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<!-- navigation toc: --> <li><a href="._BM-bs001.html#___sec0" style="font-size: 80%;">Types of Machine Learning, a repetition</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs002.html#___sec1" style="font-size: 80%;">Why Boltzmann machines?</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs003.html#___sec2" style="font-size: 80%;">An intermediate step, the Hopfield network and links to the Ising and Potts models</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs004.html#___sec3" style="font-size: 80%;">A brief review on Markov Chains, Metropolis and Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs005.html#___sec4" style="font-size: 80%;">Brownian motion and Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs006.html#___sec5" style="font-size: 80%;">Brownian motion and Markov processes, Ergodicity and Detailed balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs007.html#___sec6" style="font-size: 80%;">Brownian motion and Markov processes, jargon</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs008.html#___sec7" style="font-size: 80%;">Brownian motion and Markov processes, sequence of ingredients</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs009.html#___sec8" style="font-size: 80%;">Applications: almost every field in science</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs010.html#___sec9" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs011.html#___sec10" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs012.html#___sec11" style="font-size: 80%;">Markov processes, the probabilities</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs013.html#___sec12" style="font-size: 80%;">Markov processes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs014.html#___sec13" style="font-size: 80%;">An Illustrative Example</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs015.html#___sec14" style="font-size: 80%;">An Illustrative Example</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs016.html#___sec15" style="font-size: 80%;">An Illustrative Example, next step</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs017.html#___sec16" style="font-size: 80%;">An Illustrative Example, the steady state</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs018.html#___sec17" style="font-size: 80%;">An Illustrative Example, iterative steps</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs019.html#___sec18" style="font-size: 80%;">An Illustrative Example, what does it mean?</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs020.html#___sec19" style="font-size: 80%;">An Illustrative Example, understanding the basics</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs021.html#___sec20" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs022.html#___sec21" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs023.html#___sec22" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs024.html#___sec23" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs025.html#___sec24" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs026.html#___sec25" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs027.html#___sec26" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs028.html#___sec27" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs029.html#___sec28" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs030.html#___sec29" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs031.html#___sec30" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs032.html#___sec31" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs033.html#___sec32" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs034.html#___sec33" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs035.html#___sec34" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs036.html#___sec35" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs037.html#___sec36" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs038.html#___sec37" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs039.html#___sec38" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs040.html#___sec39" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs041.html#___sec40" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs042.html#___sec41" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs043.html#___sec42" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs044.html#___sec43" style="font-size: 80%;">Brief Summary</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs045.html#___sec44" style="font-size: 80%;">Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs046.html#___sec45" style="font-size: 80%;">Boltzmann Machines</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs047.html#___sec46" style="font-size: 80%;">Some similarities and differences from DNNs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs048.html#___sec47" style="font-size: 80%;">Boltzmann machines (BM)</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs049.html#___sec48" style="font-size: 80%;">A standard BM setup</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs050.html#___sec49" style="font-size: 80%;">The structure of the RBM network</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs051.html#___sec50" style="font-size: 80%;">The network</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs052.html#___sec51" style="font-size: 80%;">Goals</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs053.html#___sec52" style="font-size: 80%;">Joint distribution</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs054.html#___sec53" style="font-size: 80%;">Network Elements, the energy function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs055.html#___sec54" style="font-size: 80%;">Defining different types of RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs056.html#___sec55" style="font-size: 80%;">More about RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs057.html#___sec56" style="font-size: 80%;">Sampling: Metropolis sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs058.html#___sec57" style="font-size: 80%;">Sampling: Gibbs sampling</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs059.html#___sec58" style="font-size: 80%;">Gaussian RBM</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs060.html#___sec59" style="font-size: 80%;">Cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs061.html#___sec60" style="font-size: 80%;">Optimization / Training</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs062.html#___sec61" style="font-size: 80%;">More on RBMs</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs063.html#___sec62" style="font-size: 80%;">Which sampling to use</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs064.html#___sec63" style="font-size: 80%;">Kullback-Leibler relative entropy</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs065.html#___sec64" style="font-size: 80%;">Optimizing the cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs066.html#___sec65" style="font-size: 80%;">Setting up for gradient descent calculations</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs067.html#___sec66" style="font-size: 80%;">More interpretations</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs068.html#___sec67" style="font-size: 80%;">Recent examples: RBMs for the quantum many body problem</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs069.html#___sec68" style="font-size: 80%;">Choose the right RBM</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs070.html#___sec69" style="font-size: 80%;">Representing the wave function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs071.html#___sec70" style="font-size: 80%;">Choose the cost function</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs072.html#___sec71" style="font-size: 80%;">Running the codes</a></li>
<!-- navigation toc: --> <li><a href="._BM-bs073.html#___sec72" style="font-size: 80%;">Energy as function of iterations, \( N=2 \) electrons</a></li>
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<center><h1>Machine Learning and Boltzmann machines with applications</h1></center> <!-- document title -->
<p>
<!-- author(s): Morten Hjorth-Jensen -->
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<b>Morten Hjorth-Jensen</b>
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<center><b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University and Department of Physics, University of Oslo, Norway</b></center>
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<center><h4>Nov 29, 2018 </h4></center> <!-- date -->
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