Changs to BM, added Metro stuff
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mhjensen
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'___sec35'),
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|
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<a class="navbar-brand" href="BM-bs.html">Machine Learning and Boltzmann machines with applications</a>
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<li class="dropdown">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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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>
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<!-- navigation toc: --> <li><a href="._BM-bs002.html#___sec1" style="font-size: 80%;">Why Boltzmann machines?</a></li>
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<!-- 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>
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||||
<!-- 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>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs005.html#___sec4" style="font-size: 80%;">Brownian motion and Markov processes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs006.html#___sec5" style="font-size: 80%;">Brownian motion and Markov processes, Ergodicity and Detailed balance</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs007.html#___sec6" style="font-size: 80%;">Brownian motion and Markov processes, jargon</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs008.html#___sec7" style="font-size: 80%;">Brownian motion and Markov processes, sequence of ingredients</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs009.html#___sec8" style="font-size: 80%;">Applications: almost every field in science</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs010.html#___sec9" style="font-size: 80%;">Markov processes</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs011.html#___sec10" style="font-size: 80%;">Markov processes</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs012.html#___sec11" style="font-size: 80%;">Markov processes, the probabilities</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs013.html#___sec12" style="font-size: 80%;">Markov processes</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs014.html#___sec13" style="font-size: 80%;">An Illustrative Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs015.html#___sec14" style="font-size: 80%;">An Illustrative Example</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs016.html#___sec15" style="font-size: 80%;">An Illustrative Example, next step</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs017.html#___sec16" style="font-size: 80%;">An Illustrative Example, the steady state</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs018.html#___sec17" style="font-size: 80%;">An Illustrative Example, iterative steps</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs019.html#___sec18" style="font-size: 80%;">An Illustrative Example, what does it mean?</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs020.html#___sec19" style="font-size: 80%;">An Illustrative Example, understanding the basics</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs021.html#___sec20" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs022.html#___sec21" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs023.html#___sec22" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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<!-- navigation toc: --> <li><a href="._BM-bs024.html#___sec23" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs025.html#___sec24" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs026.html#___sec25" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs027.html#___sec26" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs028.html#___sec27" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs029.html#___sec28" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs030.html#___sec29" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs031.html#___sec30" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs032.html#___sec31" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs033.html#___sec32" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs034.html#___sec33" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs035.html#___sec34" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- 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>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs038.html#___sec37" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs039.html#___sec38" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs040.html#___sec39" style="font-size: 80%;">The Metropolis Algorithm and Detailed Balance</a></li>
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||||
<!-- 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>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs045.html#___sec44" style="font-size: 80%;">Gibbs sampling</a></li>
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||||
<!-- navigation toc: --> <li><a href="._BM-bs046.html#___sec45" style="font-size: 80%;">Boltzmann Machines</a></li>
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||||
<!-- 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%;">The structure of the RBM network</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs049.html#___sec48" style="font-size: 80%;">The network</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs050.html#___sec49" style="font-size: 80%;">Goals</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs051.html#___sec50" style="font-size: 80%;">Joint distribution and the Energy function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs052.html#___sec51" style="font-size: 80%;">Network Elements</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs053.html#___sec52" style="font-size: 80%;">Defining different types of RBMs</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs054.html#___sec53" style="font-size: 80%;">More about RBMs</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs055.html#___sec54" style="font-size: 80%;">Sampling: Metropolis sampling</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs056.html#___sec55" style="font-size: 80%;">Sampling: Gibbs sampling</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs057.html#___sec56" style="font-size: 80%;">Gaussian RBM</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs058.html#___sec57" style="font-size: 80%;">Cost function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs059.html#___sec58" style="font-size: 80%;">Optimization / Training</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs060.html#___sec59" style="font-size: 80%;">More on RBMs</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs061.html#___sec60" style="font-size: 80%;">Which sampling to use</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs062.html#___sec61" style="font-size: 80%;">RBMs for the quantum many body problem</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs063.html#___sec62" style="font-size: 80%;">Choose the right RBM</a></li>
|
||||
<!-- navigation toc: --> <li><a href="#___sec63" style="font-size: 80%;">Representing the wave function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs065.html#___sec64" style="font-size: 80%;">Choose the cost function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs066.html#___sec65" style="font-size: 80%;">Running the codes</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs067.html#___sec66" style="font-size: 80%;">Energy as function of iterations, \( N=2 \) electrons</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs068.html#___sec67" style="font-size: 80%;">Energy as function of iterations, \( N=6 \) electrons</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._BM-bs069.html#___sec68" style="font-size: 80%;">Conclusions and where do we stand</a></li>
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||||
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||||
</ul>
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||||
</li>
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||||
</ul>
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||||
</div>
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</div>
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</div> <!-- end of navigation bar -->
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<div class="container">
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||||
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||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
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||||
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<a name="part0064"></a>
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||||
<!-- !split -->
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||||
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<h2 id="___sec63" class="anchor">Representing the wave function </h2>
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The wavefunction should be a probability amplitude depending on \( \boldsymbol{x} \). The RBM model is given by the joint distribution of \( \boldsymbol{x} \) and \( \boldsymbol{h} \)
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$$
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\begin{align}
|
||||
F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}.
|
||||
\tag{27}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
To find the marginal distribution of \( \boldsymbol{x} \) we set:
|
||||
$$
|
||||
\begin{align}
|
||||
F_{rbm}(\mathbf{x}) &= \sum_\mathbf{h} F_{rbm}(\mathbf{x}, \mathbf{h})
|
||||
\tag{28}\\
|
||||
&= \frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}.
|
||||
\tag{29}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
Now this is what we use to represent the wave function, calling it a neural-network quantum state (NQS)
|
||||
$$
|
||||
\begin{align}
|
||||
\Psi (\mathbf{X}) &= F_{rbm}(\mathbf{x})
|
||||
\tag{30}\\
|
||||
&= \frac{1}{Z}\sum_{\boldsymbol{h}} e^{-E(\mathbf{x}, \mathbf{h})}
|
||||
\tag{31}\\
|
||||
&= \frac{1}{Z} \sum_{\{h_j\}} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2} + \sum_j^N b_j h_j + \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma^2}}
|
||||
\tag{32}\\
|
||||
&= \frac{1}{Z} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2}} \prod_j^N (1 + e^{b_j + \sum_i^M \frac{x_i w_{ij}}{\sigma^2}}).
|
||||
\tag{33}\\
|
||||
\tag{34}
|
||||
\end{align}
|
||||
$$
|
||||
|
||||
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|
||||
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||||
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|
||||
Reference in New Issue
Block a user