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class="vector-toc-numb">2</span> <span>Unit state probability</span> </div> </a> <ul id="toc-Unit_state_probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equilibrium_state" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equilibrium_state"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equilibrium state</span> </div> </a> <ul id="toc-Equilibrium_state-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Training" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Training"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Training</span> </div> </a> <ul id="toc-Training-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Problems</span> </div> </a> <ul id="toc-Problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Types</span> </div> </a> <button aria-controls="toc-Types-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types subsection</span> </button> <ul id="toc-Types-sublist" class="vector-toc-list"> <li id="toc-Restricted_Boltzmann_machine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Restricted_Boltzmann_machine"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Restricted Boltzmann machine</span> </div> </a> <ul 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class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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machine</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A0quina_de_Boltzmann" title="Màquina de Boltzmann – Catalan" lang="ca" hreflang="ca" data-title="Màquina de Boltzmann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Boltzmann-Maschine" title="Boltzmann-Maschine – German" lang="de" hreflang="de" data-title="Boltzmann-Maschine" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A1quina_de_Boltzmann" title="Máquina de Boltzmann – Spanish" lang="es" hreflang="es" data-title="Máquina de Boltzmann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%B4%DB%8C%D9%86_%D8%A8%D9%88%D9%84%D8%AA%D8%B3%D9%85%D8%A7%D9%86" title="ماشین بولتسمان – Persian" lang="fa" hreflang="fa" data-title="ماشین بولتسمان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%BC%EC%B8%A0%EB%A7%8C_%EB%A8%B8%EC%8B%A0" title="볼츠만 머신 – Korean" lang="ko" hreflang="ko" data-title="볼츠만 머신" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%95%D7%A0%D7%AA_%D7%91%D7%95%D7%9C%D7%A6%D7%9E%D7%9F" title="מכונת בולצמן – Hebrew" lang="he" hreflang="he" data-title="מכונת בולצמן" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8B%E0%B5%BE%E0%B4%B8%E0%B5%8D%E0%B4%AE%E0%B4%BE%E0%B5%BB_%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ബോൾസ്മാൻ യന്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="ബോൾസ്മാൻ യന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%AB%E3%83%84%E3%83%9E%E3%83%B3%E3%83%9E%E3%82%B7%E3%83%B3" title="ボルツマンマシン – Japanese" lang="ja" hreflang="ja" data-title="ボルツマンマシン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%88%D0%B8%D0%BD%D0%B0_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Машина Больцмана – Russian" lang="ru" hreflang="ru" data-title="Машина Больцмана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Makina_Boltzmann" title="Makina Boltzmann – Albanian" lang="sq" hreflang="sq" data-title="Makina Boltzmann" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Boltzmannin_kone" title="Boltzmannin kone – Finnish" lang="fi" hreflang="fi" data-title="Boltzmannin kone" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%88%D0%B8%D0%BD%D0%B0_%D0%91%D0%BE%D0%BB%D1%8C%D1%86%D0%BC%D0%B0%D0%BD%D0%B0" title="Машина Больцмана – Ukrainian" lang="uk" hreflang="uk" data-title="Машина Больцмана" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%A1y_Boltzmann" title="Máy Boltzmann – Vietnamese" lang="vi" hreflang="vi" data-title="Máy Boltzmann" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%8E%BB%E8%8C%B2%E6%9B%BC%E6%A9%9F" title="玻茲曼機 – Cantonese" lang="yue" hreflang="yue" data-title="玻茲曼機" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E5%85%B9%E6%9B%BC%E6%9C%BA" title="玻尔兹曼机 – Chinese" lang="zh" hreflang="zh" data-title="玻尔兹曼机" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of stochastic recurrent neural network</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Boltzmannexamplev1.png" class="mw-file-description"><img alt="A graphical representation of an example Boltzmann machine." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Boltzmannexamplev1.png/220px-Boltzmannexamplev1.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Boltzmannexamplev1.png/330px-Boltzmannexamplev1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7a/Boltzmannexamplev1.png 2x" data-file-width="433" data-file-height="411" /></a><figcaption> A graphical representation of an example Boltzmann machine. Each undirected edge represents dependency. In this example there are 3 hidden units and 4 visible units. This is not a restricted Boltzmann machine.</figcaption></figure> <p>A <b>Boltzmann machine</b> (also called <b>Sherrington–Kirkpatrick model with external field</b> or <b>stochastic Ising model</b>), named after <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> is a <a href="/wiki/Spin_glass" title="Spin glass">spin-glass</a> model with an external field, i.e., a <a href="/wiki/Spin_glass#Sherrington–Kirkpatrick_model" title="Spin glass">Sherrington–Kirkpatrick model</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> that is a stochastic <a href="/wiki/Ising_model" title="Ising model">Ising model</a>. It is a <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a> technique applied in the context of <a href="/wiki/Cognitive_science" title="Cognitive science">cognitive science</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is also classified as a <a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Boltzmann machines are theoretically intriguing because of the locality and <a href="/wiki/Hebbian" class="mw-redirect" title="Hebbian">Hebbian</a> nature of their training algorithm (being trained by Hebb's rule), and because of their <a href="/wiki/Parallelism_(computing)" class="mw-redirect" title="Parallelism (computing)">parallelism</a> and the resemblance of their dynamics to simple <a href="/wiki/Physical_process" class="mw-redirect" title="Physical process">physical processes</a>. Boltzmann machines with unconstrained connectivity have not been proven useful for practical problems in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> or <a href="/wiki/Inference" title="Inference">inference</a>, but if the connectivity is properly constrained, the learning can be made efficient enough to be useful for practical problems.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>They are named after the <a href="/wiki/Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a> in <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, which is used in their <a href="/wiki/Sampling_function" class="mw-redirect" title="Sampling function">sampling function</a>. They were heavily popularized and promoted by <a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a>, <a href="/wiki/Terry_Sejnowski" title="Terry Sejnowski">Terry Sejnowski</a> and <a href="/wiki/Yann_LeCun" title="Yann LeCun">Yann LeCun</a> in cognitive sciences communities, particularly in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>,<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as part of "<a href="/wiki/Energy-based_model" title="Energy-based model">energy-based models</a>" (EBM), because <a href="/wiki/Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonians</a> of <a href="/wiki/Spin_glasses" class="mw-redirect" title="Spin glasses">spin glasses</a> as energy are used as a starting point to define the learning task.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Structure">Structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=1" title="Edit section: Structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boltzmannexamplev2.png" class="mw-file-description"><img alt="A graphical representation of an example Boltzmann machine with weight labels." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Boltzmannexamplev2.png/220px-Boltzmannexamplev2.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Boltzmannexamplev2.png/330px-Boltzmannexamplev2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/fd/Boltzmannexamplev2.png 2x" data-file-width="433" data-file-height="411" /></a><figcaption> A graphical representation of a Boltzmann machine with a few weights labeled. Each undirected edge represents dependency and is weighted with weight <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3302ff355269436b43bc2fbe180303881c09321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.141ex; height:2.343ex;" alt="{\displaystyle w_{ij}}"></span>. In this example there are 3 hidden units (blue) and 4 visible units (white). This is not a restricted Boltzmann machine.</figcaption></figure> <p>A Boltzmann machine, like a <a href="/wiki/Spin_glass#Sherrington–Kirkpatrick_model" title="Spin glass">Sherrington–Kirkpatrick model</a>, is a network of units with a total "energy" (<a href="/wiki/Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian</a>) defined for the overall network. Its units produce <a href="/wiki/Binary_number" title="Binary number">binary</a> results. Boltzmann machine weights are <a href="/wiki/Stochastic" title="Stochastic">stochastic</a>. The global energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> in a Boltzmann machine is identical in form to that of <a href="/wiki/Hopfield_network" title="Hopfield network">Hopfield networks</a> and <a href="/wiki/Ising_model" title="Ising model">Ising models</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=-\left(\sum _{i<j}w_{ij}\,s_{i}\,s_{j}+\sum _{i}\theta _{i}\,s_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=-\left(\sum _{i<j}w_{ij}\,s_{i}\,s_{j}+\sum _{i}\theta _{i}\,s_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419a4966dae86dfb786dbeb6aadd3c38d9ec30d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.048ex; height:7.676ex;" alt="{\displaystyle E=-\left(\sum _{i<j}w_{ij}\,s_{i}\,s_{j}+\sum _{i}\theta _{i}\,s_{i}\right)}"></span></dd></dl> <p>Where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3302ff355269436b43bc2fbe180303881c09321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.141ex; height:2.343ex;" alt="{\displaystyle w_{ij}}"></span> is the connection strength between unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> and unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"></span> is the state, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}\in \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}\in \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9da1e0bc5f6f9d7f021287963bbb8b3ada491df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.415ex; height:2.843ex;" alt="{\displaystyle s_{i}\in \{0,1\}}"></span>, of unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"></span> is the bias of unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in the global energy function. (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183e14708810963ae88dbef1ab84f12367bc8574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.698ex; height:2.509ex;" alt="{\displaystyle -\theta _{i}}"></span> is the activation threshold for the unit.)</li></ul> <p>Often the weights <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3302ff355269436b43bc2fbe180303881c09321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.141ex; height:2.343ex;" alt="{\displaystyle w_{ij}}"></span> are represented as a symmetric matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=[w_{ij}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=[w_{ij}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9447624ebb1e3c8ba2dacefbb62681ea40590e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.969ex; height:3.009ex;" alt="{\displaystyle W=[w_{ij}]}"></span> with zeros along the diagonal. </p> <div class="mw-heading mw-heading2"><h2 id="Unit_state_probability">Unit state probability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=2" title="Edit section: Unit state probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The difference in the global energy that results from a single unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> equaling 0 (off) versus 1 (on), written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa40fa108e21d148e5a39c6202cf9a2ee69cb1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.451ex; height:2.509ex;" alt="{\displaystyle \Delta E_{i}}"></span>, assuming a symmetric matrix of weights, is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E_{i}=\sum _{j>i}w_{ij}\,s_{j}+\sum _{j<i}w_{ji}\,s_{j}+\theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>></mo> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo><</mo> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E_{i}=\sum _{j>i}w_{ij}\,s_{j}+\sum _{j<i}w_{ji}\,s_{j}+\theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cad4ed8631921945d7056276eaeb5e0246e7ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.661ex; height:5.843ex;" alt="{\displaystyle \Delta E_{i}=\sum _{j>i}w_{ij}\,s_{j}+\sum _{j<i}w_{ji}\,s_{j}+\theta _{i}}"></span></dd></dl> <p>This can be expressed as the difference of energies of two states: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E_{i}=E_{\text{i=off}}-E_{\text{i=on}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i=off</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i=on</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E_{i}=E_{\text{i=off}}-E_{\text{i=on}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de089322fde630248e5e9f0d4beac2a10374385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.321ex; height:2.509ex;" alt="{\displaystyle \Delta E_{i}=E_{\text{i=off}}-E_{\text{i=on}}}"></span></dd></dl> <p>Substituting the energy of each state with its relative probability according to the <a href="/wiki/Boltzmann_factor" class="mw-redirect" title="Boltzmann factor">Boltzmann factor</a> (the property of a <a href="/wiki/Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a> that the energy of a state is proportional to the negative log probability of that state) yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E_{i}=-k_{B}T\ln(p_{\text{i=off}})-(-k_{B}T\ln(p_{\text{i=on}})),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i=off</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i=on</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E_{i}=-k_{B}T\ln(p_{\text{i=off}})-(-k_{B}T\ln(p_{\text{i=on}})),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a659c1ebd238cac1d461f4d59c9f8f58c877db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.228ex; height:2.843ex;" alt="{\displaystyle \Delta E_{i}=-k_{B}T\ln(p_{\text{i=off}})-(-k_{B}T\ln(p_{\text{i=on}})),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.691ex; height:2.509ex;" alt="{\displaystyle k_{B}}"></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> and is absorbed into the artificial notion of temperature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. Noting that the probabilities of the unit being <i>on</i> or <i>off</i> sum to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> allows for the simplification: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\Delta E_{i}}{k_{B}T}}=-\ln(p_{i={\text{on}}})+\ln(p_{i={\text{off}}})=\ln {\Big (}{\frac {1-p_{i={\text{on}}}}{p_{i={\text{on}}}}}{\Big )}=\ln(p_{i={\text{on}}}^{-1}-1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>on</mtext> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>off</mtext> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>on</mtext> </mrow> </mrow> </msub> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>on</mtext> </mrow> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>on</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\Delta E_{i}}{k_{B}T}}=-\ln(p_{i={\text{on}}})+\ln(p_{i={\text{off}}})=\ln {\Big (}{\frac {1-p_{i={\text{on}}}}{p_{i={\text{on}}}}}{\Big )}=\ln(p_{i={\text{on}}}^{-1}-1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7355672491b6d4939c8d254e94fff9987705e089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.29ex; height:5.843ex;" alt="{\displaystyle -{\frac {\Delta E_{i}}{k_{B}T}}=-\ln(p_{i={\text{on}}})+\ln(p_{i={\text{off}}})=\ln {\Big (}{\frac {1-p_{i={\text{on}}}}{p_{i={\text{on}}}}}{\Big )}=\ln(p_{i={\text{on}}}^{-1}-1),}"></span></dd></dl> <p>whence the probability that the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th unit is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i={\text{on}}}={\frac {1}{1+\exp {\Big (}-{\frac {\Delta E_{i}}{k_{B}T}}{\Big )}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>on</mtext> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i={\text{on}}}={\frac {1}{1+\exp {\Big (}-{\frac {\Delta E_{i}}{k_{B}T}}{\Big )}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6a2114bd73162013561f5c136569c57bdf88fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; margin-left: -0.089ex; width:27.255ex; height:7.843ex;" alt="{\displaystyle p_{i={\text{on}}}={\frac {1}{1+\exp {\Big (}-{\frac {\Delta E_{i}}{k_{B}T}}{\Big )}}},}"></span></dd></dl> <p>where the <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is referred to as the <a href="/wiki/Temperature" title="Temperature">temperature</a> of the system. This relation is the source of the <a href="/wiki/Logistic_function" title="Logistic function">logistic function</a> found in probability expressions in variants of the Boltzmann machine. </p> <div class="mw-heading mw-heading2"><h2 id="Equilibrium_state">Equilibrium state</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=3" title="Edit section: Equilibrium state"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The network runs by repeatedly choosing a unit and resetting its state. After running for long enough at a certain temperature, the probability of a global state of the network depends only upon that global state's energy, according to a <a href="/wiki/Boltzmann_distribution" title="Boltzmann distribution">Boltzmann distribution</a>, and not on the initial state from which the process was started. This means that log-probabilities of global states become linear in their energies. This relationship is true when the machine is "at <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a>", meaning that the probability distribution of global states has converged. Running the network beginning from a high temperature, its temperature gradually decreases until reaching a <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a> at a lower temperature. It then may converge to a distribution where the energy level fluctuates around the global minimum. This process is called <a href="/wiki/Simulated_annealing" title="Simulated annealing">simulated annealing</a>. </p><p>To train the network so that the chance it will converge to a global state according to an external distribution over these states, the weights must be set so that the global states with the highest probabilities get the lowest energies. This is done by training. </p> <div class="mw-heading mw-heading2"><h2 id="Training">Training</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=4" title="Edit section: Training"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The units in the Boltzmann machine are divided into 'visible' units, V, and 'hidden' units, H. The visible units are those that receive information from the 'environment', i.e. the <a href="/wiki/Training_set" class="mw-redirect" title="Training set">training set</a> is a set of binary vectors over the set V. The distribution over the training set is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{+}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{+}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81e1d9ec5c805c37e88c74b74de6707bc7cefea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.929ex; height:3.009ex;" alt="{\displaystyle P^{+}(V)}"></span>. </p><p>The distribution over global states converges as the Boltzmann machine reaches <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a>. We denote this distribution, after we <a href="/wiki/Marginal_distribution" title="Marginal distribution">marginalize</a> it over the hidden units, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4460af507ffbc4f4cae979715fc21e0ab247b224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.929ex; height:3.009ex;" alt="{\displaystyle P^{-}(V)}"></span>. </p><p>Our goal is to approximate the "real" distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{+}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{+}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81e1d9ec5c805c37e88c74b74de6707bc7cefea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.929ex; height:3.009ex;" alt="{\displaystyle P^{+}(V)}"></span> using the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4460af507ffbc4f4cae979715fc21e0ab247b224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.929ex; height:3.009ex;" alt="{\displaystyle P^{-}(V)}"></span> produced by the machine. The similarity of the two distributions is measured by the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\sum _{v}{P^{+}(v)\ln \left({\frac {P^{+}(v)}{P^{-}(v)}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\sum _{v}{P^{+}(v)\ln \left({\frac {P^{+}(v)}{P^{-}(v)}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f7443db38ab124fe9ba987dedd47b3c8ece817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.789ex; height:6.843ex;" alt="{\displaystyle G=\sum _{v}{P^{+}(v)\ln \left({\frac {P^{+}(v)}{P^{-}(v)}}\right)}}"></span></dd></dl> <p>where the sum is over all the possible states of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is a function of the weights, since they determine the energy of a state, and the energy determines <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a26be27335709582cbbf27e2ebbce24c49ae32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.269ex; height:3.009ex;" alt="{\displaystyle P^{-}(v)}"></span>, as promised by the Boltzmann distribution. A <a href="/wiki/Gradient_descent" title="Gradient descent">gradient descent</a> algorithm over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> changes a given weight, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3302ff355269436b43bc2fbe180303881c09321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.141ex; height:2.343ex;" alt="{\displaystyle w_{ij}}"></span>, by subtracting the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> with respect to the weight. </p><p>Boltzmann machine training involves two alternating phases. One is the "positive" phase where the visible units' states are clamped to a particular binary state vector sampled from the training set (according to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7161b1309d204da1254c6fdeeec8f54d5e8c71c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle P^{+}}"></span>). The other is the "negative" phase where the network is allowed to run freely, i.e. only the input nodes have their state determined by external data, but the output nodes are allowed to float. The gradient with respect to a given weight, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3302ff355269436b43bc2fbe180303881c09321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.141ex; height:2.343ex;" alt="{\displaystyle w_{ij}}"></span>, is given by the equation:<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {G}}{\partial {w_{ij}}}}=-{\frac {1}{R}}[p_{ij}^{+}-p_{ij}^{-}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {G}}{\partial {w_{ij}}}}=-{\frac {1}{R}}[p_{ij}^{+}-p_{ij}^{-}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c83d95b86d28bcede06faaa1df7de1e8553c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.297ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial {G}}{\partial {w_{ij}}}}=-{\frac {1}{R}}[p_{ij}^{+}-p_{ij}^{-}]}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{ij}^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c5a00aaf2b5672976a1638e5c99fc4cece6db8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:2.77ex; height:3.509ex;" alt="{\displaystyle p_{ij}^{+}}"></span> is the probability that units <i>i</i> and <i>j</i> are both on when the machine is at equilibrium on the positive phase.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ij}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{ij}^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c799b8085080db94e362f3fc1671c5ce13c91daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:2.77ex; height:3.509ex;" alt="{\displaystyle p_{ij}^{-}}"></span> is the probability that units <i>i</i> and <i>j</i> are both on when the machine is at equilibrium on the negative phase.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> denotes the <a href="/wiki/Learning_rate" title="Learning rate">learning rate</a></li></ul> <p>This result follows from the fact that at <a href="/wiki/Thermal_equilibrium" title="Thermal equilibrium">thermal equilibrium</a> the probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deff9b67b67a1dccd06581b640f82f0643c29de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:3.009ex;" alt="{\displaystyle P^{-}(s)}"></span> of any global state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> when the network is free-running is given by the Boltzmann distribution. </p><p>This learning rule is biologically plausible because the only information needed to change the weights is provided by "local" information. That is, the connection (<a href="/wiki/Synapse" title="Synapse">synapse</a>, biologically) does not need information about anything other than the two neurons it connects. This is more biologically realistic than the information needed by a connection in many other neural network training algorithms, such as <a href="/wiki/Backpropagation" title="Backpropagation">backpropagation</a>. </p><p>The training of a Boltzmann machine does not use the <a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">EM algorithm</a>, which is heavily used in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>. By minimizing the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">KL-divergence</a>, it is equivalent to maximizing the log-likelihood of the data. Therefore, the training procedure performs gradient ascent on the log-likelihood of the observed data. This is in contrast to the EM algorithm, where the posterior distribution of the hidden nodes must be calculated before the maximization of the expected value of the complete data likelihood during the M-step. </p><p>Training the biases is similar, but uses only single node activity: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {G}}{\partial {\theta _{i}}}}=-{\frac {1}{R}}[p_{i}^{+}-p_{i}^{-}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {G}}{\partial {\theta _{i}}}}=-{\frac {1}{R}}[p_{i}^{+}-p_{i}^{-}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d432e59f276d5ade2f2d2a4b95071eadb57711b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.046ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial {G}}{\partial {\theta _{i}}}}=-{\frac {1}{R}}[p_{i}^{+}-p_{i}^{-}]}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Problems">Problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=5" title="Edit section: Problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Theoretically the Boltzmann machine is a rather general computational medium. For instance, if trained on photographs, the machine would theoretically model the distribution of photographs, and could use that model to, for example, <a href="/wiki/Inpainting" title="Inpainting">complete</a> a partial photograph. </p><p>Unfortunately, Boltzmann machines experience a serious practical problem, namely that it seems to stop learning correctly when the machine is scaled up to anything larger than a trivial size.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2013)">citation needed</span></a></i>]</sup> This is due to important effects, specifically: </p> <ul><li>the required time order to collect equilibrium statistics grows exponentially with the machine's size, and with the magnitude of the connection strengths<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2015)">citation needed</span></a></i>]</sup></li> <li>connection strengths are more plastic when the connected units have activation probabilities intermediate between zero and one, leading to a so-called variance trap. The net effect is that noise causes the connection strengths to follow a <a href="/wiki/Random_walk" title="Random walk">random walk</a> until the activities saturate.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=6" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Restricted_Boltzmann_machine">Restricted Boltzmann machine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=7" title="Edit section: Restricted Boltzmann machine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Restricted_Boltzmann_machine.svg" class="mw-file-description"><img alt="Graphical representation of an example restricted Boltzmann machine" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Restricted_Boltzmann_machine.svg/220px-Restricted_Boltzmann_machine.svg.png" decoding="async" width="220" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Restricted_Boltzmann_machine.svg/330px-Restricted_Boltzmann_machine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Restricted_Boltzmann_machine.svg/440px-Restricted_Boltzmann_machine.svg.png 2x" data-file-width="331" data-file-height="352" /></a><figcaption>Graphical representation of a restricted Boltzmann machine. The four blue units represent hidden units, and the three red units represent visible states. In restricted Boltzmann machines there are only connections (dependencies) between hidden and visible units, and none between units of the same type (no hidden-hidden, nor visible-visible connections).</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted Boltzmann machine</a></div> <p>Although learning is impractical in general Boltzmann machines, it can be made quite efficient in a restricted Boltzmann machine (RBM) which does not allow intralayer connections between hidden units and visible units, i.e. there is no connection between visible to visible and hidden to hidden units. After training one RBM, the activities of its hidden units can be treated as data for training a higher-level RBM. This method of stacking RBMs makes it possible to train many layers of hidden units efficiently and is one of the most common <a href="/wiki/Deep_learning" title="Deep learning">deep learning</a> strategies. As each new layer is added the generative model improves. </p><p>An extension to the restricted Boltzmann machine allows using real valued data rather than binary data.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>One example of a practical RBM application is in speech recognition.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Deep_Boltzmann_machine">Deep Boltzmann machine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=8" title="Edit section: Deep Boltzmann machine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A deep Boltzmann machine (DBM) is a type of binary pairwise <a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a> (<a href="/wiki/Graph_(discrete_mathematics)#Undirected_graph" title="Graph (discrete mathematics)">undirected</a> probabilistic <a href="/wiki/Graphical_model" title="Graphical model">graphical model</a>) with multiple layers of <a href="/wiki/Latent_variable" class="mw-redirect" title="Latent variable">hidden</a> <a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">random variables</a>. It is a network of symmetrically coupled stochastic <a href="/wiki/Binary_variable" class="mw-redirect" title="Binary variable">binary units</a>. It comprises a set of visible units <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nu }}\in \{0,1\}^{D}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nu }}\in \{0,1\}^{D}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc1e1173a24999ee667aabd24123b15a6230ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.531ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\nu }}\in \{0,1\}^{D}}"></span> and layers of hidden units <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {h}}^{(1)}\in \{0,1\}^{F_{1}},{\boldsymbol {h}}^{(2)}\in \{0,1\}^{F_{2}},\ldots ,{\boldsymbol {h}}^{(L)}\in \{0,1\}^{F_{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {h}}^{(1)}\in \{0,1\}^{F_{1}},{\boldsymbol {h}}^{(2)}\in \{0,1\}^{F_{2}},\ldots ,{\boldsymbol {h}}^{(L)}\in \{0,1\}^{F_{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6c9c277d415b9192f0c2d0eb291c3e1c1dbead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.345ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {h}}^{(1)}\in \{0,1\}^{F_{1}},{\boldsymbol {h}}^{(2)}\in \{0,1\}^{F_{2}},\ldots ,{\boldsymbol {h}}^{(L)}\in \{0,1\}^{F_{L}}}"></span>. No connection links units of the same layer (like <a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBM</a>). For the <style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><span class="rt-commentedText tooltip tooltip-dotted" title="Deep Boltzmann machine">DBM</span>, the probability assigned to vector <span class="texhtml mvar" style="font-style:italic;"><b>ν</b></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\boldsymbol {\nu }})={\frac {1}{Z}}\sum _{h}e^{\sum _{ij}W_{ij}^{(1)}\nu _{i}h_{j}^{(1)}+\sum _{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum _{lm}W_{lm}^{(3)}h_{l}^{(2)}h_{m}^{(3)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </munder> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>m</mi> </mrow> </munder> <msubsup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\boldsymbol {\nu }})={\frac {1}{Z}}\sum _{h}e^{\sum _{ij}W_{ij}^{(1)}\nu _{i}h_{j}^{(1)}+\sum _{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum _{lm}W_{lm}^{(3)}h_{l}^{(2)}h_{m}^{(3)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1119755948127041b7d6c5514472673c31bb9fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:55.585ex; height:6.343ex;" alt="{\displaystyle p({\boldsymbol {\nu }})={\frac {1}{Z}}\sum _{h}e^{\sum _{ij}W_{ij}^{(1)}\nu _{i}h_{j}^{(1)}+\sum _{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum _{lm}W_{lm}^{(3)}h_{l}^{(2)}h_{m}^{(3)}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {h}}=\{{\boldsymbol {h}}^{(1)},{\boldsymbol {h}}^{(2)},{\boldsymbol {h}}^{(3)}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {h}}=\{{\boldsymbol {h}}^{(1)},{\boldsymbol {h}}^{(2)},{\boldsymbol {h}}^{(3)}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17868581179807cb175255547b771f20577c9fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.703ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {h}}=\{{\boldsymbol {h}}^{(1)},{\boldsymbol {h}}^{(2)},{\boldsymbol {h}}^{(3)}\}}"></span> are the set of hidden units, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\{{\boldsymbol {W}}^{(1)},{\boldsymbol {W}}^{(2)},{\boldsymbol {W}}^{(3)}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\{{\boldsymbol {W}}^{(1)},{\boldsymbol {W}}^{(2)},{\boldsymbol {W}}^{(3)}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b678595c79032e3630c1f5fd2e2d6da51eaa8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.235ex; height:3.343ex;" alt="{\displaystyle \theta =\{{\boldsymbol {W}}^{(1)},{\boldsymbol {W}}^{(2)},{\boldsymbol {W}}^{(3)}\}}"></span> are the model parameters, representing visible-hidden and hidden-hidden interactions.<sup id="cite_ref-ref12_8-0" class="reference"><a href="#cite_note-ref12-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In a <a href="/wiki/Deep_belief_network" title="Deep belief network">DBN</a> only the top two layers form a restricted Boltzmann machine (which is an undirected <a href="/wiki/Graphical_model" title="Graphical model">graphical model</a>), while lower layers form a directed generative model. In a DBM all layers are symmetric and undirected. </p><p>Like <a href="/wiki/Deep_belief_network" title="Deep belief network">DBNs</a>, DBMs can learn complex and abstract internal representations of the input in tasks such as <a href="/wiki/Object_recognition" class="mw-redirect" title="Object recognition">object</a> or <a href="/wiki/Speech_recognition" title="Speech recognition">speech recognition</a>, using limited, labeled data to fine-tune the representations built using a large set of unlabeled sensory input data. However, unlike DBNs and deep <a href="/wiki/Convolutional_neural_networks" class="mw-redirect" title="Convolutional neural networks">convolutional neural networks</a>, they pursue the inference and training procedure in both directions, bottom-up and top-down, which allow the DBM to better unveil the representations of the input structures.<sup id="cite_ref-ref32_9-0" class="reference"><a href="#cite_note-ref32-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ref42_10-0" class="reference"><a href="#cite_note-ref42-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ref22_11-0" class="reference"><a href="#cite_note-ref22-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>However, the slow speed of DBMs limits their performance and functionality. Because exact maximum likelihood learning is intractable for DBMs, only approximate maximum likelihood learning is possible. Another option is to use mean-field inference to estimate data-dependent expectations and approximate the expected sufficient statistics by using <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> (MCMC).<sup id="cite_ref-ref12_8-1" class="reference"><a href="#cite_note-ref12-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This approximate inference, which must be done for each test input, is about 25 to 50 times slower than a single bottom-up pass in DBMs. This makes joint optimization impractical for large data sets, and restricts the use of DBMs for tasks such as feature representation. </p> <div class="mw-heading mw-heading3"><h3 id="Spike-and-slab_RBMs">Spike-and-slab RBMs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=9" title="Edit section: Spike-and-slab RBMs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The need for deep learning with <a href="/wiki/Real_number" title="Real number">real-valued</a> inputs, as in <a href="/wiki/Gaussian" class="mw-redirect" title="Gaussian">Gaussian</a> RBMs, led to the spike-and-slab <a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBM</a> (<i>ss</i><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBM</a>), which models continuous-valued inputs with <a href="/wiki/Binary_variable" class="mw-redirect" title="Binary variable">binary</a> <a href="/wiki/Latent_variable" class="mw-redirect" title="Latent variable">latent variables</a>.<sup id="cite_ref-ref30_12-0" class="reference"><a href="#cite_note-ref30-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Similar to basic <a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBMs</a> and its variants, a spike-and-slab RBM is a <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graph</a>, while like G<a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBMs</a>, the visible units (input) are real-valued. The difference is in the hidden layer, where each hidden unit has a binary spike variable and a real-valued slab variable. A spike is a discrete <a href="/wiki/Probability_mass" class="mw-redirect" title="Probability mass">probability mass</a> at zero, while a slab is a <a href="/wiki/Probability_density" class="mw-redirect" title="Probability density">density</a> over continuous domain;<sup id="cite_ref-ref322_13-0" class="reference"><a href="#cite_note-ref322-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> their mixture forms a <a href="/wiki/Prior_probability" title="Prior probability">prior</a>.<sup id="cite_ref-ref31_14-0" class="reference"><a href="#cite_note-ref31-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>An extension of ss<a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBM</a> called μ-ss<a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">RBM</a> provides extra modeling capacity using additional terms in the <a href="/wiki/Energy_function" class="mw-redirect" title="Energy function">energy function</a>. One of these terms enables the model to form a <a href="/wiki/Conditional_probability_distribution" title="Conditional probability distribution">conditional distribution</a> of the spike variables by <a href="/wiki/Marginalizing_out" class="mw-redirect" title="Marginalizing out">marginalizing out</a> the slab variables given an observation. </p> <div class="mw-heading mw-heading3"><h3 id="In_mathematics">In mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=10" title="Edit section: In mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a> and <a href="/wiki/Log-linear_model" title="Log-linear model">Log-linear model</a></div> <p>In more general mathematical setting, the Boltzmann distribution is also known as the <a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a>. In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> it is called a <a href="/wiki/Log-linear_model" title="Log-linear model">log-linear model</a>. In <a href="/wiki/Deep_learning" title="Deep learning">deep learning</a> the Boltzmann distribution is used in the sampling distribution of <a href="/wiki/Stochastic_neural_network" class="mw-redirect" title="Stochastic neural network">stochastic neural networks</a> such as the Boltzmann machine. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=11" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Boltzmann machine is based on the Sherrington–Kirkpatrick <a href="/wiki/Spin_glass" title="Spin glass">spin glass</a> model by <a href="/wiki/David_Sherrington_(physicist)" title="David Sherrington (physicist)">David Sherrington</a> and <a href="/wiki/Scott_Kirkpatrick" title="Scott Kirkpatrick">Scott Kirkpatrick</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The seminal publication by <a href="/wiki/John_Hopfield" title="John Hopfield">John Hopfield</a> (1982) applied methods of statistical mechanics, mainly the recently developed (1970s) theory of spin glasses, to study <a href="/wiki/Hopfield_network" title="Hopfield network">associative memory</a> (later named the "Hopfield network").<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The original contribution in applying such energy-based models in cognitive science appeared in papers by <a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a> and <a href="/wiki/Terry_Sejnowski" title="Terry Sejnowski">Terry Sejnowski</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In a 1995 interview, Hinton stated that in 1983 February or March, he was going to give a talk on <a href="/wiki/Simulated_annealing" title="Simulated annealing">simulated annealing</a> in Hopfield networks, so he had to design a learning algorithm for the talk, resulting in the Boltzmann machine learning algorithm.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The idea of applying the Ising model with annealed <a href="/wiki/Gibbs_sampling" title="Gibbs sampling">Gibbs sampling</a> was used in <a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>'s <a href="/wiki/Copycat_(software)" title="Copycat (software)">Copycat</a> project (1984).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>The explicit analogy drawn with statistical mechanics in the Boltzmann machine formulation led to the use of terminology borrowed from physics (e.g., "energy"), which became standard in the field. The widespread adoption of this terminology may have been encouraged by the fact that its use led to the adoption of a variety of concepts and methods from statistical mechanics. The various proposals to use simulated annealing for inference were apparently independent. </p><p>Similar ideas (with a change of sign in the energy function) are found in <a href="/wiki/Paul_Smolensky" title="Paul Smolensky">Paul Smolensky</a>'s "Harmony Theory".<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Ising models can be generalized to <a href="/wiki/Markov_random_field" title="Markov random field">Markov random fields</a>, which find widespread application in <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, <a href="/wiki/Robotics" title="Robotics">robotics</a>, <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a> and <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>. </p><p>In 2024, Hopfield and Hinton were awarded <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for their foundational contributions to <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, such as the Boltzmann machine.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted Boltzmann machine</a></li> <li><a href="/wiki/Helmholtz_machine" title="Helmholtz machine">Helmholtz machine</a></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a> (MRF)</li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> (Lenz–Ising model)</li> <li><a href="/wiki/Hopfield_network" title="Hopfield network">Hopfield network</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSherrington,_DavidKirkpatrick,_Scott1975" class="citation cs2">Sherrington, David; Kirkpatrick, Scott (1975), "Solvable Model of a Spin-Glass", <i>Physical Review Letters</i>, <b>35</b> (35): 1792–1796, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975PhRvL..35.1792S">1975PhRvL..35.1792S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.35.1792">10.1103/PhysRevLett.35.1792</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Solvable+Model+of+a+Spin-Glass&rft.volume=35&rft.issue=35&rft.pages=1792-1796&rft.date=1975&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.35.1792&rft_id=info%3Abibcode%2F1975PhRvL..35.1792S&rft.au=Sherrington%2C+David&rft.au=Kirkpatrick%2C+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAckleyHinton,_Geoffrey_E.Sejnowski,_Terrence_J.1985" class="citation journal cs1">Ackley, David H.; Hinton, Geoffrey E.; Sejnowski, Terrence J. (1985). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110718022336/http://learning.cs.toronto.edu/~hinton/absps/cogscibm.pdf">"A Learning Algorithm for Boltzmann Machines"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Cognitive_Science_(journal)" title="Cognitive Science (journal)">Cognitive Science</a></i>. <b>9</b> (1): 147–169. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1207%2Fs15516709cog0901_7">10.1207/s15516709cog0901_7</a></span>. Archived from <a rel="nofollow" class="external text" href="http://learning.cs.toronto.edu/~hinton/absps/cogscibm.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 18 July 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cognitive+Science&rft.atitle=A+Learning+Algorithm+for+Boltzmann+Machines&rft.volume=9&rft.issue=1&rft.pages=147-169&rft.date=1985&rft_id=info%3Adoi%2F10.1207%2Fs15516709cog0901_7&rft.aulast=Ackley&rft.aufirst=David+H.&rft.au=Hinton%2C+Geoffrey+E.&rft.au=Sejnowski%2C+Terrence+J.&rft_id=http%3A%2F%2Flearning.cs.toronto.edu%2F~hinton%2Fabsps%2Fcogscibm.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinton2007" class="citation journal cs1">Hinton, Geoffrey E. (2007-05-24). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.1668">"Boltzmann machine"</a>. <i>Scholarpedia</i>. <b>2</b> (5): 1668. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007SchpJ...2.1668H">2007SchpJ...2.1668H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.1668">10.4249/scholarpedia.1668</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1941-6016">1941-6016</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Boltzmann+machine&rft.volume=2&rft.issue=5&rft.pages=1668&rft.date=2007-05-24&rft.issn=1941-6016&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.1668&rft_id=info%3Abibcode%2F2007SchpJ...2.1668H&rft.aulast=Hinton&rft.aufirst=Geoffrey+E.&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.1668&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsborn1990" class="citation book cs1">Osborn, Thomas R. 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Retrieved <span class="nowrap">17 February</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Analyzing+Cooperative+Computation&rft.place=Rochester%2C+New+York&rft.date=1983-05&rft.aulast=Hinton&rft.aufirst=Geoffery&rft.au=Sejnowski%2C+Terrence+J.&rft_id=http%3A%2F%2Fdigitalcollections.library.cmu.edu%2Fawweb%2Fawarchive%3Ftype%3Dfile%26item%3D360445&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged June 2024">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintonSejnowski1983" class="citation conference cs1">Hinton, Geoffrey E.; Sejnowski, Terrence J. (June 1983). <i>Optimal Perceptual Inference</i>. IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Washington, D.C.: IEEE Computer Society. pp. 448–453.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Optimal+Perceptual+Inference&rft.place=Washington%2C+D.C.&rft.pages=448-453&rft.pub=IEEE+Computer+Society&rft.date=1983-06&rft.aulast=Hinton&rft.aufirst=Geoffrey+E.&rft.au=Sejnowski%2C+Terrence+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Fahlman SE, Hinton GE, Sejnowski TJ. <i><a rel="nofollow" class="external text" href="https://www.cs.toronto.edu/~fritz/absps/fahlmanBM.pdf">Massively parallel architectures for Al: NETL, Thistle, and Boltzmann machines.</a></i> In: Genesereth MR, editor. <i>AAAI-83.</i> Washington, DC: AAAI; 1983. pp. 109–113</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Chapter 16. Rosenfeld, Edward, and James A. Anderson, eds. 2000. <i>Talking Nets: An Oral History of Neural Networks</i>. Reprint edition. The MIT Press.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofstadter,_D._R.1984" class="citation book cs1">Hofstadter, D. R. (January 1984). <i>The Copycat Project: An Experiment in Nondeterminism and Creative Analogies</i>. Defense Technical Information Center. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/227617764">227617764</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Copycat+Project%3A+An+Experiment+in+Nondeterminism+and+Creative+Analogies.&rft.pub=Defense+Technical+Information+Center&rft.date=1984-01&rft_id=info%3Aoclcnum%2F227617764&rft.au=Hofstadter%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofstadter1988" class="citation book cs1">Hofstadter, Douglas R. (1988). "A Non-Deterministic Approach to Analogy, Involving the Ising Model of Ferromagnetism". In Caianiello, Eduardo R. (ed.). <i>Physics of cognitive processes</i>. Teaneck, New Jersey: World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9971-5-0255-0" title="Special:BookSources/9971-5-0255-0"><bdi>9971-5-0255-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/750950619">750950619</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Non-Deterministic+Approach+to+Analogy%2C+Involving+the+Ising+Model+of+Ferromagnetism&rft.btitle=Physics+of+cognitive+processes&rft.place=Teaneck%2C+New+Jersey&rft.pub=World+Scientific&rft.date=1988&rft_id=info%3Aoclcnum%2F750950619&rft.isbn=9971-5-0255-0&rft.aulast=Hofstadter&rft.aufirst=Douglas+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Smolensky, Paul. "Information processing in dynamical systems: Foundations of harmony theory." (1986): 194-281.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnston2024" class="citation web cs1">Johnston, Hamish (2024-10-08). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/john-hopfield-and-geoffrey-hinton-share-the-2024-nobel-prize-for-physics/">"John Hopfield and Geoffrey Hinton share the 2024 Nobel Prize for Physics"</a>. <i>Physics World</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=John+Hopfield+and+Geoffrey+Hinton+share+the+2024+Nobel+Prize+for+Physics&rft.date=2024-10-08&rft.aulast=Johnston&rft.aufirst=Hamish&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Fjohn-hopfield-and-geoffrey-hinton-share-the-2024-nobel-prize-for-physics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintonSejnowski1986" class="citation journal cs1"><a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Hinton, G. E.</a>; <a href="/wiki/Terry_Sejnowski" title="Terry Sejnowski">Sejnowski, T. J.</a> (1986). D. E. Rumelhart; J. L. McClelland (eds.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100705230134/http://learning.cs.toronto.edu/~hinton/absps/pdp7.pdf">"Learning and Relearning in Boltzmann Machines"</a> <span class="cs1-format">(PDF)</span>. <i>Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations</i>: 282–317. Archived from <a rel="nofollow" class="external text" href="http://www.cs.toronto.edu/~hinton/absps/pdp7.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-07-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Parallel+Distributed+Processing%3A+Explorations+in+the+Microstructure+of+Cognition.+Volume+1%3A+Foundations&rft.atitle=Learning+and+Relearning+in+Boltzmann+Machines&rft.pages=282-317&rft.date=1986&rft.aulast=Hinton&rft.aufirst=G.+E.&rft.au=Sejnowski%2C+T.+J.&rft_id=http%3A%2F%2Fwww.cs.toronto.edu%2F~hinton%2Fabsps%2Fpdp7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHinton2002" class="citation journal cs1"><a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Hinton, G. E.</a> (2002). <a rel="nofollow" class="external text" href="http://www.cs.toronto.edu/~hinton/absps/nccd.pdf">"Training Products of Experts by Minimizing Contrastive Divergence"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Neural_Computation_(journal)" title="Neural Computation (journal)">Neural Computation</a></i>. <b>14</b> (8): 1771–1800. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.8613">10.1.1.35.8613</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2F089976602760128018">10.1162/089976602760128018</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12180402">12180402</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207596505">207596505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neural+Computation&rft.atitle=Training+Products+of+Experts+by+Minimizing+Contrastive+Divergence&rft.volume=14&rft.issue=8&rft.pages=1771-1800&rft.date=2002&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.35.8613%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207596505%23id-name%3DS2CID&rft_id=info%3Apmid%2F12180402&rft_id=info%3Adoi%2F10.1162%2F089976602760128018&rft.aulast=Hinton&rft.aufirst=G.+E.&rft_id=http%3A%2F%2Fwww.cs.toronto.edu%2F~hinton%2Fabsps%2Fnccd.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintonOsinderoTeh2006" class="citation journal cs1"><a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Hinton, G. E.</a>; Osindero, S.; Teh, Y. (2006). <a rel="nofollow" class="external text" href="http://www.cs.toronto.edu/~hinton/absps/fastnc.pdf">"A fast learning algorithm for deep belief nets"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Neural_Computation_(journal)" title="Neural Computation (journal)">Neural Computation</a></i>. <b>18</b> (7): 1527–1554. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.1541">10.1.1.76.1541</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fneco.2006.18.7.1527">10.1162/neco.2006.18.7.1527</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16764513">16764513</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2309950">2309950</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neural+Computation&rft.atitle=A+fast+learning+algorithm+for+deep+belief+nets&rft.volume=18&rft.issue=7&rft.pages=1527-1554&rft.date=2006&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.76.1541%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2309950%23id-name%3DS2CID&rft_id=info%3Apmid%2F16764513&rft_id=info%3Adoi%2F10.1162%2Fneco.2006.18.7.1527&rft.aulast=Hinton&rft.aufirst=G.+E.&rft.au=Osindero%2C+S.&rft.au=Teh%2C+Y.&rft_id=http%3A%2F%2Fwww.cs.toronto.edu%2F~hinton%2Fabsps%2Ffastnc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.forbes.com/sites/tomtaulli/2020/02/02/coronavirus-can-ai-artificial-intelligence-make-a-difference/?sh=1eca51e55817">Kothari P (2020): https://www.forbes.com/sites/tomtaulli/2020/02/02/coronavirus-can-ai-artificial-intelligence-make-a-difference/?sh=1eca51e55817</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMontufar2018" class="citation web cs1">Montufar, Guido (2018). <a rel="nofollow" class="external text" href="https://www.mis.mpg.de/preprints/2018/preprint2018_87.pdf">"Restricted Boltzmann Machines: Introduction and Review"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/MPI_MiS" class="mw-redirect" title="MPI MiS">MPI MiS</a></i> (Preprint)<span class="reference-accessdate">. Retrieved <span class="nowrap">1 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MPI+MiS&rft.atitle=Restricted+Boltzmann+Machines%3A+Introduction+and+Review&rft.date=2018&rft.aulast=Montufar&rft.aufirst=Guido&rft_id=https%3A%2F%2Fwww.mis.mpg.de%2Fpreprints%2F2018%2Fpreprint2018_87.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABoltzmann+machine" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Boltzmann_machine&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.scholarpedia.org/article/Boltzmann_Machine">Scholarpedia article by Hinton about Boltzmann machines</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=AyzOUbkUf3M">Talk at Google by Geoffrey Hinton</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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<ul><li><a href="/wiki/Internal_energy" title="Internal energy">U</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">H</a></li> <li><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">F</a></li> <li><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">G</a></li></ul></li> <li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Statistical_model" title="Statistical model">Models</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spin_model" title="Spin model">Ferromagnetism models</a> <ul><li><a href="/wiki/Ising_model" title="Ising model">Ising</a></li> <li><a href="/wiki/Potts_model" title="Potts model">Potts</a></li> <li><a href="/wiki/Heisenberg_model_(quantum)" class="mw-redirect" title="Heisenberg model 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