref="A6.E22.m1.6.6.cmml"><mfrac id="A6.E22.m1.6.6a" xref="A6.E22.m1.6.6.cmml"><mrow id="A6.E22.m1.3.3.3" xref="A6.E22.m1.3.3.3.cmml"><mi id="A6.E22.m1.3.3.3.5" xref="A6.E22.m1.3.3.3.5.cmml">P</mi><mo lspace="0em" rspace="0em" id="A6.E22.m1.3.3.3.4" xref="A6.E22.m1.3.3.3.4.cmml">​</mo><mrow id="A6.E22.m1.3.3.3.3.2" xref="A6.E22.m1.3.3.3.3.3.cmml"><mo stretchy="false" id="A6.E22.m1.3.3.3.3.2.3" xref="A6.E22.m1.3.3.3.3.3.cmml">(</mo><mtext class="ltx_mathvariant_monospace" id="A6.E22.m1.1.1.1.1" xref="A6.E22.m1.1.1.1.1a.cmml">pos</mtext><mo id="A6.E22.m1.3.3.3.3.2.4" xref="A6.E22.m1.3.3.3.3.3.cmml">,</mo><msub id="A6.E22.m1.2.2.2.2.1.1" xref="A6.E22.m1.2.2.2.2.1.1.cmml"><mi id="A6.E22.m1.2.2.2.2.1.1.2" xref="A6.E22.m1.2.2.2.2.1.1.2.cmml">x</mi><mi id="A6.E22.m1.2.2.2.2.1.1.3" xref="A6.E22.m1.2.2.2.2.1.1.3.cmml">a</mi></msub><mo id="A6.E22.m1.3.3.3.3.2.5" xref="A6.E22.m1.3.3.3.3.3.cmml">,</mo><msub id="A6.E22.m1.3.3.3.3.2.2" xref="A6.E22.m1.3.3.3.3.2.2.cmml"><mi id="A6.E22.m1.3.3.3.3.2.2.2" xref="A6.E22.m1.3.3.3.3.2.2.2.cmml">x</mi><mi id="A6.E22.m1.3.3.3.3.2.2.3" xref="A6.E22.m1.3.3.3.3.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A6.E22.m1.3.3.3.3.2.6" xref="A6.E22.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow><mrow id="A6.E22.m1.6.6.6" xref="A6.E22.m1.6.6.6.cmml"><mi id="A6.E22.m1.6.6.6.5" xref="A6.E22.m1.6.6.6.5.cmml">P</mi><mo lspace="0em" rspace="0em" id="A6.E22.m1.6.6.6.4" xref="A6.E22.m1.6.6.6.4.cmml">​</mo><mrow id="A6.E22.m1.6.6.6.3.2" xref="A6.E22.m1.6.6.6.3.3.cmml"><mo stretchy="false" id="A6.E22.m1.6.6.6.3.2.3" xref="A6.E22.m1.6.6.6.3.3.cmml">(</mo><mtext class="ltx_mathvariant_monospace" id="A6.E22.m1.4.4.4.1" xref="A6.E22.m1.4.4.4.1a.cmml">neg</mtext><mo id="A6.E22.m1.6.6.6.3.2.4" xref="A6.E22.m1.6.6.6.3.3.cmml">,</mo><msub id="A6.E22.m1.5.5.5.2.1.1" xref="A6.E22.m1.5.5.5.2.1.1.cmml"><mi id="A6.E22.m1.5.5.5.2.1.1.2" xref="A6.E22.m1.5.5.5.2.1.1.2.cmml">x</mi><mi id="A6.E22.m1.5.5.5.2.1.1.3" xref="A6.E22.m1.5.5.5.2.1.1.3.cmml">a</mi></msub><mo id="A6.E22.m1.6.6.6.3.2.5" xref="A6.E22.m1.6.6.6.3.3.cmml">,</mo><msub id="A6.E22.m1.6.6.6.3.2.2" xref="A6.E22.m1.6.6.6.3.2.2.cmml"><mi id="A6.E22.m1.6.6.6.3.2.2.2" xref="A6.E22.m1.6.6.6.3.2.2.2.cmml">x</mi><mi id="A6.E22.m1.6.6.6.3.2.2.3" xref="A6.E22.m1.6.6.6.3.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A6.E22.m1.6.6.6.3.2.6" xref="A6.E22.m1.6.6.6.3.3.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.E22.m1.6b"><apply id="A6.E22.m1.6.7.cmml" xref="A6.E22.m1.6.7"><eq id="A6.E22.m1.6.7.1.cmml" xref="A6.E22.m1.6.7.1"></eq><csymbol cd="latexml" id="A6.E22.m1.6.7.2.cmml" xref="A6.E22.m1.6.7.2">absent</csymbol><apply id="A6.E22.m1.6.7.3.cmml" xref="A6.E22.m1.6.7.3"><log id="A6.E22.m1.6.7.3.1.cmml" xref="A6.E22.m1.6.7.3.1"></log><apply id="A6.E22.m1.6.6.cmml" xref="A6.E22.m1.6.6"><divide id="A6.E22.m1.6.6.7.cmml" xref="A6.E22.m1.6.6"></divide><apply id="A6.E22.m1.3.3.3.cmml" xref="A6.E22.m1.3.3.3"><times id="A6.E22.m1.3.3.3.4.cmml" xref="A6.E22.m1.3.3.3.4"></times><ci id="A6.E22.m1.3.3.3.5.cmml" xref="A6.E22.m1.3.3.3.5">𝑃</ci><vector id="A6.E22.m1.3.3.3.3.3.cmml" xref="A6.E22.m1.3.3.3.3.2"><ci id="A6.E22.m1.1.1.1.1a.cmml" xref="A6.E22.m1.1.1.1.1"><mtext class="ltx_mathvariant_monospace" id="A6.E22.m1.1.1.1.1.cmml" xref="A6.E22.m1.1.1.1.1">pos</mtext></ci><apply id="A6.E22.m1.2.2.2.2.1.1.cmml" xref="A6.E22.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="A6.E22.m1.2.2.2.2.1.1.1.cmml" xref="A6.E22.m1.2.2.2.2.1.1">subscript</csymbol><ci id="A6.E22.m1.2.2.2.2.1.1.2.cmml" xref="A6.E22.m1.2.2.2.2.1.1.2">𝑥</ci><ci id="A6.E22.m1.2.2.2.2.1.1.3.cmml" xref="A6.E22.m1.2.2.2.2.1.1.3">𝑎</ci></apply><apply id="A6.E22.m1.3.3.3.3.2.2.cmml" xref="A6.E22.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="A6.E22.m1.3.3.3.3.2.2.1.cmml" xref="A6.E22.m1.3.3.3.3.2.2">subscript</csymbol><ci id="A6.E22.m1.3.3.3.3.2.2.2.cmml" xref="A6.E22.m1.3.3.3.3.2.2.2">𝑥</ci><ci id="A6.E22.m1.3.3.3.3.2.2.3.cmml" xref="A6.E22.m1.3.3.3.3.2.2.3">𝑏</ci></apply></vector></apply><apply id="A6.E22.m1.6.6.6.cmml" xref="A6.E22.m1.6.6.6"><times id="A6.E22.m1.6.6.6.4.cmml" xref="A6.E22.m1.6.6.6.4"></times><ci id="A6.E22.m1.6.6.6.5.cmml" xref="A6.E22.m1.6.6.6.5">𝑃</ci><vector id="A6.E22.m1.6.6.6.3.3.cmml" xref="A6.E22.m1.6.6.6.3.2"><ci id="A6.E22.m1.4.4.4.1a.cmml" xref="A6.E22.m1.4.4.4.1"><mtext class="ltx_mathvariant_monospace" id="A6.E22.m1.4.4.4.1.cmml" xref="A6.E22.m1.4.4.4.1">neg</mtext></ci><apply id="A6.E22.m1.5.5.5.2.1.1.cmml" xref="A6.E22.m1.5.5.5.2.1.1"><csymbol cd="ambiguous" id="A6.E22.m1.5.5.5.2.1.1.1.cmml" xref="A6.E22.m1.5.5.5.2.1.1">subscript</csymbol><ci id="A6.E22.m1.5.5.5.2.1.1.2.cmml" xref="A6.E22.m1.5.5.5.2.1.1.2">𝑥</ci><ci id="A6.E22.m1.5.5.5.2.1.1.3.cmml" xref="A6.E22.m1.5.5.5.2.1.1.3">𝑎</ci></apply><apply id="A6.E22.m1.6.6.6.3.2.2.cmml" xref="A6.E22.m1.6.6.6.3.2.2"><csymbol cd="ambiguous" id="A6.E22.m1.6.6.6.3.2.2.1.cmml" xref="A6.E22.m1.6.6.6.3.2.2">subscript</csymbol><ci id="A6.E22.m1.6.6.6.3.2.2.2.cmml" xref="A6.E22.m1.6.6.6.3.2.2.2">𝑥</ci><ci id="A6.E22.m1.6.6.6.3.2.2.3.cmml" xref="A6.E22.m1.6.6.6.3.2.2.3">𝑏</ci></apply></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E22.m1.6c">\displaystyle=\log\frac{P(\texttt{pos},x_{a},x_{b})}{P(\texttt{neg},x_{a},x_{b})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td>
</tr></tbody>
<tbody id="A6.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="A6.E23.m1.3.3.3.1.1.1.1"></minus><cn type="integer" id="A6.E23.m1.3.3.3.1.1.1.2.cmml" xref="A6.E23.m1.3.3.3.1.1.1.2">1</cn><apply id="A6.E23.m1.3.3.3.1.1.1.3.cmml" xref="A6.E23.m1.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="A6.E23.m1.3.3.3.1.1.1.3.1.cmml" xref="A6.E23.m1.3.3.3.1.1.1.3">subscript</csymbol><ci id="A6.E23.m1.3.3.3.1.1.1.3.2.cmml" xref="A6.E23.m1.3.3.3.1.1.1.3.2">𝑝</ci><ci id="A6.E23.m1.3.3.3.1.1.1.3.3.cmml" xref="A6.E23.m1.3.3.3.1.1.1.3.3">𝗉𝗈𝗌</ci></apply></apply><ci id="A6.E23.m1.5.5.5.5.cmml" xref="A6.E23.m1.5.5.5.5">𝑃</ci><apply id="A6.E23.m1.4.4.4.2.1.1.cmml" xref="A6.E23.m1.4.4.4.2.1"><csymbol cd="ambiguous" id="A6.E23.m1.4.4.4.2.1.1.1.cmml" xref="A6.E23.m1.4.4.4.2.1">subscript</csymbol><ci id="A6.E23.m1.4.4.4.2.1.1.2.cmml" xref="A6.E23.m1.4.4.4.2.1.1.2">𝑥</ci><ci id="A6.E23.m1.4.4.4.2.1.1.3.cmml" xref="A6.E23.m1.4.4.4.2.1.1.3">𝑎</ci></apply><ci id="A6.E23.m1.5.5.5.6.cmml" xref="A6.E23.m1.5.5.5.6">𝑃</ci><apply id="A6.E23.m1.5.5.5.3.1.1.cmml" xref="A6.E23.m1.5.5.5.3.1"><csymbol cd="ambiguous" id="A6.E23.m1.5.5.5.3.1.1.1.cmml" xref="A6.E23.m1.5.5.5.3.1">subscript</csymbol><ci id="A6.E23.m1.5.5.5.3.1.1.2.cmml" xref="A6.E23.m1.5.5.5.3.1.1.2">𝑥</ci><ci id="A6.E23.m1.5.5.5.3.1.1.3.cmml" xref="A6.E23.m1.5.5.5.3.1.1.3">𝑏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E23.m1.5c">\displaystyle=\log\frac{p_{\mathsf{pos}}\cdot{P}_{\mathsf{coor}}(x_{a},x_{b})}{(1-p_{\mathsf{pos}})P(x_{a})P(x_{b})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td>
</tr></tbody>
<tbody id="A6.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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id="A6.E24.m1.1.1.1.1.1.1.3.cmml" xref="A6.E24.m1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E24.m1.2.2.2.2.2.2.cmml" xref="A6.E24.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.E24.m1.2.2.2.2.2.2.1.cmml" xref="A6.E24.m1.2.2.2.2.2.2">subscript</csymbol><ci id="A6.E24.m1.2.2.2.2.2.2.2.cmml" xref="A6.E24.m1.2.2.2.2.2.2.2">𝑥</ci><ci id="A6.E24.m1.2.2.2.2.2.2.3.cmml" xref="A6.E24.m1.2.2.2.2.2.2.3">𝑏</ci></apply></interval></apply><apply id="A6.E24.m1.4.4.4.cmml" xref="A6.E24.m1.4.4.4"><times id="A6.E24.m1.4.4.4.3.cmml" xref="A6.E24.m1.4.4.4.3"></times><ci id="A6.E24.m1.4.4.4.4.cmml" xref="A6.E24.m1.4.4.4.4">𝑃</ci><apply id="A6.E24.m1.3.3.3.1.1.1.cmml" xref="A6.E24.m1.3.3.3.1.1"><csymbol cd="ambiguous" id="A6.E24.m1.3.3.3.1.1.1.1.cmml" xref="A6.E24.m1.3.3.3.1.1">subscript</csymbol><ci id="A6.E24.m1.3.3.3.1.1.1.2.cmml" xref="A6.E24.m1.3.3.3.1.1.1.2">𝑥</ci><ci id="A6.E24.m1.3.3.3.1.1.1.3.cmml" xref="A6.E24.m1.3.3.3.1.1.1.3">𝑎</ci></apply><ci id="A6.E24.m1.4.4.4.5.cmml" xref="A6.E24.m1.4.4.4.5">𝑃</ci><apply id="A6.E24.m1.4.4.4.2.1.1.cmml" xref="A6.E24.m1.4.4.4.2.1"><csymbol cd="ambiguous" id="A6.E24.m1.4.4.4.2.1.1.1.cmml" xref="A6.E24.m1.4.4.4.2.1">subscript</csymbol><ci id="A6.E24.m1.4.4.4.2.1.1.2.cmml" xref="A6.E24.m1.4.4.4.2.1.1.2">𝑥</ci><ci id="A6.E24.m1.4.4.4.2.1.1.3.cmml" xref="A6.E24.m1.4.4.4.2.1.1.3">𝑏</ci></apply></apply></apply></apply><apply id="A6.E24.m1.4.5.3.3.cmml" xref="A6.E24.m1.4.5.3.3"><log id="A6.E24.m1.4.5.3.3.1.cmml" xref="A6.E24.m1.4.5.3.3.1"></log><apply id="A6.E24.m1.4.5.3.3.2.cmml" xref="A6.E24.m1.4.5.3.3.2"><divide id="A6.E24.m1.4.5.3.3.2.1.cmml" xref="A6.E24.m1.4.5.3.3.2"></divide><apply id="A6.E24.m1.4.5.3.3.2.2.cmml" xref="A6.E24.m1.4.5.3.3.2.2"><csymbol cd="ambiguous" id="A6.E24.m1.4.5.3.3.2.2.1.cmml" xref="A6.E24.m1.4.5.3.3.2.2">subscript</csymbol><ci id="A6.E24.m1.4.5.3.3.2.2.2.cmml" xref="A6.E24.m1.4.5.3.3.2.2.2">𝑝</ci><ci id="A6.E24.m1.4.5.3.3.2.2.3.cmml" xref="A6.E24.m1.4.5.3.3.2.2.3">𝗉𝗈𝗌</ci></apply><apply id="A6.E24.m1.4.5.3.3.2.3.cmml" xref="A6.E24.m1.4.5.3.3.2.3"><minus id="A6.E24.m1.4.5.3.3.2.3.1.cmml" xref="A6.E24.m1.4.5.3.3.2.3.1"></minus><cn type="integer" id="A6.E24.m1.4.5.3.3.2.3.2.cmml" xref="A6.E24.m1.4.5.3.3.2.3.2">1</cn><apply id="A6.E24.m1.4.5.3.3.2.3.3.cmml" xref="A6.E24.m1.4.5.3.3.2.3.3"><csymbol cd="ambiguous" id="A6.E24.m1.4.5.3.3.2.3.3.1.cmml" xref="A6.E24.m1.4.5.3.3.2.3.3">subscript</csymbol><ci id="A6.E24.m1.4.5.3.3.2.3.3.2.cmml" xref="A6.E24.m1.4.5.3.3.2.3.3.2">𝑝</ci><ci id="A6.E24.m1.4.5.3.3.2.3.3.3.cmml" xref="A6.E24.m1.4.5.3.3.2.3.3.3">𝗉𝗈𝗌</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E24.m1.4c">\displaystyle=\log\frac{{P}_{\mathsf{coor}}(x_{a},x_{b})}{P(x_{a})P(x_{b})}+\log\frac{p_{\mathsf{pos}}}{1-p_{\mathsf{pos}}}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td>
</tr></tbody>
<tbody id="A6.E25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A6.E25.m1.1" class="ltx_Math" alttext="\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X}." display="inline"><semantics id="A6.E25.m1.1a"><mrow id="A6.E25.m1.1.1.1" xref="A6.E25.m1.1.1.1.1.cmml"><mrow id="A6.E25.m1.1.1.1.1" xref="A6.E25.m1.1.1.1.1.cmml"><mi id="A6.E25.m1.1.1.1.1.4" xref="A6.E25.m1.1.1.1.1.4.cmml"></mi><mo id="A6.E25.m1.1.1.1.1.3" xref="A6.E25.m1.1.1.1.1.3.cmml">=</mo><mrow id="A6.E25.m1.1.1.1.1.2" xref="A6.E25.m1.1.1.1.1.2.cmml"><mrow id="A6.E25.m1.1.1.1.1.2.2" xref="A6.E25.m1.1.1.1.1.2.2.cmml"><msub id="A6.E25.m1.1.1.1.1.2.2.4" xref="A6.E25.m1.1.1.1.1.2.2.4.cmml"><mi id="A6.E25.m1.1.1.1.1.2.2.4.2" xref="A6.E25.m1.1.1.1.1.2.2.4.2.cmml">K</mi><mi id="A6.E25.m1.1.1.1.1.2.2.4.3" xref="A6.E25.m1.1.1.1.1.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E25.m1.1.1.1.1.2.2.3" xref="A6.E25.m1.1.1.1.1.2.2.3.cmml">​</mo><mrow id="A6.E25.m1.1.1.1.1.2.2.2.2" xref="A6.E25.m1.1.1.1.1.2.2.2.3.cmml"><mo stretchy="false" id="A6.E25.m1.1.1.1.1.2.2.2.2.3" xref="A6.E25.m1.1.1.1.1.2.2.2.3.cmml">(</mo><msub id="A6.E25.m1.1.1.1.1.1.1.1.1.1" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A6.E25.m1.1.1.1.1.1.1.1.1.1.2" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.E25.m1.1.1.1.1.1.1.1.1.1.3" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A6.E25.m1.1.1.1.1.2.2.2.2.4" xref="A6.E25.m1.1.1.1.1.2.2.2.3.cmml">,</mo><msub id="A6.E25.m1.1.1.1.1.2.2.2.2.2" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2.cmml"><mi id="A6.E25.m1.1.1.1.1.2.2.2.2.2.2" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2.2.cmml">x</mi><mi id="A6.E25.m1.1.1.1.1.2.2.2.2.2.3" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A6.E25.m1.1.1.1.1.2.2.2.2.5" xref="A6.E25.m1.1.1.1.1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="A6.E25.m1.1.1.1.1.2.3" xref="A6.E25.m1.1.1.1.1.2.3.cmml">+</mo><msub id="A6.E25.m1.1.1.1.1.2.4" xref="A6.E25.m1.1.1.1.1.2.4.cmml"><mi id="A6.E25.m1.1.1.1.1.2.4.2" xref="A6.E25.m1.1.1.1.1.2.4.2.cmml">c</mi><mi id="A6.E25.m1.1.1.1.1.2.4.3" xref="A6.E25.m1.1.1.1.1.2.4.3.cmml">X</mi></msub></mrow></mrow><mo lspace="0em" id="A6.E25.m1.1.1.1.2" xref="A6.E25.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.E25.m1.1b"><apply id="A6.E25.m1.1.1.1.1.cmml" xref="A6.E25.m1.1.1.1"><eq id="A6.E25.m1.1.1.1.1.3.cmml" xref="A6.E25.m1.1.1.1.1.3"></eq><csymbol cd="latexml" id="A6.E25.m1.1.1.1.1.4.cmml" xref="A6.E25.m1.1.1.1.1.4">absent</csymbol><apply id="A6.E25.m1.1.1.1.1.2.cmml" xref="A6.E25.m1.1.1.1.1.2"><plus id="A6.E25.m1.1.1.1.1.2.3.cmml" xref="A6.E25.m1.1.1.1.1.2.3"></plus><apply id="A6.E25.m1.1.1.1.1.2.2.cmml" xref="A6.E25.m1.1.1.1.1.2.2"><times id="A6.E25.m1.1.1.1.1.2.2.3.cmml" xref="A6.E25.m1.1.1.1.1.2.2.3"></times><apply id="A6.E25.m1.1.1.1.1.2.2.4.cmml" xref="A6.E25.m1.1.1.1.1.2.2.4"><csymbol cd="ambiguous" id="A6.E25.m1.1.1.1.1.2.2.4.1.cmml" xref="A6.E25.m1.1.1.1.1.2.2.4">subscript</csymbol><ci id="A6.E25.m1.1.1.1.1.2.2.4.2.cmml" xref="A6.E25.m1.1.1.1.1.2.2.4.2">𝐾</ci><ci id="A6.E25.m1.1.1.1.1.2.2.4.3.cmml" xref="A6.E25.m1.1.1.1.1.2.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="A6.E25.m1.1.1.1.1.2.2.2.3.cmml" xref="A6.E25.m1.1.1.1.1.2.2.2.2"><apply id="A6.E25.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E25.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A6.E25.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A6.E25.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A6.E25.m1.1.1.1.1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E25.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.E25.m1.1.1.1.1.2.2.2.2.2.1.cmml" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2">subscript</csymbol><ci id="A6.E25.m1.1.1.1.1.2.2.2.2.2.2.cmml" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2.2">𝑥</ci><ci id="A6.E25.m1.1.1.1.1.2.2.2.2.2.3.cmml" xref="A6.E25.m1.1.1.1.1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply><apply id="A6.E25.m1.1.1.1.1.2.4.cmml" xref="A6.E25.m1.1.1.1.1.2.4"><csymbol cd="ambiguous" id="A6.E25.m1.1.1.1.1.2.4.1.cmml" xref="A6.E25.m1.1.1.1.1.2.4">subscript</csymbol><ci id="A6.E25.m1.1.1.1.1.2.4.2.cmml" xref="A6.E25.m1.1.1.1.1.2.4.2">𝑐</ci><ci id="A6.E25.m1.1.1.1.1.2.4.3.cmml" xref="A6.E25.m1.1.1.1.1.2.4.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E25.m1.1c">\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td>
</tr></tbody>
</table>
</div>
</li>
<li id="A6.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">2.</span> 
<div id="A6.I1.i2.p1" class="ltx_para ltx_noindent">
<p id="A6.I1.i2.p1.2" class="ltx_p"><span id="A6.I1.i2.p1.2.1" class="ltx_text ltx_font_bold">The InfoNCE loss <cite class="ltx_cite ltx_citemacro_citep">(Oord et al., <a href="#bib.bib91" title="" class="ltx_ref">2018</a>)</cite></span> is defined with randomly sampling one positive pair along with <math id="A6.I1.i2.p1.1.m1.1" class="ltx_Math" alttext="K" display="inline"><semantics id="A6.I1.i2.p1.1.m1.1a"><mi id="A6.I1.i2.p1.1.m1.1.1" xref="A6.I1.i2.p1.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p1.1.m1.1b"><ci id="A6.I1.i2.p1.1.m1.1.1.cmml" xref="A6.I1.i2.p1.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p1.1.m1.1c">K</annotation></semantics></math> negative ones. With some hyperparameter <math id="A6.I1.i2.p1.2.m2.1" class="ltx_Math" alttext="\tau&gt;0" display="inline"><semantics id="A6.I1.i2.p1.2.m2.1a"><mrow id="A6.I1.i2.p1.2.m2.1.1" xref="A6.I1.i2.p1.2.m2.1.1.cmml"><mi id="A6.I1.i2.p1.2.m2.1.1.2" xref="A6.I1.i2.p1.2.m2.1.1.2.cmml">τ</mi><mo id="A6.I1.i2.p1.2.m2.1.1.1" xref="A6.I1.i2.p1.2.m2.1.1.1.cmml">&gt;</mo><mn id="A6.I1.i2.p1.2.m2.1.1.3" xref="A6.I1.i2.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p1.2.m2.1b"><apply id="A6.I1.i2.p1.2.m2.1.1.cmml" xref="A6.I1.i2.p1.2.m2.1.1"><gt id="A6.I1.i2.p1.2.m2.1.1.1.cmml" xref="A6.I1.i2.p1.2.m2.1.1.1"></gt><ci id="A6.I1.i2.p1.2.m2.1.1.2.cmml" xref="A6.I1.i2.p1.2.m2.1.1.2">𝜏</ci><cn type="integer" id="A6.I1.i2.p1.2.m2.1.1.3.cmml" xref="A6.I1.i2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p1.2.m2.1c">\tau&gt;0</annotation></semantics></math>, the loss is given by</p>
<table id="A6.E26" class="ltx_equation ltx_eqn_table">

<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 id="A6.E26.m1.10" class="ltx_Math" alttext="\mathcal{L}_{\mathsf{InfoNCE}}(g)\triangleq\mathbb{E}_{\begin{subarray}{c}(x,x_{+})\sim{P}_{\mathsf{coor}}\\
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xref="A6.E26.m1.6.6.5.3.1.1.1.1">𝑖</ci></apply></interval></apply><ci id="A6.E26.m1.8.8.7.5.3.5.cmml" xref="A6.E26.m1.8.8.7.5.3.5">𝜏</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E26.m1.10c">\mathcal{L}_{\mathsf{InfoNCE}}(g)\triangleq\mathbb{E}_{\begin{subarray}{c}(x,x_{+})\sim{P}_{\mathsf{coor}}\\
(x_{-}^{(1)},x_{-}^{(2)},\dots,x_{-}^{(K)})\overset{\text{i.i.d.}}{\sim}P\end{subarray}}\left[-\log\frac{e^{g(x,x_{+})/\tau}}{e^{g(x,x_{+})/\tau}+\sum_{i=1}^{K}e^{g(x,x_{-}^{(i)})/\tau}}\right].</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.I1.i2.p2" class="ltx_para ltx_noindent">
<p id="A6.I1.i2.p2.1" class="ltx_p">The Bayes optimal solution is given by</p>
<table id="A6.EGx19" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A6.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A6.E27.m1.7.7.7.5.3.3.1.1.1"><csymbol cd="ambiguous" id="A6.E27.m1.7.7.7.5.3.3.1.1.1.1.cmml" xref="A6.E27.m1.7.7.7.5.3.3.1.1.1">superscript</csymbol><apply id="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.cmml" xref="A6.E27.m1.7.7.7.5.3.3.1.1.1"><csymbol cd="ambiguous" id="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.1.cmml" xref="A6.E27.m1.7.7.7.5.3.3.1.1.1">subscript</csymbol><ci id="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.2.cmml" xref="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.2">𝑥</ci><minus id="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.3.cmml" xref="A6.E27.m1.7.7.7.5.3.3.1.1.1.2.3"></minus></apply><ci id="A6.E27.m1.5.5.5.3.1.1.1.1.cmml" xref="A6.E27.m1.5.5.5.3.1.1.1.1">𝑖</ci></apply></interval></apply><ci id="A6.E27.m1.7.7.7.5.3.5.cmml" xref="A6.E27.m1.7.7.7.5.3.5">𝜏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E27.m1.7c">\displaystyle\frac{e^{g(x,x_{+})/\tau}}{e^{g(x,x_{+})/\tau}+\sum_{i=1}^{K}e^{g(x,x_{-}^{(i)})/\tau}}</annotation></semantics></math></td>
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td>
</tr></tbody>
<tbody id="A6.E28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="A6.E28.m1.7.7.7.5.1.1.1.1.1.1.3">𝑥</ci></apply></apply><ci id="A6.E28.m1.7.7.7.5.1.1.3.cmml" xref="A6.E28.m1.7.7.7.5.1.1.3">𝑃</ci></apply><apply id="A6.E28.m1.8.8.8.6.2.2.1.1.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1"><csymbol cd="ambiguous" id="A6.E28.m1.8.8.8.6.2.2.1.1.1.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1">superscript</csymbol><apply id="A6.E28.m1.8.8.8.6.2.2.1.1.2.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1"><csymbol cd="ambiguous" id="A6.E28.m1.8.8.8.6.2.2.1.1.2.1.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1">subscript</csymbol><ci id="A6.E28.m1.8.8.8.6.2.2.1.1.2.2.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1.1.2.2">𝑥</ci><minus id="A6.E28.m1.8.8.8.6.2.2.1.1.2.3.cmml" xref="A6.E28.m1.8.8.8.6.2.2.1.1.2.3"></minus></apply><ci id="A6.E28.m1.4.4.4.2.1.1.cmml" xref="A6.E28.m1.4.4.4.2.1.1">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E28.m1.9c">\displaystyle=\frac{{P}_{\mathsf{coor}}(x_{+}\mathrel{|}x)/P(x_{+})}{{P}_{\mathsf{coor}}(x_{+}\mathrel{|}x)/P(x_{+})+\sum_{i}{P}_{\mathsf{coor}}(x_{-}^{(i)}\mathrel{|}x)/P(x_{-}^{(i)})}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.I1.i2.p3" class="ltx_para ltx_noindent">
<p id="A6.I1.i2.p3.2" class="ltx_p">For <math id="A6.I1.i2.p3.1.m1.1" class="ltx_Math" alttext="\tau=1" display="inline"><semantics id="A6.I1.i2.p3.1.m1.1a"><mrow id="A6.I1.i2.p3.1.m1.1.1" xref="A6.I1.i2.p3.1.m1.1.1.cmml"><mi id="A6.I1.i2.p3.1.m1.1.1.2" xref="A6.I1.i2.p3.1.m1.1.1.2.cmml">τ</mi><mo id="A6.I1.i2.p3.1.m1.1.1.1" xref="A6.I1.i2.p3.1.m1.1.1.1.cmml">=</mo><mn id="A6.I1.i2.p3.1.m1.1.1.3" xref="A6.I1.i2.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p3.1.m1.1b"><apply id="A6.I1.i2.p3.1.m1.1.1.cmml" xref="A6.I1.i2.p3.1.m1.1.1"><eq id="A6.I1.i2.p3.1.m1.1.1.1.cmml" xref="A6.I1.i2.p3.1.m1.1.1.1"></eq><ci id="A6.I1.i2.p3.1.m1.1.1.2.cmml" xref="A6.I1.i2.p3.1.m1.1.1.2">𝜏</ci><cn type="integer" id="A6.I1.i2.p3.1.m1.1.1.3.cmml" xref="A6.I1.i2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p3.1.m1.1c">\tau=1</annotation></semantics></math>, this optima corresponds to <math id="A6.I1.i2.p3.2.m2.1" class="ltx_Math" alttext="g" display="inline"><semantics id="A6.I1.i2.p3.2.m2.1a"><mi id="A6.I1.i2.p3.2.m2.1.1" xref="A6.I1.i2.p3.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p3.2.m2.1b"><ci id="A6.I1.i2.p3.2.m2.1.1.cmml" xref="A6.I1.i2.p3.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p3.2.m2.1c">g</annotation></semantics></math> choices where</p>
<table id="A6.EGx20" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A6.E29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A6.E29.m1.2" class="ltx_Math" alttext="\displaystyle g(x_{a},x_{b})" display="inline"><semantics id="A6.E29.m1.2a"><mrow id="A6.E29.m1.2.2" xref="A6.E29.m1.2.2.cmml"><mi id="A6.E29.m1.2.2.4" xref="A6.E29.m1.2.2.4.cmml">g</mi><mo lspace="0em" rspace="0em" id="A6.E29.m1.2.2.3" xref="A6.E29.m1.2.2.3.cmml">​</mo><mrow id="A6.E29.m1.2.2.2.2" xref="A6.E29.m1.2.2.2.3.cmml"><mo stretchy="false" id="A6.E29.m1.2.2.2.2.3" xref="A6.E29.m1.2.2.2.3.cmml">(</mo><msub id="A6.E29.m1.1.1.1.1.1" xref="A6.E29.m1.1.1.1.1.1.cmml"><mi id="A6.E29.m1.1.1.1.1.1.2" xref="A6.E29.m1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.E29.m1.1.1.1.1.1.3" xref="A6.E29.m1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="A6.E29.m1.2.2.2.2.4" xref="A6.E29.m1.2.2.2.3.cmml">,</mo><msub id="A6.E29.m1.2.2.2.2.2" xref="A6.E29.m1.2.2.2.2.2.cmml"><mi id="A6.E29.m1.2.2.2.2.2.2" xref="A6.E29.m1.2.2.2.2.2.2.cmml">x</mi><mi id="A6.E29.m1.2.2.2.2.2.3" xref="A6.E29.m1.2.2.2.2.2.3.cmml">b</mi></msub><mo stretchy="false" id="A6.E29.m1.2.2.2.2.5" xref="A6.E29.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.E29.m1.2b"><apply id="A6.E29.m1.2.2.cmml" xref="A6.E29.m1.2.2"><times id="A6.E29.m1.2.2.3.cmml" xref="A6.E29.m1.2.2.3"></times><ci id="A6.E29.m1.2.2.4.cmml" xref="A6.E29.m1.2.2.4">𝑔</ci><interval closure="open" id="A6.E29.m1.2.2.2.3.cmml" xref="A6.E29.m1.2.2.2.2"><apply id="A6.E29.m1.1.1.1.1.1.cmml" xref="A6.E29.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E29.m1.1.1.1.1.1.1.cmml" xref="A6.E29.m1.1.1.1.1.1">subscript</csymbol><ci id="A6.E29.m1.1.1.1.1.1.2.cmml" xref="A6.E29.m1.1.1.1.1.1.2">𝑥</ci><ci id="A6.E29.m1.1.1.1.1.1.3.cmml" xref="A6.E29.m1.1.1.1.1.1.3">𝑎</ci></apply><apply id="A6.E29.m1.2.2.2.2.2.cmml" xref="A6.E29.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.E29.m1.2.2.2.2.2.1.cmml" xref="A6.E29.m1.2.2.2.2.2">subscript</csymbol><ci id="A6.E29.m1.2.2.2.2.2.2.cmml" xref="A6.E29.m1.2.2.2.2.2.2">𝑥</ci><ci id="A6.E29.m1.2.2.2.2.2.3.cmml" xref="A6.E29.m1.2.2.2.2.2.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E29.m1.2c">\displaystyle g(x_{a},x_{b})</annotation></semantics></math></td>
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xref="A6.E29.m2.2.2.2.1.1">subscript</csymbol><ci id="A6.E29.m2.2.2.2.1.1.1.2.cmml" xref="A6.E29.m2.2.2.2.1.1.1.2">𝑥</ci><ci id="A6.E29.m2.2.2.2.1.1.1.3.cmml" xref="A6.E29.m2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></apply><apply id="A6.E29.m2.3.3.1.1.cmml" xref="A6.E29.m2.3.3.1.1"><times id="A6.E29.m2.3.3.1.1.2.cmml" xref="A6.E29.m2.3.3.1.1.2"></times><apply id="A6.E29.m2.3.3.1.1.3.cmml" xref="A6.E29.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="A6.E29.m2.3.3.1.1.3.1.cmml" xref="A6.E29.m2.3.3.1.1.3">subscript</csymbol><ci id="A6.E29.m2.3.3.1.1.3.2.cmml" xref="A6.E29.m2.3.3.1.1.3.2">𝑐</ci><ci id="A6.E29.m2.3.3.1.1.3.3.cmml" xref="A6.E29.m2.3.3.1.1.3.3">𝑋</ci></apply><apply id="A6.E29.m2.3.3.1.1.1.1.1.cmml" xref="A6.E29.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A6.E29.m2.3.3.1.1.1.1.1.1.cmml" xref="A6.E29.m2.3.3.1.1.1.1">subscript</csymbol><ci id="A6.E29.m2.3.3.1.1.1.1.1.2.cmml" xref="A6.E29.m2.3.3.1.1.1.1.1.2">𝑥</ci><ci id="A6.E29.m2.3.3.1.1.1.1.1.3.cmml" xref="A6.E29.m2.3.3.1.1.1.1.1.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E29.m2.3c">\displaystyle=\log\frac{{P}_{\mathsf{coor}}(x_{b}\mathrel{|}x_{a})}{P(x_{b})}+c_{X}(x_{a})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td>
</tr></tbody>
<tbody id="A6.E30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A6.E30.m1.1" class="ltx_Math" alttext="\displaystyle=K_{\mathsf{PMI}}(x_{a},x_{b})+c_{X}(x_{a})." display="inline"><semantics id="A6.E30.m1.1a"><mrow id="A6.E30.m1.1.1.1" xref="A6.E30.m1.1.1.1.1.cmml"><mrow id="A6.E30.m1.1.1.1.1" xref="A6.E30.m1.1.1.1.1.cmml"><mi id="A6.E30.m1.1.1.1.1.5" xref="A6.E30.m1.1.1.1.1.5.cmml"></mi><mo id="A6.E30.m1.1.1.1.1.4" xref="A6.E30.m1.1.1.1.1.4.cmml">=</mo><mrow id="A6.E30.m1.1.1.1.1.3" xref="A6.E30.m1.1.1.1.1.3.cmml"><mrow id="A6.E30.m1.1.1.1.1.2.2" xref="A6.E30.m1.1.1.1.1.2.2.cmml"><msub id="A6.E30.m1.1.1.1.1.2.2.4" xref="A6.E30.m1.1.1.1.1.2.2.4.cmml"><mi id="A6.E30.m1.1.1.1.1.2.2.4.2" xref="A6.E30.m1.1.1.1.1.2.2.4.2.cmml">K</mi><mi id="A6.E30.m1.1.1.1.1.2.2.4.3" xref="A6.E30.m1.1.1.1.1.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E30.m1.1.1.1.1.2.2.3" xref="A6.E30.m1.1.1.1.1.2.2.3.cmml">​</mo><mrow id="A6.E30.m1.1.1.1.1.2.2.2.2" xref="A6.E30.m1.1.1.1.1.2.2.2.3.cmml"><mo 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.I1.i2.p4" class="ltx_para ltx_noindent">
<p id="A6.I1.i2.p4.6" class="ltx_p">For the general <math id="A6.I1.i2.p4.1.m1.1" class="ltx_Math" alttext="\tau\neq 1" display="inline"><semantics id="A6.I1.i2.p4.1.m1.1a"><mrow id="A6.I1.i2.p4.1.m1.1.1" xref="A6.I1.i2.p4.1.m1.1.1.cmml"><mi id="A6.I1.i2.p4.1.m1.1.1.2" xref="A6.I1.i2.p4.1.m1.1.1.2.cmml">τ</mi><mo id="A6.I1.i2.p4.1.m1.1.1.1" xref="A6.I1.i2.p4.1.m1.1.1.1.cmml">≠</mo><mn id="A6.I1.i2.p4.1.m1.1.1.3" xref="A6.I1.i2.p4.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.1.m1.1b"><apply id="A6.I1.i2.p4.1.m1.1.1.cmml" xref="A6.I1.i2.p4.1.m1.1.1"><neq id="A6.I1.i2.p4.1.m1.1.1.1.cmml" xref="A6.I1.i2.p4.1.m1.1.1.1"></neq><ci id="A6.I1.i2.p4.1.m1.1.1.2.cmml" xref="A6.I1.i2.p4.1.m1.1.1.2">𝜏</ci><cn type="integer" id="A6.I1.i2.p4.1.m1.1.1.3.cmml" xref="A6.I1.i2.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.1.m1.1c">\tau\neq 1</annotation></semantics></math> case, we have <math id="A6.I1.i2.p4.2.m2.1" class="ltx_Math" alttext="g" display="inline"><semantics id="A6.I1.i2.p4.2.m2.1a"><mi id="A6.I1.i2.p4.2.m2.1.1" xref="A6.I1.i2.p4.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.2.m2.1b"><ci id="A6.I1.i2.p4.2.m2.1.1.cmml" xref="A6.I1.i2.p4.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.2.m2.1c">g</annotation></semantics></math> (and corresponding <math id="A6.I1.i2.p4.3.m3.1" class="ltx_Math" alttext="f_{X}" display="inline"><semantics id="A6.I1.i2.p4.3.m3.1a"><msub id="A6.I1.i2.p4.3.m3.1.1" xref="A6.I1.i2.p4.3.m3.1.1.cmml"><mi id="A6.I1.i2.p4.3.m3.1.1.2" xref="A6.I1.i2.p4.3.m3.1.1.2.cmml">f</mi><mi id="A6.I1.i2.p4.3.m3.1.1.3" xref="A6.I1.i2.p4.3.m3.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.3.m3.1b"><apply id="A6.I1.i2.p4.3.m3.1.1.cmml" xref="A6.I1.i2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="A6.I1.i2.p4.3.m3.1.1.1.cmml" xref="A6.I1.i2.p4.3.m3.1.1">subscript</csymbol><ci id="A6.I1.i2.p4.3.m3.1.1.2.cmml" xref="A6.I1.i2.p4.3.m3.1.1.2">𝑓</ci><ci id="A6.I1.i2.p4.3.m3.1.1.3.cmml" xref="A6.I1.i2.p4.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.3.m3.1c">f_{X}</annotation></semantics></math>) recovers <math id="A6.I1.i2.p4.4.m4.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A6.I1.i2.p4.4.m4.1a"><msub id="A6.I1.i2.p4.4.m4.1.1" xref="A6.I1.i2.p4.4.m4.1.1.cmml"><mi id="A6.I1.i2.p4.4.m4.1.1.2" xref="A6.I1.i2.p4.4.m4.1.1.2.cmml">K</mi><mi id="A6.I1.i2.p4.4.m4.1.1.3" xref="A6.I1.i2.p4.4.m4.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.4.m4.1b"><apply id="A6.I1.i2.p4.4.m4.1.1.cmml" xref="A6.I1.i2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="A6.I1.i2.p4.4.m4.1.1.1.cmml" xref="A6.I1.i2.p4.4.m4.1.1">subscript</csymbol><ci id="A6.I1.i2.p4.4.m4.1.1.2.cmml" xref="A6.I1.i2.p4.4.m4.1.1.2">𝐾</ci><ci id="A6.I1.i2.p4.4.m4.1.1.3.cmml" xref="A6.I1.i2.p4.4.m4.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.4.m4.1c">K_{\mathsf{PMI}}</annotation></semantics></math> up to an offset and a scale. Our main argument in <a href="#S4" title="4 What representation are we converging to? ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a> that <math id="A6.I1.i2.p4.5.m5.1" class="ltx_Math" alttext="f_{X}" display="inline"><semantics id="A6.I1.i2.p4.5.m5.1a"><msub id="A6.I1.i2.p4.5.m5.1.1" xref="A6.I1.i2.p4.5.m5.1.1.cmml"><mi id="A6.I1.i2.p4.5.m5.1.1.2" xref="A6.I1.i2.p4.5.m5.1.1.2.cmml">f</mi><mi id="A6.I1.i2.p4.5.m5.1.1.3" xref="A6.I1.i2.p4.5.m5.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.5.m5.1b"><apply id="A6.I1.i2.p4.5.m5.1.1.cmml" xref="A6.I1.i2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="A6.I1.i2.p4.5.m5.1.1.1.cmml" xref="A6.I1.i2.p4.5.m5.1.1">subscript</csymbol><ci id="A6.I1.i2.p4.5.m5.1.1.2.cmml" xref="A6.I1.i2.p4.5.m5.1.1.2">𝑓</ci><ci id="A6.I1.i2.p4.5.m5.1.1.3.cmml" xref="A6.I1.i2.p4.5.m5.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.5.m5.1c">f_{X}</annotation></semantics></math> recovers <math id="A6.I1.i2.p4.6.m6.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A6.I1.i2.p4.6.m6.1a"><msub id="A6.I1.i2.p4.6.m6.1.1" xref="A6.I1.i2.p4.6.m6.1.1.cmml"><mi id="A6.I1.i2.p4.6.m6.1.1.2" xref="A6.I1.i2.p4.6.m6.1.1.2.cmml">K</mi><mi id="A6.I1.i2.p4.6.m6.1.1.3" xref="A6.I1.i2.p4.6.m6.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A6.I1.i2.p4.6.m6.1b"><apply id="A6.I1.i2.p4.6.m6.1.1.cmml" xref="A6.I1.i2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="A6.I1.i2.p4.6.m6.1.1.1.cmml" xref="A6.I1.i2.p4.6.m6.1.1">subscript</csymbol><ci id="A6.I1.i2.p4.6.m6.1.1.2.cmml" xref="A6.I1.i2.p4.6.m6.1.1.2">𝐾</ci><ci id="A6.I1.i2.p4.6.m6.1.1.3.cmml" xref="A6.I1.i2.p4.6.m6.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.I1.i2.p4.6.m6.1c">K_{\mathsf{PMI}}</annotation></semantics></math> still holds.</p>
</div>
</li>
</ol>
</div>
</section>
<section id="A6.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">F.2 </span>Contrastive learners can represent <math id="A6.SS2.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A6.SS2.1.m1.1b"><msub id="A6.SS2.1.m1.1.1" xref="A6.SS2.1.m1.1.1.cmml"><mi id="A6.SS2.1.m1.1.1.2" xref="A6.SS2.1.m1.1.1.2.cmml">K</mi><mi id="A6.SS2.1.m1.1.1.3" xref="A6.SS2.1.m1.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A6.SS2.1.m1.1c"><apply id="A6.SS2.1.m1.1.1.cmml" xref="A6.SS2.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS2.1.m1.1.1.1.cmml" xref="A6.SS2.1.m1.1.1">subscript</csymbol><ci id="A6.SS2.1.m1.1.1.2.cmml" xref="A6.SS2.1.m1.1.1.2">𝐾</ci><ci id="A6.SS2.1.m1.1.1.3.cmml" xref="A6.SS2.1.m1.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.1.m1.1d">K_{\mathsf{PMI}}</annotation></semantics></math> exactly under smoothness conditions</h3>

<div id="A6.SS2.p1" class="ltx_para ltx_noindent">
<p id="A6.SS2.p1.6" class="ltx_p">We want to express <math id="A6.SS2.p1.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}+C" display="inline"><semantics id="A6.SS2.p1.1.m1.1a"><mrow id="A6.SS2.p1.1.m1.1.1" xref="A6.SS2.p1.1.m1.1.1.cmml"><msub id="A6.SS2.p1.1.m1.1.1.2" xref="A6.SS2.p1.1.m1.1.1.2.cmml"><mi id="A6.SS2.p1.1.m1.1.1.2.2" xref="A6.SS2.p1.1.m1.1.1.2.2.cmml">K</mi><mi id="A6.SS2.p1.1.m1.1.1.2.3" xref="A6.SS2.p1.1.m1.1.1.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo id="A6.SS2.p1.1.m1.1.1.1" xref="A6.SS2.p1.1.m1.1.1.1.cmml">+</mo><mi id="A6.SS2.p1.1.m1.1.1.3" xref="A6.SS2.p1.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS2.p1.1.m1.1b"><apply id="A6.SS2.p1.1.m1.1.1.cmml" xref="A6.SS2.p1.1.m1.1.1"><plus id="A6.SS2.p1.1.m1.1.1.1.cmml" xref="A6.SS2.p1.1.m1.1.1.1"></plus><apply id="A6.SS2.p1.1.m1.1.1.2.cmml" xref="A6.SS2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A6.SS2.p1.1.m1.1.1.2.1.cmml" xref="A6.SS2.p1.1.m1.1.1.2">subscript</csymbol><ci id="A6.SS2.p1.1.m1.1.1.2.2.cmml" xref="A6.SS2.p1.1.m1.1.1.2.2">𝐾</ci><ci id="A6.SS2.p1.1.m1.1.1.2.3.cmml" xref="A6.SS2.p1.1.m1.1.1.2.3">𝖯𝖬𝖨</ci></apply><ci id="A6.SS2.p1.1.m1.1.1.3.cmml" xref="A6.SS2.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.1.m1.1c">K_{\mathsf{PMI}}+C</annotation></semantics></math> using some representation function <math id="A6.SS2.p1.2.m2.1" class="ltx_Math" alttext="f_{X}\colon\mathcal{X}\rightarrow\mathbb{R}^{n}" display="inline"><semantics id="A6.SS2.p1.2.m2.1a"><mrow id="A6.SS2.p1.2.m2.1.1" xref="A6.SS2.p1.2.m2.1.1.cmml"><msub id="A6.SS2.p1.2.m2.1.1.2" xref="A6.SS2.p1.2.m2.1.1.2.cmml"><mi id="A6.SS2.p1.2.m2.1.1.2.2" xref="A6.SS2.p1.2.m2.1.1.2.2.cmml">f</mi><mi id="A6.SS2.p1.2.m2.1.1.2.3" xref="A6.SS2.p1.2.m2.1.1.2.3.cmml">X</mi></msub><mo lspace="0.278em" rspace="0.278em" id="A6.SS2.p1.2.m2.1.1.1" xref="A6.SS2.p1.2.m2.1.1.1.cmml">:</mo><mrow id="A6.SS2.p1.2.m2.1.1.3" xref="A6.SS2.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.SS2.p1.2.m2.1.1.3.2" xref="A6.SS2.p1.2.m2.1.1.3.2.cmml">𝒳</mi><mo stretchy="false" id="A6.SS2.p1.2.m2.1.1.3.1" xref="A6.SS2.p1.2.m2.1.1.3.1.cmml">→</mo><msup id="A6.SS2.p1.2.m2.1.1.3.3" xref="A6.SS2.p1.2.m2.1.1.3.3.cmml"><mi mathvariant="normal" id="A6.SS2.p1.2.m2.1.1.3.3.2" xref="A6.SS2.p1.2.m2.1.1.3.3.2.cmml">ℝ</mi><mi id="A6.SS2.p1.2.m2.1.1.3.3.3" xref="A6.SS2.p1.2.m2.1.1.3.3.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS2.p1.2.m2.1b"><apply id="A6.SS2.p1.2.m2.1.1.cmml" xref="A6.SS2.p1.2.m2.1.1"><ci id="A6.SS2.p1.2.m2.1.1.1.cmml" xref="A6.SS2.p1.2.m2.1.1.1">:</ci><apply id="A6.SS2.p1.2.m2.1.1.2.cmml" xref="A6.SS2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.SS2.p1.2.m2.1.1.2.1.cmml" xref="A6.SS2.p1.2.m2.1.1.2">subscript</csymbol><ci id="A6.SS2.p1.2.m2.1.1.2.2.cmml" xref="A6.SS2.p1.2.m2.1.1.2.2">𝑓</ci><ci id="A6.SS2.p1.2.m2.1.1.2.3.cmml" xref="A6.SS2.p1.2.m2.1.1.2.3">𝑋</ci></apply><apply id="A6.SS2.p1.2.m2.1.1.3.cmml" xref="A6.SS2.p1.2.m2.1.1.3"><ci id="A6.SS2.p1.2.m2.1.1.3.1.cmml" xref="A6.SS2.p1.2.m2.1.1.3.1">→</ci><ci id="A6.SS2.p1.2.m2.1.1.3.2.cmml" xref="A6.SS2.p1.2.m2.1.1.3.2">𝒳</ci><apply id="A6.SS2.p1.2.m2.1.1.3.3.cmml" xref="A6.SS2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="A6.SS2.p1.2.m2.1.1.3.3.1.cmml" xref="A6.SS2.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="A6.SS2.p1.2.m2.1.1.3.3.2.cmml" xref="A6.SS2.p1.2.m2.1.1.3.3.2">ℝ</ci><ci id="A6.SS2.p1.2.m2.1.1.3.3.3.cmml" xref="A6.SS2.p1.2.m2.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.2.m2.1c">f_{X}\colon\mathcal{X}\rightarrow\mathbb{R}^{n}</annotation></semantics></math> so that <math id="A6.SS2.p1.3.m3.2" class="ltx_Math" alttext="&lt;f_{X}(x_{a}),f_{X}(x_{b})&gt;=K_{\mathsf{PMI}}(x_{a},x_{b})+C" display="inline"><semantics id="A6.SS2.p1.3.m3.2a"><mrow id="A6.SS2.p1.3.m3.2.2.2" xref="A6.SS2.p1.3.m3.2.2.3.cmml"><mrow id="A6.SS2.p1.3.m3.1.1.1.1" xref="A6.SS2.p1.3.m3.1.1.1.1.cmml"><mi id="A6.SS2.p1.3.m3.1.1.1.1.3" xref="A6.SS2.p1.3.m3.1.1.1.1.3.cmml"></mi><mo id="A6.SS2.p1.3.m3.1.1.1.1.2" xref="A6.SS2.p1.3.m3.1.1.1.1.2.cmml">&lt;</mo><mrow id="A6.SS2.p1.3.m3.1.1.1.1.1" xref="A6.SS2.p1.3.m3.1.1.1.1.1.cmml"><msub id="A6.SS2.p1.3.m3.1.1.1.1.1.3" xref="A6.SS2.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="A6.SS2.p1.3.m3.1.1.1.1.1.3.2" xref="A6.SS2.p1.3.m3.1.1.1.1.1.3.2.cmml">f</mi><mi id="A6.SS2.p1.3.m3.1.1.1.1.1.3.3" xref="A6.SS2.p1.3.m3.1.1.1.1.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="A6.SS2.p1.3.m3.1.1.1.1.1.2" xref="A6.SS2.p1.3.m3.1.1.1.1.1.2.cmml">​</mo><mrow id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.2" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.2" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.3" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.3.cmml">a</mi></msub><mo stretchy="false" id="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.3" xref="A6.SS2.p1.3.m3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A6.SS2.p1.3.m3.2.2.2.3" xref="A6.SS2.p1.3.m3.2.2.3a.cmml">,</mo><mrow id="A6.SS2.p1.3.m3.2.2.2.2" xref="A6.SS2.p1.3.m3.2.2.2.2.cmml"><mrow id="A6.SS2.p1.3.m3.2.2.2.2.1" xref="A6.SS2.p1.3.m3.2.2.2.2.1.cmml"><msub id="A6.SS2.p1.3.m3.2.2.2.2.1.3" xref="A6.SS2.p1.3.m3.2.2.2.2.1.3.cmml"><mi id="A6.SS2.p1.3.m3.2.2.2.2.1.3.2" xref="A6.SS2.p1.3.m3.2.2.2.2.1.3.2.cmml">f</mi><mi id="A6.SS2.p1.3.m3.2.2.2.2.1.3.3" xref="A6.SS2.p1.3.m3.2.2.2.2.1.3.3.cmml">X</mi></msub><mo lspace="0em" rspace="0em" id="A6.SS2.p1.3.m3.2.2.2.2.1.2" xref="A6.SS2.p1.3.m3.2.2.2.2.1.2.cmml">​</mo><mrow id="A6.SS2.p1.3.m3.2.2.2.2.1.1.1" xref="A6.SS2.p1.3.m3.2.2.2.2.1.1.1.1.cmml"><mo stretchy="false" id="A6.SS2.p1.3.m3.2.2.2.2.1.1.1.2" 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xref="A6.SS2.p1.3.m3.2.2.2.2.2.1.1.1.1.2">𝑥</ci><ci id="A6.SS2.p1.3.m3.2.2.2.2.2.1.1.1.1.3.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.2.1.1.1.1.3">𝑎</ci></apply><apply id="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2"><csymbol cd="ambiguous" id="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.1.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2">subscript</csymbol><ci id="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.2.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.2">𝑥</ci><ci id="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.3.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.3.2.2.2.2.3">𝑏</ci></apply></interval></apply><ci id="A6.SS2.p1.3.m3.2.2.2.2.3.4.cmml" xref="A6.SS2.p1.3.m3.2.2.2.2.3.4">𝐶</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.3.m3.2c">&lt;f_{X}(x_{a}),f_{X}(x_{b})&gt;=K_{\mathsf{PMI}}(x_{a},x_{b})+C</annotation></semantics></math>, for some <math id="A6.SS2.p1.4.m4.1" class="ltx_Math" alttext="C" display="inline"><semantics id="A6.SS2.p1.4.m4.1a"><mi id="A6.SS2.p1.4.m4.1.1" xref="A6.SS2.p1.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A6.SS2.p1.4.m4.1b"><ci id="A6.SS2.p1.4.m4.1.1.cmml" xref="A6.SS2.p1.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.4.m4.1c">C</annotation></semantics></math>. For such <math id="A6.SS2.p1.5.m5.1" class="ltx_Math" alttext="f" display="inline"><semantics id="A6.SS2.p1.5.m5.1a"><mi id="A6.SS2.p1.5.m5.1.1" xref="A6.SS2.p1.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A6.SS2.p1.5.m5.1b"><ci id="A6.SS2.p1.5.m5.1.1.cmml" xref="A6.SS2.p1.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.5.m5.1c">f</annotation></semantics></math> to exist, an equivalent criterion is that <math id="A6.SS2.p1.6.m6.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}+C" display="inline"><semantics id="A6.SS2.p1.6.m6.1a"><mrow id="A6.SS2.p1.6.m6.1.1" xref="A6.SS2.p1.6.m6.1.1.cmml"><msub id="A6.SS2.p1.6.m6.1.1.2" xref="A6.SS2.p1.6.m6.1.1.2.cmml"><mi id="A6.SS2.p1.6.m6.1.1.2.2" xref="A6.SS2.p1.6.m6.1.1.2.2.cmml">K</mi><mi id="A6.SS2.p1.6.m6.1.1.2.3" xref="A6.SS2.p1.6.m6.1.1.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo id="A6.SS2.p1.6.m6.1.1.1" xref="A6.SS2.p1.6.m6.1.1.1.cmml">+</mo><mi id="A6.SS2.p1.6.m6.1.1.3" xref="A6.SS2.p1.6.m6.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS2.p1.6.m6.1b"><apply id="A6.SS2.p1.6.m6.1.1.cmml" xref="A6.SS2.p1.6.m6.1.1"><plus id="A6.SS2.p1.6.m6.1.1.1.cmml" xref="A6.SS2.p1.6.m6.1.1.1"></plus><apply id="A6.SS2.p1.6.m6.1.1.2.cmml" xref="A6.SS2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="A6.SS2.p1.6.m6.1.1.2.1.cmml" xref="A6.SS2.p1.6.m6.1.1.2">subscript</csymbol><ci id="A6.SS2.p1.6.m6.1.1.2.2.cmml" xref="A6.SS2.p1.6.m6.1.1.2.2">𝐾</ci><ci id="A6.SS2.p1.6.m6.1.1.2.3.cmml" xref="A6.SS2.p1.6.m6.1.1.2.3">𝖯𝖬𝖨</ci></apply><ci id="A6.SS2.p1.6.m6.1.1.3.cmml" xref="A6.SS2.p1.6.m6.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p1.6.m6.1c">K_{\mathsf{PMI}}+C</annotation></semantics></math> is positive semi-definite (PSD), as can be seen from eigendecomposition.</p>
</div>
<div id="A6.Thmtheorem1" class="ltx_theorem ltx_theorem_proposition">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="A6.Thmtheorem1.1.1.1" class="ltx_text ltx_font_bold">Proposition F.1</span></span><span id="A6.Thmtheorem1.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="A6.Thmtheorem1.p1" class="ltx_para">
<p id="A6.Thmtheorem1.p1.4" class="ltx_p"><span id="A6.Thmtheorem1.p1.4.4" class="ltx_text ltx_font_italic">Suppose that the off-diagonal elements of <math id="A6.Thmtheorem1.p1.1.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A6.Thmtheorem1.p1.1.1.m1.1a"><msub id="A6.Thmtheorem1.p1.1.1.m1.1.1" xref="A6.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="A6.Thmtheorem1.p1.1.1.m1.1.1.2" xref="A6.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">K</mi><mi id="A6.Thmtheorem1.p1.1.1.m1.1.1.3" xref="A6.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem1.p1.1.1.m1.1b"><apply id="A6.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="A6.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="A6.Thmtheorem1.p1.1.1.m1.1.1">subscript</csymbol><ci id="A6.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="A6.Thmtheorem1.p1.1.1.m1.1.1.2">𝐾</ci><ci id="A6.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="A6.Thmtheorem1.p1.1.1.m1.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem1.p1.1.1.m1.1c">K_{\mathsf{PMI}}</annotation></semantics></math> are bounded within <math id="A6.Thmtheorem1.p1.2.2.m2.4" class="ltx_Math" alttext="[\log\rho_{\mathsf{min}},\log\rho_{\mathsf{min}}+\delta]\in(-\infty,0]" display="inline"><semantics id="A6.Thmtheorem1.p1.2.2.m2.4a"><mrow id="A6.Thmtheorem1.p1.2.2.m2.4.4" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.cmml"><mrow id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.3.cmml"><mo stretchy="false" id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.3" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.3.cmml">[</mo><mrow id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.cmml"><mi id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.1" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.1.cmml">log</mi><mo lspace="0.167em" id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1a" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.cmml">⁡</mo><msub id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.cmml"><mi id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.2.cmml">ρ</mi><mi id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.3" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.3.cmml">𝗆𝗂𝗇</mi></msub></mrow><mo id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.4" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.3.cmml">,</mo><mrow id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.cmml"><mrow id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.cmml"><mi id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.1" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.1.cmml">log</mi><mo lspace="0.167em" id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2a" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.cmml">⁡</mo><msub id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.cmml"><mi id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.2" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.2.cmml">ρ</mi><mi id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.3" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.3.cmml">𝗆𝗂𝗇</mi></msub></mrow><mo id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.1" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.1.cmml">+</mo><mi id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.3" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.3.cmml">δ</mi></mrow><mo stretchy="false" id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.5" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.3.cmml">]</mo></mrow><mo id="A6.Thmtheorem1.p1.2.2.m2.4.4.4" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.4.cmml">∈</mo><mrow id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.2.cmml"><mo stretchy="false" id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.2" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.2.cmml">(</mo><mrow id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.cmml"><mo id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1a" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.cmml">−</mo><mi mathvariant="normal" id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.2" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.2.cmml">∞</mi></mrow><mo id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.3" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.2.cmml">,</mo><mn id="A6.Thmtheorem1.p1.2.2.m2.1.1" xref="A6.Thmtheorem1.p1.2.2.m2.1.1.cmml">0</mn><mo stretchy="false" id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.4" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem1.p1.2.2.m2.4b"><apply id="A6.Thmtheorem1.p1.2.2.m2.4.4.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4"><in id="A6.Thmtheorem1.p1.2.2.m2.4.4.4.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.4"></in><interval closure="closed" id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.3.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2"><apply id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1"><log id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.1"></log><apply id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2">subscript</csymbol><ci id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.2">𝜌</ci><ci id="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.3.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.2.2.1.1.1.2.3">𝗆𝗂𝗇</ci></apply></apply><apply id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2"><plus id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.1"></plus><apply id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2"><log id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.1"></log><apply id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2">subscript</csymbol><ci id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.2">𝜌</ci><ci id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.3.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.2.2.3">𝗆𝗂𝗇</ci></apply></apply><ci id="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.3.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.3.3.2.2.2.3">𝛿</ci></apply></interval><interval closure="open-closed" id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1"><apply id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1"><minus id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1"></minus><infinity id="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.2.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.4.4.3.1.1.2"></infinity></apply><cn type="integer" id="A6.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="A6.Thmtheorem1.p1.2.2.m2.1.1">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem1.p1.2.2.m2.4c">[\log\rho_{\mathsf{min}},\log\rho_{\mathsf{min}}+\delta]\in(-\infty,0]</annotation></semantics></math>. We have <math id="A6.Thmtheorem1.p1.3.3.m3.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}+C" display="inline"><semantics id="A6.Thmtheorem1.p1.3.3.m3.1a"><mrow id="A6.Thmtheorem1.p1.3.3.m3.1.1" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.cmml"><msub id="A6.Thmtheorem1.p1.3.3.m3.1.1.2" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2.cmml"><mi id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.2" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2.2.cmml">K</mi><mi id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.3" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo id="A6.Thmtheorem1.p1.3.3.m3.1.1.1" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.1.cmml">+</mo><mi id="A6.Thmtheorem1.p1.3.3.m3.1.1.3" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem1.p1.3.3.m3.1b"><apply id="A6.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1"><plus id="A6.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.1"></plus><apply id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.1.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.2.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2.2">𝐾</ci><ci id="A6.Thmtheorem1.p1.3.3.m3.1.1.2.3.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.2.3">𝖯𝖬𝖨</ci></apply><ci id="A6.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="A6.Thmtheorem1.p1.3.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem1.p1.3.3.m3.1c">K_{\mathsf{PMI}}+C</annotation></semantics></math> is positive semi-definite (PSD) for some <math id="A6.Thmtheorem1.p1.4.4.m4.1" class="ltx_Math" alttext="C" display="inline"><semantics id="A6.Thmtheorem1.p1.4.4.m4.1a"><mi id="A6.Thmtheorem1.p1.4.4.m4.1.1" xref="A6.Thmtheorem1.p1.4.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem1.p1.4.4.m4.1b"><ci id="A6.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="A6.Thmtheorem1.p1.4.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem1.p1.4.4.m4.1c">C</annotation></semantics></math> if the joint distribution is sufficiently smooth:</span></p>
<table id="A6.E31" class="ltx_equation ltx_eqn_table">

<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 id="A6.E31.m1.2" class="ltx_math_unparsed" alttext="\frac{{P}_{\mathsf{coor}}(z_{i}\mathrel{|}z_{i})}{{P}_{\mathsf{coor}}(z_{i})}\geq e^{N\delta}\rho_{\mathsf{min}}\mathrlap{,\qquad\forall i.}" display="block"><semantics id="A6.E31.m1.2a"><mrow id="A6.E31.m1.2.3"><mfrac id="A6.E31.m1.2.2"><mrow id="A6.E31.m1.1.1.1"><msub id="A6.E31.m1.1.1.1.3"><mi id="A6.E31.m1.1.1.1.3.2">P</mi><mi id="A6.E31.m1.1.1.1.3.3">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="A6.E31.m1.1.1.1.2">​</mo><mrow id="A6.E31.m1.1.1.1.1.1"><mo stretchy="false" id="A6.E31.m1.1.1.1.1.1.2">(</mo><mrow id="A6.E31.m1.1.1.1.1.1.1"><msub id="A6.E31.m1.1.1.1.1.1.1.2"><mi id="A6.E31.m1.1.1.1.1.1.1.2.2">z</mi><mi id="A6.E31.m1.1.1.1.1.1.1.2.3">i</mi></msub><mo fence="false" id="A6.E31.m1.1.1.1.1.1.1.1">|</mo><msub id="A6.E31.m1.1.1.1.1.1.1.3"><mi id="A6.E31.m1.1.1.1.1.1.1.3.2">z</mi><mi id="A6.E31.m1.1.1.1.1.1.1.3.3">i</mi></msub></mrow><mo stretchy="false" id="A6.E31.m1.1.1.1.1.1.3">)</mo></mrow></mrow><mrow id="A6.E31.m1.2.2.2"><msub id="A6.E31.m1.2.2.2.3"><mi id="A6.E31.m1.2.2.2.3.2">P</mi><mi id="A6.E31.m1.2.2.2.3.3">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="A6.E31.m1.2.2.2.2">​</mo><mrow id="A6.E31.m1.2.2.2.1.1"><mo stretchy="false" id="A6.E31.m1.2.2.2.1.1.2">(</mo><msub id="A6.E31.m1.2.2.2.1.1.1"><mi id="A6.E31.m1.2.2.2.1.1.1.2">z</mi><mi id="A6.E31.m1.2.2.2.1.1.1.3">i</mi></msub><mo stretchy="false" id="A6.E31.m1.2.2.2.1.1.3">)</mo></mrow></mrow></mfrac><mo id="A6.E31.m1.2.3.1">≥</mo><mrow id="A6.E31.m1.2.3.2"><msup id="A6.E31.m1.2.3.2.2"><mi id="A6.E31.m1.2.3.2.2.2">e</mi><mrow id="A6.E31.m1.2.3.2.2.3"><mi id="A6.E31.m1.2.3.2.2.3.2">N</mi><mo lspace="0em" rspace="0em" id="A6.E31.m1.2.3.2.2.3.1">​</mo><mi id="A6.E31.m1.2.3.2.2.3.3">δ</mi></mrow></msup><mo lspace="0em" rspace="0em" id="A6.E31.m1.2.3.2.1">​</mo><msub id="A6.E31.m1.2.3.2.3"><mi id="A6.E31.m1.2.3.2.3.2">ρ</mi><mi id="A6.E31.m1.2.3.2.3.3">𝗆𝗂𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A6.E31.m1.2.3.2.1a">​</mo><mpadded width="0pt" id="A6.E31.m1.2.3.2.4"><mo rspace="2.167em" id="A6.E31.m1.2.3.2.4.1">,</mo><mo rspace="0.167em" id="A6.E31.m1.2.3.2.4.2">∀</mo><mi id="A6.E31.m1.2.3.2.4.3">i</mi><mo lspace="0em" id="A6.E31.m1.2.3.2.4.4">.</mo></mpadded></mrow></mrow><annotation encoding="application/x-tex" id="A6.E31.m1.2b">\frac{{P}_{\mathsf{coor}}(z_{i}\mathrel{|}z_{i})}{{P}_{\mathsf{coor}}(z_{i})}\geq e^{N\delta}\rho_{\mathsf{min}}\mathrlap{,\qquad\forall i.}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td>
</tr></tbody>
</table>
</div>
</div>
<div id="A6.SS2.5" class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6>
<div id="A6.SS2.2.p1" class="ltx_para ltx_noindent">
<p id="A6.SS2.2.p1.1" class="ltx_p">Note that <math id="A6.SS2.2.p1.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}+C" display="inline"><semantics id="A6.SS2.2.p1.1.m1.1a"><mrow id="A6.SS2.2.p1.1.m1.1.1" xref="A6.SS2.2.p1.1.m1.1.1.cmml"><msub id="A6.SS2.2.p1.1.m1.1.1.2" xref="A6.SS2.2.p1.1.m1.1.1.2.cmml"><mi id="A6.SS2.2.p1.1.m1.1.1.2.2" xref="A6.SS2.2.p1.1.m1.1.1.2.2.cmml">K</mi><mi id="A6.SS2.2.p1.1.m1.1.1.2.3" xref="A6.SS2.2.p1.1.m1.1.1.2.3.cmml">𝖯𝖬𝖨</mi></msub><mo id="A6.SS2.2.p1.1.m1.1.1.1" xref="A6.SS2.2.p1.1.m1.1.1.1.cmml">+</mo><mi id="A6.SS2.2.p1.1.m1.1.1.3" xref="A6.SS2.2.p1.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS2.2.p1.1.m1.1b"><apply id="A6.SS2.2.p1.1.m1.1.1.cmml" xref="A6.SS2.2.p1.1.m1.1.1"><plus id="A6.SS2.2.p1.1.m1.1.1.1.cmml" xref="A6.SS2.2.p1.1.m1.1.1.1"></plus><apply id="A6.SS2.2.p1.1.m1.1.1.2.cmml" xref="A6.SS2.2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A6.SS2.2.p1.1.m1.1.1.2.1.cmml" xref="A6.SS2.2.p1.1.m1.1.1.2">subscript</csymbol><ci id="A6.SS2.2.p1.1.m1.1.1.2.2.cmml" xref="A6.SS2.2.p1.1.m1.1.1.2.2">𝐾</ci><ci id="A6.SS2.2.p1.1.m1.1.1.2.3.cmml" xref="A6.SS2.2.p1.1.m1.1.1.2.3">𝖯𝖬𝖨</ci></apply><ci id="A6.SS2.2.p1.1.m1.1.1.3.cmml" xref="A6.SS2.2.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.2.p1.1.m1.1c">K_{\mathsf{PMI}}+C</annotation></semantics></math> still only has non-positive off-diagonal elements if</p>
<table id="A6.E32" class="ltx_equation ltx_eqn_table">

<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 id="A6.E32.m1.1" class="ltx_Math" alttext="-C\geq\log\rho_{\mathsf{min}}+\delta." display="block"><semantics id="A6.E32.m1.1a"><mrow id="A6.E32.m1.1.1.1" xref="A6.E32.m1.1.1.1.1.cmml"><mrow id="A6.E32.m1.1.1.1.1" xref="A6.E32.m1.1.1.1.1.cmml"><mrow id="A6.E32.m1.1.1.1.1.2" xref="A6.E32.m1.1.1.1.1.2.cmml"><mo id="A6.E32.m1.1.1.1.1.2a" xref="A6.E32.m1.1.1.1.1.2.cmml">−</mo><mi id="A6.E32.m1.1.1.1.1.2.2" xref="A6.E32.m1.1.1.1.1.2.2.cmml">C</mi></mrow><mo id="A6.E32.m1.1.1.1.1.1" xref="A6.E32.m1.1.1.1.1.1.cmml">≥</mo><mrow id="A6.E32.m1.1.1.1.1.3" xref="A6.E32.m1.1.1.1.1.3.cmml"><mrow id="A6.E32.m1.1.1.1.1.3.2" xref="A6.E32.m1.1.1.1.1.3.2.cmml"><mi id="A6.E32.m1.1.1.1.1.3.2.1" xref="A6.E32.m1.1.1.1.1.3.2.1.cmml">log</mi><mo lspace="0.167em" id="A6.E32.m1.1.1.1.1.3.2a" xref="A6.E32.m1.1.1.1.1.3.2.cmml">⁡</mo><msub id="A6.E32.m1.1.1.1.1.3.2.2" xref="A6.E32.m1.1.1.1.1.3.2.2.cmml"><mi id="A6.E32.m1.1.1.1.1.3.2.2.2" xref="A6.E32.m1.1.1.1.1.3.2.2.2.cmml">ρ</mi><mi id="A6.E32.m1.1.1.1.1.3.2.2.3" xref="A6.E32.m1.1.1.1.1.3.2.2.3.cmml">𝗆𝗂𝗇</mi></msub></mrow><mo id="A6.E32.m1.1.1.1.1.3.1" xref="A6.E32.m1.1.1.1.1.3.1.cmml">+</mo><mi id="A6.E32.m1.1.1.1.1.3.3" xref="A6.E32.m1.1.1.1.1.3.3.cmml">δ</mi></mrow></mrow><mo lspace="0em" id="A6.E32.m1.1.1.1.2" xref="A6.E32.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.E32.m1.1b"><apply id="A6.E32.m1.1.1.1.1.cmml" xref="A6.E32.m1.1.1.1"><geq id="A6.E32.m1.1.1.1.1.1.cmml" xref="A6.E32.m1.1.1.1.1.1"></geq><apply id="A6.E32.m1.1.1.1.1.2.cmml" xref="A6.E32.m1.1.1.1.1.2"><minus id="A6.E32.m1.1.1.1.1.2.1.cmml" xref="A6.E32.m1.1.1.1.1.2"></minus><ci id="A6.E32.m1.1.1.1.1.2.2.cmml" xref="A6.E32.m1.1.1.1.1.2.2">𝐶</ci></apply><apply id="A6.E32.m1.1.1.1.1.3.cmml" xref="A6.E32.m1.1.1.1.1.3"><plus id="A6.E32.m1.1.1.1.1.3.1.cmml" xref="A6.E32.m1.1.1.1.1.3.1"></plus><apply id="A6.E32.m1.1.1.1.1.3.2.cmml" xref="A6.E32.m1.1.1.1.1.3.2"><log id="A6.E32.m1.1.1.1.1.3.2.1.cmml" xref="A6.E32.m1.1.1.1.1.3.2.1"></log><apply id="A6.E32.m1.1.1.1.1.3.2.2.cmml" xref="A6.E32.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="A6.E32.m1.1.1.1.1.3.2.2.1.cmml" xref="A6.E32.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="A6.E32.m1.1.1.1.1.3.2.2.2.cmml" xref="A6.E32.m1.1.1.1.1.3.2.2.2">𝜌</ci><ci id="A6.E32.m1.1.1.1.1.3.2.2.3.cmml" xref="A6.E32.m1.1.1.1.1.3.2.2.3">𝗆𝗂𝗇</ci></apply></apply><ci id="A6.E32.m1.1.1.1.1.3.3.cmml" xref="A6.E32.m1.1.1.1.1.3.3">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E32.m1.1c">-C\geq\log\rho_{\mathsf{min}}+\delta.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td>
</tr></tbody>
</table>
<p id="A6.SS2.2.p1.2" class="ltx_p">For such <math id="A6.SS2.2.p1.2.m1.1" class="ltx_Math" alttext="C" display="inline"><semantics id="A6.SS2.2.p1.2.m1.1a"><mi id="A6.SS2.2.p1.2.m1.1.1" xref="A6.SS2.2.p1.2.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A6.SS2.2.p1.2.m1.1b"><ci id="A6.SS2.2.p1.2.m1.1.1.cmml" xref="A6.SS2.2.p1.2.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.2.p1.2.m1.1c">C</annotation></semantics></math>, it is diagonally dominant (and thus PSD) if,</p>
<table id="A6.E33" class="ltx_equation ltx_eqn_table">

<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 id="A6.E33.m1.1" class="ltx_math_unparsed" alttext="\mathllap{\forall i,\qquad}K_{\mathsf{PMI}}(z_{i},z_{i})+C\geq\sum_{j\neq i}\left\lvert K_{\mathsf{PMI}}(z_{i},z_{j})+C\right\rvert=-(N-1)C-\sum_{j\neq i}K_{\mathsf{PMI}}(z_{i},z_{j})," display="block"><semantics id="A6.E33.m1.1a"><mrow id="A6.E33.m1.1.1.1"><mrow id="A6.E33.m1.1.1.1.1"><mrow id="A6.E33.m1.1.1.1.1.2"><mrow id="A6.E33.m1.1.1.1.1.2.2"><mpadded lspace="-35.1pt" width="0pt" id="A6.E33.m1.1.1.1.1.2.2.4"><mo rspace="0.167em" id="A6.E33.m1.1.1.1.1.2.2.4.1">∀</mo><mi id="A6.E33.m1.1.1.1.1.2.2.4.2">i</mi><mo id="A6.E33.m1.1.1.1.1.2.2.4.3">,</mo></mpadded><mo lspace="0em" rspace="0em" id="A6.E33.m1.1.1.1.1.2.2.3">​</mo><msub id="A6.E33.m1.1.1.1.1.2.2.5"><mi id="A6.E33.m1.1.1.1.1.2.2.5.2">K</mi><mi id="A6.E33.m1.1.1.1.1.2.2.5.3">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E33.m1.1.1.1.1.2.2.3a">​</mo><mrow id="A6.E33.m1.1.1.1.1.2.2.2.2"><mo stretchy="false" id="A6.E33.m1.1.1.1.1.2.2.2.2.3">(</mo><msub id="A6.E33.m1.1.1.1.1.1.1.1.1.1"><mi id="A6.E33.m1.1.1.1.1.1.1.1.1.1.2">z</mi><mi id="A6.E33.m1.1.1.1.1.1.1.1.1.1.3">i</mi></msub><mo id="A6.E33.m1.1.1.1.1.2.2.2.2.4">,</mo><msub id="A6.E33.m1.1.1.1.1.2.2.2.2.2"><mi id="A6.E33.m1.1.1.1.1.2.2.2.2.2.2">z</mi><mi id="A6.E33.m1.1.1.1.1.2.2.2.2.2.3">i</mi></msub><mo stretchy="false" id="A6.E33.m1.1.1.1.1.2.2.2.2.5">)</mo></mrow></mrow><mo id="A6.E33.m1.1.1.1.1.2.3">+</mo><mi id="A6.E33.m1.1.1.1.1.2.4">C</mi></mrow><mo rspace="0.111em" id="A6.E33.m1.1.1.1.1.8">≥</mo><mrow id="A6.E33.m1.1.1.1.1.3"><munder id="A6.E33.m1.1.1.1.1.3.2"><mo movablelimits="false" rspace="0em" id="A6.E33.m1.1.1.1.1.3.2.2">∑</mo><mrow id="A6.E33.m1.1.1.1.1.3.2.3"><mi id="A6.E33.m1.1.1.1.1.3.2.3.2">j</mi><mo id="A6.E33.m1.1.1.1.1.3.2.3.1">≠</mo><mi id="A6.E33.m1.1.1.1.1.3.2.3.3">i</mi></mrow></munder><mrow id="A6.E33.m1.1.1.1.1.3.1.1"><mo id="A6.E33.m1.1.1.1.1.3.1.1.2">|</mo><mrow id="A6.E33.m1.1.1.1.1.3.1.1.1"><mrow id="A6.E33.m1.1.1.1.1.3.1.1.1.2"><msub id="A6.E33.m1.1.1.1.1.3.1.1.1.2.4"><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.2.4.2">K</mi><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.2.4.3">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E33.m1.1.1.1.1.3.1.1.1.2.3">​</mo><mrow id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2"><mo stretchy="false" id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.3">(</mo><msub id="A6.E33.m1.1.1.1.1.3.1.1.1.1.1.1.1"><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.1.1.1.1.2">z</mi><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.1.1.1.1.3">i</mi></msub><mo id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.4">,</mo><msub id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.2"><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.2.2">z</mi><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.2.3">j</mi></msub><mo stretchy="false" id="A6.E33.m1.1.1.1.1.3.1.1.1.2.2.2.5">)</mo></mrow></mrow><mo id="A6.E33.m1.1.1.1.1.3.1.1.1.3">+</mo><mi id="A6.E33.m1.1.1.1.1.3.1.1.1.4">C</mi></mrow><mo id="A6.E33.m1.1.1.1.1.3.1.1.3">|</mo></mrow></mrow><mo id="A6.E33.m1.1.1.1.1.9">=</mo><mrow id="A6.E33.m1.1.1.1.1.6"><mrow id="A6.E33.m1.1.1.1.1.4.1"><mo id="A6.E33.m1.1.1.1.1.4.1a">−</mo><mrow id="A6.E33.m1.1.1.1.1.4.1.1"><mrow id="A6.E33.m1.1.1.1.1.4.1.1.1.1"><mo stretchy="false" id="A6.E33.m1.1.1.1.1.4.1.1.1.1.2">(</mo><mrow id="A6.E33.m1.1.1.1.1.4.1.1.1.1.1"><mi id="A6.E33.m1.1.1.1.1.4.1.1.1.1.1.2">N</mi><mo id="A6.E33.m1.1.1.1.1.4.1.1.1.1.1.1">−</mo><mn id="A6.E33.m1.1.1.1.1.4.1.1.1.1.1.3">1</mn></mrow><mo stretchy="false" id="A6.E33.m1.1.1.1.1.4.1.1.1.1.3">)</mo></mrow><mo lspace="0em" rspace="0em" id="A6.E33.m1.1.1.1.1.4.1.1.2">​</mo><mi id="A6.E33.m1.1.1.1.1.4.1.1.3">C</mi></mrow></mrow><mo rspace="0.055em" id="A6.E33.m1.1.1.1.1.6.4">−</mo><mrow id="A6.E33.m1.1.1.1.1.6.3"><munder id="A6.E33.m1.1.1.1.1.6.3.3"><mo movablelimits="false" id="A6.E33.m1.1.1.1.1.6.3.3.2">∑</mo><mrow id="A6.E33.m1.1.1.1.1.6.3.3.3"><mi id="A6.E33.m1.1.1.1.1.6.3.3.3.2">j</mi><mo id="A6.E33.m1.1.1.1.1.6.3.3.3.1">≠</mo><mi id="A6.E33.m1.1.1.1.1.6.3.3.3.3">i</mi></mrow></munder><mrow id="A6.E33.m1.1.1.1.1.6.3.2"><msub id="A6.E33.m1.1.1.1.1.6.3.2.4"><mi id="A6.E33.m1.1.1.1.1.6.3.2.4.2">K</mi><mi id="A6.E33.m1.1.1.1.1.6.3.2.4.3">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E33.m1.1.1.1.1.6.3.2.3">​</mo><mrow id="A6.E33.m1.1.1.1.1.6.3.2.2.2"><mo stretchy="false" id="A6.E33.m1.1.1.1.1.6.3.2.2.2.3">(</mo><msub id="A6.E33.m1.1.1.1.1.5.2.1.1.1.1"><mi id="A6.E33.m1.1.1.1.1.5.2.1.1.1.1.2">z</mi><mi id="A6.E33.m1.1.1.1.1.5.2.1.1.1.1.3">i</mi></msub><mo id="A6.E33.m1.1.1.1.1.6.3.2.2.2.4">,</mo><msub id="A6.E33.m1.1.1.1.1.6.3.2.2.2.2"><mi id="A6.E33.m1.1.1.1.1.6.3.2.2.2.2.2">z</mi><mi id="A6.E33.m1.1.1.1.1.6.3.2.2.2.2.3">j</mi></msub><mo stretchy="false" id="A6.E33.m1.1.1.1.1.6.3.2.2.2.5">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="A6.E33.m1.1.1.1.2">,</mo></mrow><annotation encoding="application/x-tex" id="A6.E33.m1.1b">\mathllap{\forall i,\qquad}K_{\mathsf{PMI}}(z_{i},z_{i})+C\geq\sum_{j\neq i}\left\lvert K_{\mathsf{PMI}}(z_{i},z_{j})+C\right\rvert=-(N-1)C-\sum_{j\neq i}K_{\mathsf{PMI}}(z_{i},z_{j}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td>
</tr></tbody>
</table>
<p id="A6.SS2.2.p1.3" class="ltx_p">or equivalently,</p>
<table id="A6.E34" class="ltx_equation ltx_eqn_table">

<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 id="A6.E34.m1.1" class="ltx_math_unparsed" alttext="\mathllap{\forall i,\qquad}NC+\sum_{j}K_{\mathsf{PMI}}(z_{i},z_{j})\geq 0." display="block"><semantics id="A6.E34.m1.1a"><mrow id="A6.E34.m1.1.1.1"><mrow id="A6.E34.m1.1.1.1.1"><mrow id="A6.E34.m1.1.1.1.1.2"><mrow id="A6.E34.m1.1.1.1.1.2.4"><mpadded lspace="-35.1pt" width="0pt" id="A6.E34.m1.1.1.1.1.2.4.2"><mo rspace="0.167em" id="A6.E34.m1.1.1.1.1.2.4.2.1">∀</mo><mi id="A6.E34.m1.1.1.1.1.2.4.2.2">i</mi><mo id="A6.E34.m1.1.1.1.1.2.4.2.3">,</mo></mpadded><mo lspace="0em" rspace="0em" id="A6.E34.m1.1.1.1.1.2.4.1">​</mo><mi id="A6.E34.m1.1.1.1.1.2.4.3">N</mi><mo lspace="0em" rspace="0em" id="A6.E34.m1.1.1.1.1.2.4.1a">​</mo><mi id="A6.E34.m1.1.1.1.1.2.4.4">C</mi></mrow><mo rspace="0.055em" id="A6.E34.m1.1.1.1.1.2.3">+</mo><mrow id="A6.E34.m1.1.1.1.1.2.2"><munder id="A6.E34.m1.1.1.1.1.2.2.3"><mo movablelimits="false" id="A6.E34.m1.1.1.1.1.2.2.3.2">∑</mo><mi id="A6.E34.m1.1.1.1.1.2.2.3.3">j</mi></munder><mrow id="A6.E34.m1.1.1.1.1.2.2.2"><msub id="A6.E34.m1.1.1.1.1.2.2.2.4"><mi id="A6.E34.m1.1.1.1.1.2.2.2.4.2">K</mi><mi id="A6.E34.m1.1.1.1.1.2.2.2.4.3">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E34.m1.1.1.1.1.2.2.2.3">​</mo><mrow id="A6.E34.m1.1.1.1.1.2.2.2.2.2"><mo stretchy="false" id="A6.E34.m1.1.1.1.1.2.2.2.2.2.3">(</mo><msub id="A6.E34.m1.1.1.1.1.1.1.1.1.1.1"><mi id="A6.E34.m1.1.1.1.1.1.1.1.1.1.1.2">z</mi><mi id="A6.E34.m1.1.1.1.1.1.1.1.1.1.1.3">i</mi></msub><mo id="A6.E34.m1.1.1.1.1.2.2.2.2.2.4">,</mo><msub id="A6.E34.m1.1.1.1.1.2.2.2.2.2.2"><mi id="A6.E34.m1.1.1.1.1.2.2.2.2.2.2.2">z</mi><mi id="A6.E34.m1.1.1.1.1.2.2.2.2.2.2.3">j</mi></msub><mo stretchy="false" id="A6.E34.m1.1.1.1.1.2.2.2.2.2.5">)</mo></mrow></mrow></mrow></mrow><mo id="A6.E34.m1.1.1.1.1.3">≥</mo><mn id="A6.E34.m1.1.1.1.1.4">0</mn></mrow><mo lspace="0em" id="A6.E34.m1.1.1.1.2">.</mo></mrow><annotation encoding="application/x-tex" id="A6.E34.m1.1b">\mathllap{\forall i,\qquad}NC+\sum_{j}K_{\mathsf{PMI}}(z_{i},z_{j})\geq 0.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.SS2.3.p2" class="ltx_para ltx_noindent">
<p id="A6.SS2.3.p2.1" class="ltx_p">The following choice of <math id="A6.SS2.3.p2.1.m1.1" class="ltx_Math" alttext="C" display="inline"><semantics id="A6.SS2.3.p2.1.m1.1a"><mi id="A6.SS2.3.p2.1.m1.1.1" xref="A6.SS2.3.p2.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A6.SS2.3.p2.1.m1.1b"><ci id="A6.SS2.3.p2.1.m1.1.1.cmml" xref="A6.SS2.3.p2.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.3.p2.1.m1.1c">C</annotation></semantics></math> readily satisfies the above <a href="#A6.E34" title="In Proof. ‣ F.2 Contrastive learners can represent 𝐾_𝖯𝖬𝖨 exactly under smoothness conditions ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">34</span></a>:</p>
<table id="A6.E35" class="ltx_equation ltx_eqn_table">

<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 id="A6.E35.m1.1" class="ltx_Math" alttext="C\triangleq-\min_{i}\frac{1}{N}\sum_{k}K_{\mathsf{PMI}}(z_{i},z_{j})." display="block"><semantics id="A6.E35.m1.1a"><mrow id="A6.E35.m1.1.1.1" xref="A6.E35.m1.1.1.1.1.cmml"><mrow id="A6.E35.m1.1.1.1.1" xref="A6.E35.m1.1.1.1.1.cmml"><mi id="A6.E35.m1.1.1.1.1.4" xref="A6.E35.m1.1.1.1.1.4.cmml">C</mi><mo id="A6.E35.m1.1.1.1.1.3" xref="A6.E35.m1.1.1.1.1.3.cmml">≜</mo><mrow id="A6.E35.m1.1.1.1.1.2" xref="A6.E35.m1.1.1.1.1.2.cmml"><mo rspace="0.167em" id="A6.E35.m1.1.1.1.1.2a" xref="A6.E35.m1.1.1.1.1.2.cmml">−</mo><mrow id="A6.E35.m1.1.1.1.1.2.2" xref="A6.E35.m1.1.1.1.1.2.2.cmml"><mrow id="A6.E35.m1.1.1.1.1.2.2.4" xref="A6.E35.m1.1.1.1.1.2.2.4.cmml"><munder id="A6.E35.m1.1.1.1.1.2.2.4.1" xref="A6.E35.m1.1.1.1.1.2.2.4.1.cmml"><mi id="A6.E35.m1.1.1.1.1.2.2.4.1.2" xref="A6.E35.m1.1.1.1.1.2.2.4.1.2.cmml">min</mi><mi id="A6.E35.m1.1.1.1.1.2.2.4.1.3" xref="A6.E35.m1.1.1.1.1.2.2.4.1.3.cmml">i</mi></munder><mo lspace="0.167em" id="A6.E35.m1.1.1.1.1.2.2.4a" xref="A6.E35.m1.1.1.1.1.2.2.4.cmml">⁡</mo><mfrac id="A6.E35.m1.1.1.1.1.2.2.4.2" xref="A6.E35.m1.1.1.1.1.2.2.4.2.cmml"><mn id="A6.E35.m1.1.1.1.1.2.2.4.2.2" xref="A6.E35.m1.1.1.1.1.2.2.4.2.2.cmml">1</mn><mi id="A6.E35.m1.1.1.1.1.2.2.4.2.3" xref="A6.E35.m1.1.1.1.1.2.2.4.2.3.cmml">N</mi></mfrac></mrow><mo lspace="0em" rspace="0em" id="A6.E35.m1.1.1.1.1.2.2.3" xref="A6.E35.m1.1.1.1.1.2.2.3.cmml">​</mo><mrow id="A6.E35.m1.1.1.1.1.2.2.2" xref="A6.E35.m1.1.1.1.1.2.2.2.cmml"><munder id="A6.E35.m1.1.1.1.1.2.2.2.3" xref="A6.E35.m1.1.1.1.1.2.2.2.3.cmml"><mo movablelimits="false" id="A6.E35.m1.1.1.1.1.2.2.2.3.2" xref="A6.E35.m1.1.1.1.1.2.2.2.3.2.cmml">∑</mo><mi id="A6.E35.m1.1.1.1.1.2.2.2.3.3" xref="A6.E35.m1.1.1.1.1.2.2.2.3.3.cmml">k</mi></munder><mrow id="A6.E35.m1.1.1.1.1.2.2.2.2" xref="A6.E35.m1.1.1.1.1.2.2.2.2.cmml"><msub id="A6.E35.m1.1.1.1.1.2.2.2.2.4" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4.cmml"><mi id="A6.E35.m1.1.1.1.1.2.2.2.2.4.2" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4.2.cmml">K</mi><mi id="A6.E35.m1.1.1.1.1.2.2.2.2.4.3" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4.3.cmml">𝖯𝖬𝖨</mi></msub><mo lspace="0em" rspace="0em" id="A6.E35.m1.1.1.1.1.2.2.2.2.3" xref="A6.E35.m1.1.1.1.1.2.2.2.2.3.cmml">​</mo><mrow id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.3.cmml"><mo stretchy="false" id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.3" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.3.cmml">(</mo><msub id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">z</mi><mi id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.4" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.3.cmml">,</mo><msub id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.cmml"><mi id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.2" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.2.cmml">z</mi><mi id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.3" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.5" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></mrow></mrow><mo lspace="0em" id="A6.E35.m1.1.1.1.2" xref="A6.E35.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.E35.m1.1b"><apply id="A6.E35.m1.1.1.1.1.cmml" xref="A6.E35.m1.1.1.1"><ci id="A6.E35.m1.1.1.1.1.3.cmml" xref="A6.E35.m1.1.1.1.1.3">≜</ci><ci id="A6.E35.m1.1.1.1.1.4.cmml" xref="A6.E35.m1.1.1.1.1.4">𝐶</ci><apply id="A6.E35.m1.1.1.1.1.2.cmml" xref="A6.E35.m1.1.1.1.1.2"><minus id="A6.E35.m1.1.1.1.1.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2"></minus><apply id="A6.E35.m1.1.1.1.1.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2"><times id="A6.E35.m1.1.1.1.1.2.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.3"></times><apply id="A6.E35.m1.1.1.1.1.2.2.4.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4"><apply id="A6.E35.m1.1.1.1.1.2.2.4.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.1"><csymbol cd="ambiguous" id="A6.E35.m1.1.1.1.1.2.2.4.1.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.1">subscript</csymbol><min id="A6.E35.m1.1.1.1.1.2.2.4.1.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.1.2"></min><ci id="A6.E35.m1.1.1.1.1.2.2.4.1.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.1.3">𝑖</ci></apply><apply id="A6.E35.m1.1.1.1.1.2.2.4.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.2"><divide id="A6.E35.m1.1.1.1.1.2.2.4.2.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.2"></divide><cn type="integer" id="A6.E35.m1.1.1.1.1.2.2.4.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.2.2">1</cn><ci id="A6.E35.m1.1.1.1.1.2.2.4.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.4.2.3">𝑁</ci></apply></apply><apply id="A6.E35.m1.1.1.1.1.2.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2"><apply id="A6.E35.m1.1.1.1.1.2.2.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="A6.E35.m1.1.1.1.1.2.2.2.3.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.3">subscript</csymbol><sum id="A6.E35.m1.1.1.1.1.2.2.2.3.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.3.2"></sum><ci id="A6.E35.m1.1.1.1.1.2.2.2.3.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.3.3">𝑘</ci></apply><apply id="A6.E35.m1.1.1.1.1.2.2.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2"><times id="A6.E35.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.3"></times><apply id="A6.E35.m1.1.1.1.1.2.2.2.2.4.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4"><csymbol cd="ambiguous" id="A6.E35.m1.1.1.1.1.2.2.2.2.4.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4">subscript</csymbol><ci id="A6.E35.m1.1.1.1.1.2.2.2.2.4.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4.2">𝐾</ci><ci id="A6.E35.m1.1.1.1.1.2.2.2.2.4.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.4.3">𝖯𝖬𝖨</ci></apply><interval closure="open" id="A6.E35.m1.1.1.1.1.2.2.2.2.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2"><apply id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A6.E35.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.1.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2">subscript</csymbol><ci id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.2.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.2">𝑧</ci><ci id="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.3.cmml" xref="A6.E35.m1.1.1.1.1.2.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E35.m1.1c">C\triangleq-\min_{i}\frac{1}{N}\sum_{k}K_{\mathsf{PMI}}(z_{i},z_{j}).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.SS2.4.p3" class="ltx_para ltx_noindent">
<p id="A6.SS2.4.p3.1" class="ltx_p">Therefore, it remains to show that <a href="#A6.E32" title="In Proof. ‣ F.2 Contrastive learners can represent 𝐾_𝖯𝖬𝖨 exactly under smoothness conditions ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">32</span></a> is true. Note that</p>
<table id="A6.E36" class="ltx_equation ltx_eqn_table">

<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 id="A6.E36.m1.1" class="ltx_Math" alttext="-C\triangleq\min_{i}\frac{1}{N}\sum_{k}K_{\mathsf{PMI}}(z_{i},z_{j})\geq\frac{N-1}{N}\log\rho_{\mathsf{min}}+\frac{1}{N}(\min_{i}K_{\mathsf{PMI}}(z_{i},z_{i}))." display="block"><semantics id="A6.E36.m1.1a"><mrow id="A6.E36.m1.1.1.1" xref="A6.E36.m1.1.1.1.1.cmml"><mrow id="A6.E36.m1.1.1.1.1" xref="A6.E36.m1.1.1.1.1.cmml"><mrow id="A6.E36.m1.1.1.1.1.5" xref="A6.E36.m1.1.1.1.1.5.cmml"><mo id="A6.E36.m1.1.1.1.1.5a" xref="A6.E36.m1.1.1.1.1.5.cmml">−</mo><mi id="A6.E36.m1.1.1.1.1.5.2" xref="A6.E36.m1.1.1.1.1.5.2.cmml">C</mi></mrow><mo id="A6.E36.m1.1.1.1.1.6" xref="A6.E36.m1.1.1.1.1.6.cmml">≜</mo><mrow id="A6.E36.m1.1.1.1.1.2" xref="A6.E36.m1.1.1.1.1.2.cmml"><mrow id="A6.E36.m1.1.1.1.1.2.4" xref="A6.E36.m1.1.1.1.1.2.4.cmml"><munder id="A6.E36.m1.1.1.1.1.2.4.1" xref="A6.E36.m1.1.1.1.1.2.4.1.cmml"><mi id="A6.E36.m1.1.1.1.1.2.4.1.2" xref="A6.E36.m1.1.1.1.1.2.4.1.2.cmml">min</mi><mi 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xref="A6.E36.m1.1.1.1.1.3.1.1.1.1.2.2.2.3">𝑖</ci></apply></interval></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E36.m1.1c">-C\triangleq\min_{i}\frac{1}{N}\sum_{k}K_{\mathsf{PMI}}(z_{i},z_{j})\geq\frac{N-1}{N}\log\rho_{\mathsf{min}}+\frac{1}{N}(\min_{i}K_{\mathsf{PMI}}(z_{i},z_{i})).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td>
</tr></tbody>
</table>
</div>
<div id="A6.SS2.5.p4" class="ltx_para">
<p id="A6.SS2.5.p4.1" class="ltx_p">Therefore, it suffices to have</p>
<table id="A6.E37" class="ltx_equation ltx_eqn_table">

<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 id="A6.E37.m1.1" class="ltx_Math" alttext="\log\rho_{\mathsf{min}}+\delta\leq\frac{N-1}{N}\log\rho_{\mathsf{min}}+\frac{1}{N}(\min_{i}K_{\mathsf{PMI}}(z_{i},z_{i}))." display="block"><semantics id="A6.E37.m1.1a"><mrow id="A6.E37.m1.1.1.1" xref="A6.E37.m1.1.1.1.1.cmml"><mrow id="A6.E37.m1.1.1.1.1" xref="A6.E37.m1.1.1.1.1.cmml"><mrow id="A6.E37.m1.1.1.1.1.3" xref="A6.E37.m1.1.1.1.1.3.cmml"><mrow id="A6.E37.m1.1.1.1.1.3.2" xref="A6.E37.m1.1.1.1.1.3.2.cmml"><mi id="A6.E37.m1.1.1.1.1.3.2.1" xref="A6.E37.m1.1.1.1.1.3.2.1.cmml">log</mi><mo lspace="0.167em" id="A6.E37.m1.1.1.1.1.3.2a" xref="A6.E37.m1.1.1.1.1.3.2.cmml">⁡</mo><msub id="A6.E37.m1.1.1.1.1.3.2.2" xref="A6.E37.m1.1.1.1.1.3.2.2.cmml"><mi id="A6.E37.m1.1.1.1.1.3.2.2.2" xref="A6.E37.m1.1.1.1.1.3.2.2.2.cmml">ρ</mi><mi id="A6.E37.m1.1.1.1.1.3.2.2.3" xref="A6.E37.m1.1.1.1.1.3.2.2.3.cmml">𝗆𝗂𝗇</mi></msub></mrow><mo id="A6.E37.m1.1.1.1.1.3.1" xref="A6.E37.m1.1.1.1.1.3.1.cmml">+</mo><mi 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xref="A6.E37.m1.1.1.1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A6.E37.m1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml" xref="A6.E37.m1.1.1.1.1.1.1.1.1.1.2.2.2.2">𝑧</ci><ci id="A6.E37.m1.1.1.1.1.1.1.1.1.1.2.2.2.3.cmml" xref="A6.E37.m1.1.1.1.1.1.1.1.1.1.2.2.2.3">𝑖</ci></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.E37.m1.1c">\log\rho_{\mathsf{min}}+\delta\leq\frac{N-1}{N}\log\rho_{\mathsf{min}}+\frac{1}{N}(\min_{i}K_{\mathsf{PMI}}(z_{i},z_{i})).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td>
</tr></tbody>
</table>
<p id="A6.SS2.5.p4.2" class="ltx_p">Rearranging terms gives the desired condition</p>
<table id="A6.E38" class="ltx_equation ltx_eqn_table">

<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 id="A6.E38.m1.2" class="ltx_math_unparsed" alttext="\frac{{P}_{\mathsf{coor}}(z_{i}\mathrel{|}z_{i})}{{P}_{\mathsf{coor}}(z_{i})}\geq e^{N\delta}\rho_{\mathsf{min}}\mathrlap{,\qquad\forall i.}" display="block"><semantics id="A6.E38.m1.2a"><mrow id="A6.E38.m1.2.3"><mfrac id="A6.E38.m1.2.2"><mrow id="A6.E38.m1.1.1.1"><msub id="A6.E38.m1.1.1.1.3"><mi id="A6.E38.m1.1.1.1.3.2">P</mi><mi id="A6.E38.m1.1.1.1.3.3">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="A6.E38.m1.1.1.1.2">​</mo><mrow id="A6.E38.m1.1.1.1.1.1"><mo stretchy="false" id="A6.E38.m1.1.1.1.1.1.2">(</mo><mrow id="A6.E38.m1.1.1.1.1.1.1"><msub id="A6.E38.m1.1.1.1.1.1.1.2"><mi id="A6.E38.m1.1.1.1.1.1.1.2.2">z</mi><mi id="A6.E38.m1.1.1.1.1.1.1.2.3">i</mi></msub><mo fence="false" id="A6.E38.m1.1.1.1.1.1.1.1">|</mo><msub id="A6.E38.m1.1.1.1.1.1.1.3"><mi id="A6.E38.m1.1.1.1.1.1.1.3.2">z</mi><mi id="A6.E38.m1.1.1.1.1.1.1.3.3">i</mi></msub></mrow><mo stretchy="false" id="A6.E38.m1.1.1.1.1.1.3">)</mo></mrow></mrow><mrow id="A6.E38.m1.2.2.2"><msub id="A6.E38.m1.2.2.2.3"><mi id="A6.E38.m1.2.2.2.3.2">P</mi><mi id="A6.E38.m1.2.2.2.3.3">𝖼𝗈𝗈𝗋</mi></msub><mo lspace="0em" rspace="0em" id="A6.E38.m1.2.2.2.2">​</mo><mrow id="A6.E38.m1.2.2.2.1.1"><mo stretchy="false" id="A6.E38.m1.2.2.2.1.1.2">(</mo><msub id="A6.E38.m1.2.2.2.1.1.1"><mi id="A6.E38.m1.2.2.2.1.1.1.2">z</mi><mi id="A6.E38.m1.2.2.2.1.1.1.3">i</mi></msub><mo stretchy="false" id="A6.E38.m1.2.2.2.1.1.3">)</mo></mrow></mrow></mfrac><mo id="A6.E38.m1.2.3.1">≥</mo><mrow id="A6.E38.m1.2.3.2"><msup id="A6.E38.m1.2.3.2.2"><mi id="A6.E38.m1.2.3.2.2.2">e</mi><mrow id="A6.E38.m1.2.3.2.2.3"><mi id="A6.E38.m1.2.3.2.2.3.2">N</mi><mo lspace="0em" rspace="0em" id="A6.E38.m1.2.3.2.2.3.1">​</mo><mi id="A6.E38.m1.2.3.2.2.3.3">δ</mi></mrow></msup><mo lspace="0em" rspace="0em" id="A6.E38.m1.2.3.2.1">​</mo><msub id="A6.E38.m1.2.3.2.3"><mi id="A6.E38.m1.2.3.2.3.2">ρ</mi><mi id="A6.E38.m1.2.3.2.3.3">𝗆𝗂𝗇</mi></msub><mo lspace="0em" rspace="0em" id="A6.E38.m1.2.3.2.1a">​</mo><mpadded width="0pt" id="A6.E38.m1.2.3.2.4"><mo rspace="2.167em" id="A6.E38.m1.2.3.2.4.1">,</mo><mo rspace="0.167em" id="A6.E38.m1.2.3.2.4.2">∀</mo><mi id="A6.E38.m1.2.3.2.4.3">i</mi><mo lspace="0em" id="A6.E38.m1.2.3.2.4.4">.</mo></mpadded></mrow></mrow><annotation encoding="application/x-tex" id="A6.E38.m1.2b">\frac{{P}_{\mathsf{coor}}(z_{i}\mathrel{|}z_{i})}{{P}_{\mathsf{coor}}(z_{i})}\geq e^{N\delta}\rho_{\mathsf{min}}\mathrlap{,\qquad\forall i.}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td>
</tr></tbody>
</table>
<p id="A6.SS2.5.p4.3" class="ltx_p">∎</p>
</div>
</div>
<div id="A6.Thmtheorem2" class="ltx_theorem ltx_theorem_remark">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="A6.Thmtheorem2.1.1.1" class="ltx_text ltx_font_italic">Remark F.2</span></span><span id="A6.Thmtheorem2.2.2" class="ltx_text ltx_font_italic">.</span>
</h6>
<div id="A6.Thmtheorem2.p1" class="ltx_para">
<p id="A6.Thmtheorem2.p1.2" class="ltx_p"><a href="#A6.Thmtheorem1" title="Proposition F.1. ‣ F.2 Contrastive learners can represent 𝐾_𝖯𝖬𝖨 exactly under smoothness conditions ‣ Appendix F Analysis of Contrastive Learners ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Proposition</span> <span class="ltx_text ltx_ref_tag">F.1</span></a> is one example that a sufficiently smooth world or a sufficiently high sampling rate allows the PMI kernel <math id="A6.Thmtheorem2.p1.1.m1.1" class="ltx_Math" alttext="K_{\mathsf{PMI}}" display="inline"><semantics id="A6.Thmtheorem2.p1.1.m1.1a"><msub id="A6.Thmtheorem2.p1.1.m1.1.1" xref="A6.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="A6.Thmtheorem2.p1.1.m1.1.1.2" xref="A6.Thmtheorem2.p1.1.m1.1.1.2.cmml">K</mi><mi id="A6.Thmtheorem2.p1.1.m1.1.1.3" xref="A6.Thmtheorem2.p1.1.m1.1.1.3.cmml">𝖯𝖬𝖨</mi></msub><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem2.p1.1.m1.1b"><apply id="A6.Thmtheorem2.p1.1.m1.1.1.cmml" xref="A6.Thmtheorem2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="A6.Thmtheorem2.p1.1.m1.1.1">subscript</csymbol><ci id="A6.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="A6.Thmtheorem2.p1.1.m1.1.1.2">𝐾</ci><ci id="A6.Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="A6.Thmtheorem2.p1.1.m1.1.1.3">𝖯𝖬𝖨</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem2.p1.1.m1.1c">K_{\mathsf{PMI}}</annotation></semantics></math> to be <em id="A6.Thmtheorem2.p1.2.1" class="ltx_emph ltx_font_italic">exactly</em> represented as inner products of a learned feature space (up to a scale). The condition here can be satisfied, for example, if the off-diagonal terms decay linearly with respect to <math id="A6.Thmtheorem2.p1.2.m2.1" class="ltx_Math" alttext="N" display="inline"><semantics id="A6.Thmtheorem2.p1.2.m2.1a"><mi id="A6.Thmtheorem2.p1.2.m2.1.1" xref="A6.Thmtheorem2.p1.2.m2.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="A6.Thmtheorem2.p1.2.m2.1b"><ci id="A6.Thmtheorem2.p1.2.m2.1.1.cmml" xref="A6.Thmtheorem2.p1.2.m2.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.Thmtheorem2.p1.2.m2.1c">N</annotation></semantics></math> and stay sufficiently close to each other. While the condition is somewhat strict, it
captures the essence that smoothness and continuity allow easier learning. Nonetheless, we note that exact representation is not necessary for convergence, and thus this requirement can likely be relaxed. Please see <a href="#S6" title="6 Counterexamples and limitations ‣ The Platonic Representation Hypothesis" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a> for discussions on practical settings.</p>
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2405.07987
We argue that representations in AI models, particularly deep networks, are converging. First, we survey many examples of convergence in the literature: over time and across multiple domains, the ways by which different neural networks represent data are becoming more aligned.
Next, we demonstrate convergence across data modalities: as vision models and language models get larger, they measure distance between datapoints in a more and more alike way.
We hypothesize that this convergence is driving toward a shared statistical model of reality, akin to Plato’s concept of an ideal reality. We term such a representation theplatonic representationand discuss several possible selective pressures toward it. Finally, we discuss the implications of these trends, their limitations, and counterexamples to our analysis.
abstract
The Platonic Representation Hypothesis
Abstract
2405.07987
1IntroductionAI systems are rapidly evolving into highly multifunctional entities. For example, whereas in the past we had special-purpose solutions for different language processing tasks (e.g., sentiment analysis, parsing, dialogue) , modern large language models (LLMs)  are competent at all these tasks using a single set of weights. Unified systems are also being built across data modalities: instead of using a different architecture for processing images versus text, recent models, such as GPT4-V, Gemini, and LLaVA, handle both modalities with a combined architecture.
More and more systems are built off of general-purpose pretrained backbones, sometimes called foundation models, that support a large range of tasks, including robotics, bioinformatics, and healthcare.
In short, AI systems are becoming increasingly homogeneous in both their architectures and their capabilities.The Platonic Representation HypothesisNeural networks, trained with different objectives on different data and modalities, are converging to a shared statistical model of reality in their representation spaces.Figure 1:The Platonic Representation Hypothesis:Images ()  and text ()  are projections of a common underlying reality () . We conjecture that representation learning algorithms will converge on a shared representation of, and scaling model size, as well as data and task diversity, drives this convergence.This paper explores one aspect of this trend: representational convergence. We argue that there is a growing similarity in how datapoints are represented in different neural network models. This similarity spans across different model architectures, training objectives, and even data modalities.What has led to this convergence? Will it continue? And ultimately, where does it end?Our central hypothesis, stated above inFigure1, is that there is indeed an endpoint to this convergence and a principle that drives it: different models are all trying to arrive at arepresentation of reality, meaning a representation of the joint distribution over events in the world that generate the data we observe. Figure1conveys this hypothesis: there exists a real world (labeled) , which we measure with various sensors, such as the camera shown to the left () . Otherprojectionsof these measurements, such as the textual description shown, can be produced from the first set of measurements or mediated by some other set of measurements,e.g., touch or other camera views (dotted arrow fromto) 111Touch could convey the shapes in this example but not the colors. This is an important limitation to our hypothesis that we discuss at several points in the paper: different sensors and views might capture different information, which may limit their potential to converge to identical representations.. Representation learning algorithms find vector embeddings that statistically model the various measurements and projections. The resulting vector embeddings are all derived from the underlying reality inand thereby become aligned. As models are trained on more data and for more tasks, they require representations that capture more and more information about, and hence alignment towardincreases toward a convergent point as a function of scale.We call this converged hypothetical representation the “platonic representation” in reference to Plato’s Allegory of the Cave, and his idea of an ideal reality that underlies our sensations. The training data for our algorithms are shadows on the cave wall, yet, we hypothesize, models are recovering ever better representations of the actual world outside the cave. This idea is not unique to Plato; our hypothesis is also related to the notion of “convergent realism”in the philosophy of science (i.e., that science is converging on truth) , and to many arguments that have been put forth in the representation learning literature (e.g.,) .Also closely related to our hypothesis is the “Anna Karenina scenario” described by, referring to the possibility that all well-performing neural nets represent the world in the same way. We discuss the evidence they give for this possibility in Section2.222Borrowed from, similar analogies have been made in other domains, such as the “Anna Karenina principle” popularized byto explain animal domestication.The platonic representation hypothesis refers to the situation where we are in an Anna Karenina scenarioandthe “happy representation” that is converged upon is one that reflects a statistical model of the underlying reality. We discuss the potential nature of this statistical model in more detail in Section4.
introduction
The Platonic Representation Hypothesis
1Introduction
2405.07987
2Representations are convergingPreliminariesWe restrict our attention to representations that arevector embeddings. We characterize such a representation by the similarity structure it induces, referred to as its kernel. Kernels are commonly used to assess representations; this can be justified by the fact that they capture the relative structures among data samples, which are also the learning signal for many machine learning algorithms. Following prior literature, we definerepresentational alignmentas a measure of the similarity of the similarity structures induced by two representations,i.e., a similarity metric over kernels.
We give the mathematical definition of these concepts below:•Arepresentationis a functionthat assigns a feature vector to each input in some data domain.•Akernel,, characterizes how a representation measures distance/similarity between datapoints., wheredenotes inner product,and.•Akernel-alignment metric,, measures the similarity between two kernels,i.e., how similar is the distance measure induced by one representation to the distance measure induced by another. Examples include Centered Kernel Distance (CKA) , SVCCA, and nearest-neighbor metrics.In our experiments, we use amutual nearest-neighbor metricthat measures the mean intersection of the-nearest neighbor sets induced by two kernels,and, normalized by.
This metric is a variant of those proposed inand.
See AppendixAfor the exact definition and AppendixBfor comparisons with alternative alignment metrics.Next, we explore several ways in which representations are converging. First, we argue that different neural networks are converging to aligned representations. Then, we show that this continues to hold across modalities, where image embeddings in vision models align with text embeddings in language models.Figure 2:VISION models converge as COMPETENCE increases:We measure alignment amongmodels using mutual nearest-neighbors on Places-365, and evaluate their performance on downstream tasks from the Visual Task Adaptation Benchmark (VTAB;) .LEFT:Models that solve more VTAB tasks tend to be more aligned with each other. Error bars show standard error.RIGHT:We use UMAP to embedmodelsinto a 2D space, based on. More competent and general models (blue)  have more similar representations.2.1Different models, with different architectures and objectives, can have aligned representationsOne indication of representational convergence is the rising number of systems built on top of pre-trained foundation models. These models are becoming standard backbones across a growing spectrum of tasks. Their versatility across numerous applications implies a level of universality in the way they represent data.While this trend implies convergence toward a relatively small set of foundation models, it does not imply thatdifferentfoundation models will arrive at the same representation. Yet that is what has been observed by several recent papers.conducted one such study, in which they measured representational similarity through a technique calledmodel stitching. Given two models,and, each composed of multiple layers (,) , an intermediate representation fromis integrated intovia a learned affine stitching layer, resulting in a new stitched model. Ifhas good performance, it indicates thatandhave compatible representations at layer, up to the transform.In their study,made two notable findings: (1)  A vision model trained on ImageNetcan be aligned with a model trained on the Places-365 datasetwhile maintaining good performance; (2)  The early layers of these convolutional networks are more interchangeable than later layers. The first finding illustrates a level of data independence where distinct image datasets lead to similar representations. The second finding agrees with extensive research that oriented Gabor-like filters are common in both artificial and biological vision systems. This suggests a convergence to a similar initial layer of representation across various neural network architectures.expanded on the idea of model stitching, uncovering that models trained using self-supervised objectives align closely with their supervised counterparts.further demonstrated the feasibility of “zero-shot” model stitching without learning a stitching layer.
Despite the fact that different text models were trained on different modalities, they found that the models often
embed data in remarkably similar ways. In particular, they considered the kerneldefined by learned representations and showed thatserves as a bridge between models, allowing an encoder trained in one language, like English, to work effectively with a decoder in another, like French.extended this idea to individual neurons, and found “Rosetta Neurons” that are activated by the same pattern across a range of vision models. Such neurons form a common dictionary independently discovered by all models.Figure 3:LANGUAGE and VISION models align:We measure alignment using mutual nearest-neighbor on the Wikipedia caption dataset (WIT) . The x-axis is the language model performance measured over 4M tokens from the OpenWebText dataset(see AppendixBfor plots with model names) .
We measure performance using, wherebits-per-bytenormalizes the cross-entropy by the total bytes in the input text string.
The results show a linear relationship between language-vision alignment and language modeling score, where a general trend is that more capable language models align better with more capable vision models.
We find that CLIP models, which are trained with explicit language supervision, exhibit a higher level of alignment. However, this alignment decreases after being fine-tuned on ImageNet classification (labeled CLIP (I12K ft) ) .2.2Alignment increases with scale and performanceobserved model alignment not only exists but also increases with model scale. On CIFAR-10 classification, they found that larger models exhibit greater alignment with each other compared to smaller ones.
Theoretically,showed that models with similar outputs (e.g., as a result of having high performance)  also have similar internal activations.
With the continuing trend of models scaling up, this suggests model alignment will increase over time – we might expect that the next generation of bigger, better models will be even more aligned with each other.We expand upon this observation by evaluating the transfer performance ofvision models. These models were trained with varying architectures, training objectives, and datasets (detailed inSectionC.1) . In Figure2(left) , we bin these models based on their average transfer performance on the VTAB dataset, and then measure the average kernel alignment of the models within each bin. The results indicate that models with high transfer performance form a tightly clustered set of representations, while models with weak performance have more variable representations. We further visualize this structure with UMAPover models representation in Figure2(right) . This suggests that models that are competent all represent data in a similar way. Echoingand, we might say: all strong models are alike, each weak model is weak in its own way.The discussion so far indicates that various models are aligning toward a unified representation. But does the convergence extend to model weights? While models with different architectures might not have compatible weight spaces, there exists ample evidence that models with the same architecture will often converge to the same basin of weights. This holds even for models with different initializations, up to permutations over weight space. Because of this, it is possible to merge separately trained models of the same architecture, and achieve some of the capabilities of all models in the mixture.Figure 4:Alignment predicts downstream performance:We visualize correlation between LLM alignment score to DINOv2and downstream task performance on Hellaswag (common-sense) and GSM8K (math) . LLMs are plotted with radii proportional to the size of the model, and color-coded by their rank order in language modeling scores () . We observe that models aligned more closely with vision also show better performance on downstream language tasks. For Hellaswag, there is a linear relationship with alignment score, while GSM8K exhibits an “emergence”-esque trend.2.3Representations are converging across modalitiesDo models trained on different data modalities also converge?
Several works indicate that the answer isyes.extended model stitching to the cross-modal setting, finding that a single linear projection is sufficient to stitch a vision model to an LLM and achieve good performance on visual question answering and image captioning.showed that linear stitching can also work in the opposite direction, aligning text inputs to visual outputs.
In fact, many recent language-vision models stitch pre-trained language and vision models together. For example, LLaVAdemonstrated state-of-the-art results by projecting visual features into a language model with a 2-layer MLP.Other works show further kinds of evidence of cross-modal synergy.found that jointly training a language model with a vision model improves performance on language tasks, compared to training the language model on its own.probed the visual knowledge of LLMs trainedonlyon language data, by converting images into code that an LLM can process. They found that LLMs have rich knowledge of visual structures, to the extent that decent visual representations can be trained on images generated solely by querying an LLM to produce code and rendering the response. In visual generation, LLMs show abilities to augment captions with visual structures (e.g., bounding boxes and locations)  and improve generation quality. In other modalities,showed auditory models are also roughly aligned with language models up to a linear transformation, anddemonstrated the effectiveness of using pre-trained language models for facial motion prediction.We set out to address these claims in a broader scope to determine whether models are indeed learning an increasingly modality-agnostic representation of the world. We sampled a variety of models trained either solely on vision or solely on language, and compared their representations as they became larger and more competent over many tasks.InFigure3, we assess alignment between a suite of language models and vision models. So far we have only defined alignment for two kernels defined over the same input space. To measure cross-modal alignment, we use paired datasets to bridge the two modalities. For vision and text, we use the Wikipedia captions dataset, composed of images from Wikipedia ()  and their corresponding captions () . We then measure alignment between a language modeland a vision modelas the alignment of the two following kernels:(1) (2) Using this analysis, we find that the better an LLM is at language modeling, the more it tends to aligns with vision models, as shown inFigure3. The converse effect also holds: the better a vision models is, the more it tends to align with LLMs. SeeSectionC.2for more details.2.4Models are increasingly aligning to brainsNeural networks also show substantial alignment with biological representations in the brain. This commonality may be due to similarities in the task and data constraints both systems are confronted with.
Even though the mediums may differ – silicon transistors versus biological neurons – the fundamental problem faced by brains and machines is the same: efficiently extracting and understanding the underlying structure in images, text, sounds, etc. The tasks that the human visual system has been honed to perform through evolution – like segmentation, detection, and whole-image classification – are also the ones that we train our neural nets to perform.went as far as to title their work in the spirit that performance over such tasks implies brain alignment. Further,showed that training data plays a large role in alignment. Psychophysical studies have also shown agreement between how humans perceive visual similarity and how models do, even when the models are trained on tasks, such as self-supervised prediction, that are seemingly unrelated to mimicking human perception.2.5Does alignment predict downstream performance?If models are converging towards a more accurate representation of reality, we expect that alignment should correspond to improved performance on downstream tasks.Figure4supports this hypothesis by demonstrating improved performance on commonsense reasoning (Hellaswag;)  and mathematical problem solving (GSM8K;)  as alignment improves.
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3Why are representations converging?Modern machine learning models are generally trained to minimize the empirical risk with possible implicit and/or explicit regularization:In the following sections, we lay out how each colored component in this optimization process potentially plays a role in facilitating representational convergence.Figure 5:The Capacity Hypothesis:If an optimal representation exists in function space, larger hypothesis spaces are more likely to cover it.LEFT:Two small models might not cover the optimum and thus finddifferentsolutions (marked by outlined) .RIGHT:As the models become larger, they cover the optimum and converge to the same solution (marked by filled) .Figure 6:The Multitask Scaling Hypothesis:Models trained with an increasing number of tasks are subjected to pressure to learn a representation that can solve all the tasks.3.1Convergence viaTask GeneralityEach training datapoint and objective (task)  places an additional constraint on the model. As data and tasks scale,
the volume of representations that satisfy these constraints must proportionately grow smaller, as visualized in Figure6and stated below:The Multitask Scaling HypothesisThere are fewer representations that are competent fortasks than there are fortasks. As we train more general models that solve more tasks at once, we should expect fewer possible solutions.This has been previously termed as the Contravariance principle by, which states that the set of solutions to an easy goal is large, while the set of solutions to a challenging goal is comparatively smaller. Moreover, we argue that this narrower solution set also generalizes better. As data scales, models that optimize the empirical riskalso improve on the population risk, and become better at capturing statistical structures of the true data generating process () .Recent work has demonstrated a power law relationship between data scale and model performance. This implies that with enough data (e.g., consisting of the entire internet and all offline scientific measurements)  one ought to converge to a very small solution set with irreducible error – the inherent epistemic uncertainty of the world.
As more models are trained on internet-scale data, the set of solutions that satisfies all data constraints must become relatively small.In addition to data-scaling, many modern representation learning objectivesdirectly optimize for multi-task solving. Contrastive learning finds a distance structure over data samples that optimizes many classification tasks. Masked Autoencodersoptimize randomly sampled reconstruction tasks. In fact, autoregressive language modeling can also be seen as optimizing a diverse set of tasks. Such multi-task objectives may be more effective than single-task ones (e.g., ImageNet classification)  due to the fact that they impose more task constraints on the representation, leading to a smaller and higher-quality solution space.3.2Convergence viaModel CapacitySuppose there is a globally optimal representation for standard learning objectives. Then, under sufficient data,scalinga model (i.e., using larger function classes) , as well asimproved optimization, should be more effective at finding better approximations to this optimum, as illustrated inFigure5.
With the same training objective, larger models, even of different architectures, will thus tend to converge toward this optimum. When different training objectives share similar minimizers, larger models are better at finding these minimizers,
and will train to similar solutions over the training tasks. We summarize this hypothesis as follows:The Capacity HypothesisBigger models are more likely to converge to a shared representation than smaller models.3.3Convergence viaSimplicity BiasArriving at the same mapping on thetraining datadoes not prohibit the models from developing distinct internal representations. It is not unreasonable to posit that the representations used to detect a dog in a 1M parameter model could be quite different than that used by a 1B parameter model. What would stop a billion-parameter (and counting)  model from learning an overly complicated and distinct representation? One key factor might be simplicity bias:The Simplicity Bias HypothesisDeep networks are biased toward finding simple fits to the data, and the bigger the model, the stronger the bias. Therefore, as models get bigger, we should expect convergence to a smaller solution space.Such simplicity bias could be coming from explicit regularizationcommonly used in deep learning (e.g., weight decay and dropout) . However, even in the absence of external influences, deep networks naturally adhere to Occam’s razor,implicitly favoring simple solutionsthat fit the data.
Figure7visualizes how simplicity bias can drive convergence.Figure 7:The Simplicity Bias Hypothesis:Larger models have larger coverage of all possible ways to fit the same data. However, the implicit simplicity biases of deep networks encourage larger models to find the simplest of these solutions.
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4What representation are we converging to?By now, we hope to have convinced the reader that task and data pressures, combined with increasing model capacity, can lead to convergence. We next turn our attention towhatexactly is the endpoint of all this convergence.Our central hypothesis, stated inFigure1, is that the representation we are converging toward is a statistical model of the underlying reality that generates our observations. Consistent with the multitask scaling hypothesis, such a representation would naturally be useful toward many tasks (or at least toward any task grounded in reality) . Additionally, this representation might be relatively simple, assuming that scientists
are correct in suggesting that the fundamental laws of nature are indeed simple functions, in line with the simplicity bias hypothesis.But what exactly do we mean by “a statistical model of the underlying reality.” In this section, we formalize one definition with concrete mathematical statements.Importantly, this section should be read as just one concrete candidate for the form of the platonic representation; other candidates could be arrived at from other modeling assumptions.4.1An idealized worldWe consider a world that works as follows, consistent with the cartoon in Figure1. The world consists of a sequence ofdiscrete events, denoted as, sampled from some unknown distribution. Each event can be observed in various ways. An observation is a bijective, deterministic functionthat maps events to an arbitrary measurement space, such as pixels, sounds, mass, force, torque, words, etc.
Later, inSection6, we discuss limitations and potential extensions to continuous and unbounded worlds, and stochastic observations, that could yield a model that better reflects real learning scenarios.One can think of an event as corresponding to the state of the world at some point in time333Here we only analyze temporal sequences, but note that the same could be done with respect to events laid out in space instead., but it is also fine to simply consider an event as any variable that indexes observations, with no further physical meaning444This latter interpretation may be more consistent with Plato’s intent. Scholars have argued that his allegory of the cave rejects any notion of a true world state. Instead, we could say that the joint distribution of observation indices isitselfthe platonic reality..In this idealized world, knowingwould be useful for many kinds of predictions; this would constitute a world model over the events that cause our observations. We will next show that a particular representation ofis recovered by certain contrastive learners.Figure 8:Color cooccurrence in VISION and LANGUAGE yields perceptual organization:Similar representations of color are obtained via,from LEFT to RIGHT, the perceptual layout from CIELAB color space, cooccurrence in CIFAR-10 images, and language cooccurrence modeling (; computed roughly following) . Details inAppendixD.4.2A family of contrastive learners converge to a representation ofConsider a contrastive learner that models observations thatcooccurtogether. For simplicity, we ground our discussion with the following definition of thecooccurrence probability,, of two observationsandboth occurring within some window:Analogously, we can defineforand other observation modalities. Note thatis symmetric.Considerpositive pairsas two observations nearby in time (sampled from)  andnegative pairsas observations drawn from any point in time (sampled independently from the marginal) . Our contrastive learner tries to classify if a pair is positive or negative by learning a representationsuch that the dot-product kernel approximates the log odds ratio up to some offset:(3) (4) (5) whereis the pointwise mutual information (PMI)  kernel, andis constant in. We note that is a common setting for self-supervised contrastive learners with NCE objectives, such as SimCLRand SimCSE. (SeeandSectionF.1for details.) Under mild conditions that the world is smooth enough (seeSectionF.2) , a choice ofcan exactly represent:(6) where we observed thatfromEquation5must be a constant since both sides are symmetric.Therefore, the contrastive learners we consider are minimized by a representationwhose kernel is(up to a constant offset) . With sufficient data and optimization, we will observe convergence to this point.Thus we have convergence to a representation of the statistics of, but what about? Recall that our idealized world consists ofbijectiveobservation functions, which, over discrete random variables, preserve probabilities. So we have:where we useandin a modality-agnostic way to emphasize that different modalities share the same such quantities.All these arguments hold not just forbut also for(or any other bijective, discrete modality) , implying:(7) (8) Therefore, for any modality in our idealized world, we observe representational convergence to the same kernel, which represents certain pairwise statistics of.This analysis suggests that certain representation learning algorithms may boil down to a simple rule:find an embedding in which similarity equals PMI. We note that this idea is consistent with prior works that have used PMI as a similarity measure for clustering in vision and language (e.g.,) .A study in colorWe conduct a case study to verify that convergence does happen on real data.discovered that color distances in learned language representations, when trained to predict cooccurrences intext, closely mirror human perception of these distances, which we reproduce inFigure8with both contrastive and predictive models. Interestingly, they noted an increasing similarity as models scale larger and become better at modelingtextcooccurrences. InFigure8, we also learn representations of color based onfrom cooccurrences inimages. Indeed, learning cooccurrence statistics in either domain recovers roughly thesameperceptual representation. Details of this experiment are described in AppendixD.We believe that our simple model encapsulates essential aspects of complex real-world systems, and offers a path toward understanding the representation that models are converging to—a unified model that is proficient across various domains and modalities, grounded in the statistical properties of the underlying world.Section6further elaborates some limitations.
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5What are the implications of convergence?Scaling is sufficient, but not necessarily efficientOur arguments are roughly in line with the claim that “scale is all you need” to reach high levels of intelligence. We have argued that as resources are scaled (# parameters, # datapoints, # flops) , representations are converging, regardless of other modeling choices and even data modality. Does this mean that scale is all that matters? Not quite: different methods can scale with different levels ofefficiency, and successful methods must still satisfy some general requirements (e.g., be a consistent estimator, model pairwise statistics of) .Training data can be shared across modalitiesSuppose you have access toimages andsentences, and want to learn the best representation. If there is indeed a modality-agnostic platonic representation, then the image data should help find it, and so should the language data. The implication is that if you want to train the best vision model, you should train not just onimages but also onsentences. This is already becoming common practice. Many vision models are finetuned from pre-trained LLMs. The other direction is less common, but also is implied by our hypothesis: if you want to build the best LLM,you should also train it on image data. Indeed,claim evidence that this is true, where training on images improved performance on text. In theory, there should be some conversion ratio: a pixel is worthwords for training LLMs, and a word is worthpixels for training vision models.Ease of translation and adaptation across modalitiesWhen two representations are aligned, transitioning from one to the other should be a simple function that’s easily obtained. Our hypothesis could explain the phenomenon that conditional generation is easier than unconditional, as the data we condition on may have the same platonic structure as the data we are generating. In line with this, recent work has found that representation-conditioning is even easier. Similarly, representational convergence could act as a bridge that lets us find mappings between domains even without paired data; this may underlie the success of unpaired translation in visionand language.
We emphasize that this doesn’t mean that models trained on a single modality (e.g., language)  can immediately process raw data from another (e.g., vision) . What makes them adaptable to the new modalities is that they share a common modality-agnostic representation, and can readily processrepresentationsof new modalities.
Furthermore, this implies that language models would achieve some notion of grounding in the visual domain even in the absence of cross-modal data.555In 1688, William Molyneux posed the question: could someone born blind, upon being given sight, be able to distinguish shapes by vision alone?Our arguments suggest an answer: not immediately, but after a bit of visual experience (to form a visual representation)  it should be easy (by mapping to prior touch-based representations) . Empirical data shows that indeed congentially blind children given sight can quickly learn such abilities.The primary advantage of cross-modal data could then simply be sample efficiency.Scaling may reduce hallucination and biasA prominent shortcoming of current LLMs is their propensity to hallucinate, or output false statements. If models are indeed converging toward an accurate model of reality, and scale powers this convergence, then we may expect hallucinations to decrease with scale. Of course, our hypothesis is conditioned on the training data for future models constituting a sufficiently lossless and diverse set of measurements. This may not come to pass, but it is an implication of our hypothesis worth pointing out. A similar argument can be made about certain kinds of bias. It has been shown that large models can exacerbate existing biases present in their training data. Our hypothesis implies that, while this may be true, we should expectlargermodels to amplify biasless. This does not mean bias will be removed, rather that the model’s biases will more accurately reflect the data’s biases, rather than exacerbating them.
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6Counterexamples and limitationsDifferent modalities may contain different informationOne immediate objection to our hypothesis is: what about the information that is unique to a given modality? Can language really describe the ineffable experience of watching a total solar eclipse? Or, how could an image convey the a concept like “I believe in the freedom of speech,” which is easy to write in English? Two different models cannot converge to the same representation if they have access to fundamentally different information.More precisely, our mathematical argument inSection4only strictly holds for bijective projections of, so that the information in all the projections is equivalent to the information in the underlying world. This will not hold true for either lossy or stochastic observation functions. Nonetheless, similar arguments have been made theoretically and empirically that cooccurrence relations are learned by practical contrastiveand predictive learners.andalso showed that models trained to autoregressively generate text also capture statistical relations in many other modalities, including symbolic reasoning,
vision, protein folding, and robotics.Figure 9:Increasing caption density improves alignment:We vary caption length using the Densely-Captioned-Images (DCI)  dataset. Starting from a dense caption, we used LLaMA3-8B-Instructto summarize and generate coarse-grained captions. We compute the average alignment score across all vision and language models with standard deviation measured over the language models we evaluated. With denser captions, the mapping may become more bijective, leading to improved language-vision alignment scores.A more nuanced version of our hypothesis will need to be developed to handle the case of non-bijective observations and abstract concepts. A starting point could be: different models will converge to the same representationwhen the input signals are sufficiently high information and the models are sufficiently high capacity; when they are not, the lower-information representation will only align with the higher-information one up to a level capped by the mutual information between the input signals and by the capacity of each model. This cap might or might not be practically important. Popular representations like CLIP are explicitly optimized to only capture the shared information between vision and language, yet are highly successful on many pure vision tasks.
We perform a preliminary test of the effect of information level in Figure9(detailed in AppendixE) , and find that the more descriptive (higher information)  a caption is, the better its LLM representation aligns with the visual representation of the corresponding image.Not all representations are presently convergingOur argument has mainly focused on two modalities: vision and language. While we do expect other modalities will follow similar trends, we have yet to see the same level of convergence across all domains. For example, in robotics there is not yet a standardized approach to representing world states in the same way as there is for representing images and text. One limitation lies in the hardware used in robotics, which is often expensive and slow. This creates a bottleneck in the quantity and diversity of training data.Sociological bias in producing AI modelsResearcher bias and collective preferences within the AI community have shaped the trajectory of model development. There is often an explicit or implicit goal of designing AI systems that mimic human reasoning and performance, and this could lead to convergence toward human-like representations even if other kinds of intelligence are in fact possible.
Additionally, the “hardware lottery”suggests that the success of AI models can also depend on the compatibility of their design with available computational architectures, further contributing to convergent trends.Special-purpose intelligences might not convergeDifferent intelligent systems can be designed to accomplish different tasks. For instance:
A bioinformatics systems might predict protein structure;
an autonomous vehicle might follow lanes on highways. It’s possible that not much is shared between these two narrow tasks. Our argument only holds for intelligences that are optimized to perform well onmanytasks. We have argued that a representation ofrealityis a structure that is useful across many tasks, but for any special purpose there may be shortcuts, or even effective representations detached from reality. Such shortcuts may be more efficient and necessary for continued improvements in specific domains. This will become more relevant if continued scaling comes up against boundary conditions around resources like energy and compute.How do we measure alignment?We focused on one particular alignment measure, mutual nearest-neighbor, in our experiments, and cited experiments using several others. However, there is active debate on the merits and deficiencies of all these ways of measuring alignment. We discuss our choice and show results for other alignment metrics in AppendixA.Lots left to explainWe have shown results where different models arrive atsimilarbut not thesamerepresentations. For example, inFigure3, alignment clearly increases but only reaches a score of, according to our mutual nearest-neighbor metric. The maximum theoretical value for this metric is. Is a score ofindicative of strong alignment with the remaining gap being “noise” or does it signify poor alignment with major differences left to explain? We leave this as an open question.
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AcknowledgementsWe thankfor sharing their data for our experiments shown inFigure8. We thank the anonymous reviewers for helpful feedback, and for providing the counterexample on how to visually convey “I believe in the freedom of speech.” Thanks for Yonglong Tian, Dilip Krishnan, Anna Decker, Yoon Kim, Jyo Pari, Ani Nrusimha, Dave Epstein, Victor Butoi, and Seungwook Han for helpful discussions and suggestions. This work was supported by a Packard Fellowship and a Sloan Research Fellowship to P.I., by the MIT-IBM Watson AI Lab, by ONR MURI grant N00014-22-1-2740, by the Center for Brains, Minds, and Machines, the MIT Quest for Intelligence, NSF STC award CCF-1231216, the DARPA Knowledge Management at Scale and Speed (KMASS)  program, and the DARPA Machine Common Sense (MCS)  program.
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