Machine Learning and Boltzmann machines with applications
+
+
+
+
+
+Morten Hjorth-Jensen
+
+
+
+
+
+
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University and Department of Physics, University of Oslo, Norway
+The approaches to machine learning are many, but are often split into two main categories.
+In supervised learning we know the answer to a problem,
+and let the computer deduce the logic behind it. On the other hand, unsupervised learning
+is a method for finding patterns and relationship in data sets without any prior knowledge of the system.
+Some authours also operate with a third category, namely reinforcement learning. This is a paradigm
+of learning inspired by behavioural psychology, where learning is achieved by trial-and-error,
+solely from rewards and punishment.
+
+
+Another way to categorize machine learning tasks is to consider the desired output of a system.
+Some of the most common tasks are:
+
+
+
Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.
+
Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.
+
Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.
+
Other unsupervised learning algortihms, here Boltzmann machines
+What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately.
+One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.
+
+
+The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images.
+
+
+Furthermore, they have been used to solve complicated quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models.
+
+
A brief review on Markov Chains, Metropolis and Gibbs sampling
+
+
+
We want to study a physical system which evolves towards equilibrium, from given initial conditions.
+
We start with a PDF \( w(x_0,t_0) \) and we want to understand how the system evolves with time.
+
We want to reach a situation where after a given number of time steps we obtain a steady state. This means that the system reaches its most likely state (equilibrium situation)
+
Our PDF is normally a multidimensional object whose normalization constant is impossible to find.
+
Analytical calculations from \( w(x,t) \) are not possible.
+
To sample directly from from \( w(x,t) \) is not possible/difficult.
+
The transition probability \( W \) is also not known.
+
How can we establish that we have reached a steady state? Sounds impossible!
+A Markov process is a random walk with a selected probability for making a
+move. The new move is independent of the previous history of the system.
+
+
+The Markov process is used repeatedly in Monte Carlo simulations in order to generate
+new random states.
+
+
+The reason for choosing a Markov process is that when it is run for a
+long enough time starting with a random state, we will eventually reach the most likely state of the system.
+
+
+In thermodynamics, this means that after a certain number of Markov processes
+we reach an equilibrium distribution.
+
+
+This mimicks the way a real system reaches
+its most likely state at a given temperature of the surroundings.
+
+
Brownian motion and Markov processes, Ergodicity and Detailed balance
+
+
+To reach this distribution, the Markov process needs to obey two important conditions, that of
+ergodicity and detailed balance. These conditions impose then constraints on our algorithms
+for accepting or rejecting new random states.
+
+
+The Metropolis algorithm discussed here
+abides to both these constraints.
+
+
+The Metropolis algorithm is widely used in Monte Carlo
+simulations and the understanding of it rests within
+the interpretation of random walks and Markov processes.
+
+
+In a random walk one defines a mathematical entity called a walker,
+whose attributes
+completely define the state of the system in question.
+
+
+The state of the system can refer to any physical quantities,
+from the vibrational state of a molecule specified by a set of quantum numbers,
+to the brands of coffee in your favourite supermarket.
+
+
+The walker moves in an appropriate state space by a combination of
+deterministic and random displacements from its previous
+position.
+
+
Brownian motion and Markov processes, sequence of ingredients
+
+
+
We want to study a physical system which evolves towards equilibrium, from given initial conditions.
+
Markov chains are intimately linked with the physical process of diffusion.
+
From a Markov chain we can then derive the conditions for detailed balance and ergodicity. These are the conditions needed for obtaining a steady state.
+
The widely used algorithm for doing this is the so-called Metropolis algorithm, in its refined form the Metropolis-Hastings algorithm.
Financial engineering, see for example Patriarca et al, Physica 340, page 334 (2004).
+
Neuroscience, see for example Lipinski, Physics Medical Biology 35, page 441 (1990) or Farnell and Gibson, Journal of Computational Physics 208, page 253 (2005)
+
Tons of applications in physics
+
and chemistry
+
and biology, medicine
+
Nobel prize in economy to Black and Scholes
+
+
+$$
+\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2} V}{\partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0.
+$$
+
+The Black and Scholes equation is a partial differential equation, which describes the price
+of the option over time. It is a diffusion equation with a random term.
+
+
+A Markov process allows in principle for a microscopic description of Brownian motion.
+As with the random walk studied in the previous section, we consider a particle
+which moves along the \( x \)-axis in the form of a series of jumps with step length
+\( \Delta x = l \). Time and space are discretized and the subsequent moves are
+statistically independent, i.e., the new move depends only on the previous step
+and not on the results from earlier trials.
+We start at a position \( x=jl=j\Delta x \) and move to
+a new position \( x =i\Delta x \) during a step \( \Delta t=\epsilon \), where
+\( i\ge 0 \) and \( j\ge 0 \) are integers.
+The original probability distribution function (PDF) of the particles is given by
+\( w_i(t=0) \) where \( i \) refers to a specific position on the grid in
+
+
+
+The function \( w_i(t=0) \) is now the discretized version of \( w(x,t) \).
+We can regard the discretized PDF as a vector.
+
+
+For the Markov process we have a transition probability from a position
+\( x=jl \) to a position \( x=il \) given by
+$$
+\begin{equation*}
+ W_{ij}(\epsilon)=W(il-jl,\epsilon)=\left\{\begin{array}{cc}\frac{1}{2} & |i-j| = 1\\
+ 0 & \mathrm{else} \end{array} \right. ,
+\end{equation*}
+$$
+
+where \( W_{ij} \) is normally called
+the transition probability and we can represent it, see below,
+as a matrix.
+Here we have specialized to a case where the transition probability is known.
+
+
+Our new PDF \( w_i(t=\epsilon) \) is now related to the PDF at
+\( t=0 \) through the relation
+
+$$
+\begin{equation*}
+ w_i(t=\epsilon) =\sum_{j} W(j\rightarrow i)w_j(t=0).
+\end{equation*}
+$$
+
+
+This equation represents the discretized time-development of an original
+PDF with equal probability of jumping left or right.
+
+Since both \( W \) and \( w \) represent probabilities, they have to be normalized, i.e., we require
+that at each time step we have
+
+$$
+\begin{equation*}
+ \sum_i w_i(t) = 1,
+\end{equation*}
+$$
+
+and
+
+$$
+\begin{equation*}
+ \sum_j W(j\rightarrow i) = 1,
+\end{equation*}
+$$
+
+which applies for all \( j \)-values.
+The further constraints are
+\( 0 \le W_{ij} \le 1 \) and \( 0 \le w_{j} \le 1 \).
+Note that the probability for remaining at the same place is in general
+not necessarily equal zero.
+
+The time development of our initial PDF can now be represented through the action of
+the transition probability matrix applied \( n \) times. At a
+time \( t_n=n\epsilon \) our initial distribution has developed into
+
+$$
+\begin{equation*}
+ w_i(t_n) = \sum_jW_{ij}(t_n)w_j(0),
+\end{equation*}
+$$
+
+and defining
+
+$$
+\begin{equation*}
+ W(il-jl,n\epsilon)=(W^n(\epsilon))_{ij}
+\end{equation*}
+$$
+
+we obtain
+
+$$
+\begin{equation*}
+ w_i(n\epsilon) = \sum_j(W^n(\epsilon))_{ij}w_j(0),
+\end{equation*}
+$$
+
+or in matrix form
+$$
+\begin{equation} \tag{1}
+ \hat{w}(n\epsilon) = \hat{W}^n(\epsilon)\hat{w}(0).
+\end{equation}
+$$
+
+We note that both the vector and the matrix are properly normalized. Summing the vector elements gives one and
+summing over columns for the matrix results also in one. Furthermore, the largest eigenvalue is one.
+We act then on \( \hat{w} \) with \( \hat{W} \).
+The first iteration is
+
+$$
+\begin{equation*}
+ \hat{w}(t=\epsilon) = \hat{W}\hat{w}(t=0),
+\end{equation*}
+$$
+
+
+The table here demonstrates the convergence as a function of the number of iterations or
+time steps. After twelve iterations we have reached the exact value with six leading digits.
+
+
+We have after \( t \)-steps
+
+$$
+\begin{equation*}
+ \hat{w}(t) = \hat{W}^t\hat{w}(0),
+\end{equation*}
+$$
+
+with \( \hat{w}(0) \) the distribution at \( t=0 \) and \( \hat{W} \) representing the
+transition probability matrix.
+
+
+We can always expand \( \hat{w}(0) \) in terms of the right eigenvectors
+\( \hat{v} \) of \( \hat{W} \) as
+
+$$
+\begin{equation*}
+ \hat{w}(0) = \sum_i\alpha_i\hat{v}_i,
+\end{equation*}
+$$
+
+resulting in
+
+$$
+\begin{equation*}
+ \hat{w}(t) = \hat{W}^t\hat{w}(0)=\hat{W}^t\sum_i\alpha_i\hat{v}_i=
+\sum_i\lambda_i^t\alpha_i\hat{v}_i,
+\end{equation*}
+$$
+
+with \( \lambda_i \) the \( i^{\mathrm{th}} \) eigenvalue corresponding to
+the eigenvector \( \hat{v}_i \).
+
+
+If we assume that \( \lambda_0 \) is the largest eigenvector we see that in the limit \( t\rightarrow \infty \),
+\( \hat{w}(t) \) becomes proportional to the corresponding eigenvector
+\( \hat{v}_0 \). This is our steady state or final distribution.
+
+
+Let us recapitulate some of our results about Markov chains and random walks.
+
+
+
The time development of our PDF \( w(t) \), after
+
+
+one time-step from \( t=0 \) is given by
+$$
+\begin{equation*}
+w_i(t=\epsilon) = W(j\rightarrow i)w_j(t=0).
+\end{equation*}
+$$
+
+
+This equation represents the discretized time-development of an original
+PDF. We can rewrite this as a
+
+$$
+\begin{equation*}
+ w_i(t=\epsilon) = W_{ij}w_j(t=0).
+\end{equation*}
+$$
+
+with the transition matrix \( W \) for a random walk given by
+
+$$
+\begin{equation*}
+ W_{ij}(\epsilon)=W(il-jl,\epsilon)=\left\{\begin{array}{cc}\frac{1}{2} & |i-j| = 1\\
+ 0 & \mathrm{else} \end{array} \right.
+\end{equation*}
+$$
+
+
+The further constraints are
+\( 0 \le W_{ij} \le 1 \) and \( 0 \le w_{j} \le 1 \).
+
+
+
We can thus write the action of \( W \) as
+
+
+$$
+\begin{equation*}
+w_i(t+1) = \sum_jW_{ij}w_j(t),
+\end{equation*}
+$$
+
+or as vector-matrix relation
+
+$$
+\begin{equation*}
+ \hat{w}(t+1) = \hat{W\hat{w}}(t),
+\end{equation*}
+$$
+
+and if we have that \( ||\hat{w}(t+1)-\hat{w}(t)||\rightarrow 0 \), we say that
+we have reached the most likely state of the system, the so-called steady state or equilibrium state.
+
+The question then is how can we model anything under such a severe lack of knowledge? The Metropolis algorithm comes to our rescue here. Since \( W(j\rightarrow i) \) is unknown, we model it as the product of two probabilities,
+a probability for accepting the proposed move from the state \( j \) to the state \( j \), and a probability for making the transition to the state \( i \) being in the state \( j \). We label these probabilities \( A(j\rightarrow i) \) and \( T(j\rightarrow i) \), respectively. Our total transition probability is then
+
+$$
+\begin{equation*}
+W(j\rightarrow i)=T(j\rightarrow i)A(j\rightarrow i).
+\end{equation*}
+$$
+
+The algorithm can then be expressed as
+
+
+
We make a suggested move to the new state \( i \) with some transition or moving probability \( T_{j\rightarrow i} \).
+
We accept this move to the new state with an acceptance probability \( A_{j \rightarrow i} \). The new state \( i \) is in turn used as our new starting point for the next move. We reject this proposed moved with a \( 1-A_{j\rightarrow i} \) and the original state \( j \) is used again as a sample.
+We wish to derive the required properties of the probabilities \( T \) and \( A \) such that
+\( w_i^{(t\rightarrow \infty)} \rightarrow w_i \), starting
+from any distribution, will lead us to the correct distribution.
+
+
+We can now derive the dynamical process towards
+equilibrium. To obtain this equation we note that after \( t \) time steps the probability for being in a state \( i \) is related
+to the probability of being in a state \( j \) and performing a transition to the new state together with the probability of actually being in the state \( i \) and making a move to any of the possible states \( j \) from the previous time step.
+
+The last equation is very similar to the so-called Master equation, which relates the temporal dependence of
+a PDF \( w_i(t) \) to various transition rates. The equation can be derived from the so-called
+Chapman-Einstein-Enskog-Kolmogorov equation. The equation is given as
+$$
+\begin{equation}
+\tag{4}
+\frac{d w_i(t)}{dt} = \sum_j\left[ W(j\rightarrow i)w_j-W(i\rightarrow j)w_i\right],
+\end{equation}
+$$
+
+which simply states that the rate at which the systems moves from a state \( j \)
+to a final state \( i \) (the first term on the right-hand side of the last equation) is balanced by the rate at which the system undergoes transitions from the state \( i \) to a state \( j \) (the second term). If we have reached the so-called steady state, then the temporal development is zero. This means that in equilibrium we have
+
+$$
+\begin{equation*}
+\frac{d w_i(t)}{dt} = 0.
+\end{equation*}
+$$
+
+
+In the limit \( t\rightarrow \infty \) we require that the two distributions \( w_i(t+1)=w_i \) and \( w_i(t)=w_i \)
+and we have
+
+$$
+\begin{equation*}
+ \sum_j w_jT_{j\rightarrow i} A_{j\rightarrow i}= \sum_j w_iT_{i\rightarrow j}A_{i\rightarrow j},
+\end{equation*}
+$$
+
+which is the condition for balance when the most likely state (or steady state) has been reached.
+We see also that the right-hand side can be rewritten as
+
+$$
+\begin{equation*}
+\sum_j w_iT_{i\rightarrow j}A_{i\rightarrow j}= \sum_j w_iW_{i\rightarrow j},
+\end{equation*}
+$$
+
+and using the property that \( \sum_j W_{i\rightarrow j}=1 \), we can rewrite our equation
+as
+
+$$
+\begin{equation*}
+w_i= \sum_j w_jT_{j\rightarrow i} A_{j\rightarrow i}= \sum_j w_j W_{j\rightarrow i},
+\end{equation*}
+$$
+
+which is nothing but the standard equation for a Markov chain when the steady state has been reached.
+
+
+However, the condition that the rates should equal each other is in general not sufficient
+to guarantee that we, after many simulations, generate the correct distribution.
+We may risk to end up with so-called cyclic solutions. To avoid this
+we therefore introduce an additional condition, namely that of detailed balance
+
+$$
+\begin{equation*} W(j\rightarrow i)w_j= W(i\rightarrow j)w_i. \end{equation*}
+$$
+
+These equations were derived by Lars Onsager when studying irreversible processes.
+At equilibrium detailed balance gives thus
+
+$$
+\begin{equation*} \frac{W(j\rightarrow i)}{W(i\rightarrow j)}=\frac{w_i}{w_j}. \end{equation*}
+$$
+
+Rewriting the last equation in terms of our transition probabilities \( T \) and
+acceptance probobalities \( A \) we obtain
+
+$$
+\begin{equation*}
+w_j(t)T_{j\rightarrow i}A_{j\rightarrow i}= w_i(t)T_{i\rightarrow j}A_{i\rightarrow j}.
+\end{equation*}
+$$
+
+Since we normally have an expression
+for the probability distribution functions \( w_i \), we can rewrite the last equation as
+
+$$
+\begin{equation*}
+\frac{T_{j\rightarrow i}A_{j\rightarrow i}}{T_{i\rightarrow j}A_{i\rightarrow j}}= \frac{w_i}{w_j}.
+\end{equation*}
+$$
+
+In statistical physics this condition ensures that it is e.g., the
+Boltzmann distribution which is generated when equilibrium is reached.
+
+
+We introduce now the Boltzmann distribution
+
+$$
+\begin{equation*}
+ w_i= \frac{\exp{(-\beta(E_i))}}{Z},
+\end{equation*}
+$$
+
+which states that the probability of finding the system in a state \( i \) with energy \( E_i \)
+at an inverse temperature \( \beta = 1/k_BT \) is \( w_i\propto \exp{(-\beta(E_i))} \).
+The denominator \( Z \) is a normalization constant which ensures that the sum of all
+probabilities is normalized to one. It is defined as the sum of probabilities over all microstates
+\( j \) of the system
+
+$$
+\begin{equation*}
+ Z=\sum_j \exp{(-\beta(E_i))}.
+\end{equation*}
+$$
+
+From the partition function we can in principle generate all interesting quantities
+for a given system in equilibrium with its surroundings at a temperature \( T \).
+
+
+With the probability distribution given by the Boltzmann distribution we are now in a position
+where we can generate expectation values for a given variable \( A \) through the
+definition
+
+$$
+\begin{equation*}
+ \langle A \rangle = \sum_jA_jw_j=
+ \frac{\sum_jA_j\exp{(-\beta(E_j)}}{Z}.
+\end{equation*}
+$$
+
+In general, most systems have an infinity of microstates making thereby the computation
+of \( Z \) practically impossible and
+a brute force Monte Carlo calculation over a given number of randomly selected microstates
+may therefore not yield those microstates which are important
+at equilibrium.
+To select the most important contributions we need to
+use the condition for detailed balance. Since this is just given by the ratios of probabilities,
+we never need to evaluate the partition function \( Z \).
+
+
+For the
+Boltzmann distribution, detailed balance results in
+
+$$
+\begin{equation*} \frac{w_i}{w_j}= \exp{(-\beta(E_i-E_j))}. \end{equation*}
+$$
+
+
+Let us now specialize to a system whose energy is defined by the orientation of single spins.
+Consider the state \( i \), with given energy \( E_i \) represented by the following \( N \) spins
+
+$$
+\begin{equation*}
+\begin{array}{cccccccccc}
+\uparrow&\uparrow&\uparrow&\dots&\uparrow&\downarrow&\uparrow&\dots&\uparrow&\downarrow\\
+1&2&3&\dots& k-1&k&k+1&\dots&N-1&N\end{array}
+\end{equation*}
+$$
+
+We are interested in the transition with one single spinflip to a new state \( j \) with energy \( E_j \)
+
+$$
+\begin{equation*}
+\begin{array}{cccccccccc}
+\uparrow&\uparrow&\uparrow&\dots&\uparrow&\uparrow&\uparrow&\dots&\uparrow&\downarrow\\
+1&2&3&\dots& k-1&k&k+1&\dots&N-1&N\end{array}
+\end{equation*}
+$$
+
+This change from one microstate \( i \) (or spin configuration) to another microstate \( j \) is the
+configuration space analogue to a random walk on a lattice. Instead of jumping from
+one place to another in space, we 'jump' from one microstate to another.
+
+However, the selection of states has to generate a final distribution which is the
+Boltzmann distribution. This is again the same we saw for a random walker, for the discrete case we had
+always a binomial distribution, whereas for the continuous case we had a normal distribution.
+The way we sample configurations should result, when equilibrium is established, in the
+Boltzmann distribution. Else, our algorithm for selecting microstates is wrong.
+
+
+As stated above, we do in general not know the closed-form expression of the transition rate and we are free to model it as
+ \( W(i\rightarrow j)=T(i\rightarrow j)A(i\rightarrow j) \).
+Our ratio between probabilities gives us
+
+$$
+\begin{equation*}
+\frac{A_{j\rightarrow i}}{A_{i\rightarrow j}}= \frac{w_iT_{i\rightarrow j}}{w_jT_{j\rightarrow i}}.
+\end{equation*}
+$$
+
+The simplest form of the Metropolis algorithm (sometimes called for brute force Metropolis) assumes that
+the transition probability \( T(i\rightarrow j) \) is symmetric, implying that \( T(i\rightarrow j)=T(j\rightarrow i) \).
+
+
+We obtain then (using the Boltzmann distribution)
+
+$$
+\begin{equation*}
+\frac{A(j\rightarrow i)}{A(i\rightarrow j)}= \exp{(-\beta(E_i-E_j))} .
+\end{equation*}
+$$
+
+We are in this case interested in a new state \( E_j \) whose energy is lower than
+\( E_i \), viz., \( \Delta E = E_j-E_i \le 0 \). A simple test would then be to accept only those
+microstates which lower the energy.
+Suppose we have ten microstates with energy \( E_0 \le E_1 \le E_2 \le E_3 \le \dots \le E_9 \).
+Our desired energy is \( E_0 \).
+
+
+At a given temperature \( T \) we start our simulation by randomly choosing state
+\( E_9 \). Flipping spins we may then find a path from \( E_9\rightarrow E_8 \rightarrow E_7 \dots \rightarrow E_1 \rightarrow E_0 \).
+This would however lead to biased statistical averages since it would violate the ergodic hypothesis discussed
+in the previous section. This principle states that
+it should be possible for any Markov process to reach every possible state of the system
+from any starting point if the simulations is carried out for a long enough time.
+
+
+Any state in a Boltzmann distribution has a probability different from zero and if such
+a state cannot be reached from a given starting point, then the system is not ergodic.
+This means that another possible path to \( E_0 \) could be
+\( E_9\rightarrow E_7 \rightarrow E_8 \dots \rightarrow E_9 \rightarrow E_5 \rightarrow E_0 \) and so forth.
+Even though such a path could have a negligible probability it is still a possibility, and if
+we simulate long enough it should be included in our computation of an expectation value.
+
+Thus, we require that our algorithm should satisfy the principle of detailed balance and be ergodic.
+The problem with our ratio
+
+$$
+\begin{equation*}
+\frac{A(j\rightarrow i)}{A(i\rightarrow j)}= \exp{(-\beta(E_i-E_j))},
+\end{equation*}
+$$
+
+is that we do not know the acceptance probability. This equation only specifies the ratio of pairs of probabilities. Normally we want an algorithm which is as efficient as possible and maximizes the number of accepted moves.
+Moreover, we know that the acceptance probability has \( 0 \) as its smallest value and \( 1 \) as its largest.
+If we assume that the largest possible acceptance probability is \( 1 \), we adjust thereafter the other acceptance probability
+to this constraint.
+
+To understand this better, assume that we have two energies, \( E_i \) and \( E_j \), with \( E_i < E_j \). This means that the largest acceptance value must be
+\( A(j\rightarrow i) \) since we move to a state with lower energy. It follows from also from the fact that the probability \( w_i \) is larger than \( w_j \).
+The trick then is to fix this value to \( A(j\rightarrow i)=1 \). It means that
+the other acceptance probability has to be
+
+$$
+\begin{equation*}
+A(i\rightarrow j)= \exp{(-\beta(E_j-E_i))}.
+\end{equation*}
+$$
+
+
+One possible way to encode this equation reads
+
+$$
+\begin{equation*}
+A(j\rightarrow i)=\left\{\begin{array}{cc}
+\exp{(-\beta(E_i-E_j))} & E_i-E_j > 0 \\ 1 & else \end{array} \right.,
+\end{equation*}
+$$
+
+implying that if we move to a state with a lower energy, we always accept
+this move with acceptance probability \( A(j\rightarrow i)=1 \). If the energy is higher, we need to check
+this acceptance probability with the ratio between the probabilities from our PDF. From a practical point of view,
+the above ratio is compared with a random number.
+If the ratio is smaller than a given random number we accept the move to a higher energy, else we stay in the same state.
+
+
+Nothing hinders us obviously in choosing another acceptance ratio, like a weighting of the two energies via
+
+$$
+\begin{equation*}
+A(j\rightarrow i)=\exp{(-\frac{1}{2}\beta(E_i-E_j))}.
+\end{equation*}
+$$
+
+However, it is easy to see that such an acceptance ratio would result in
+fewer accepted moves.
+
+The Monte Carlo approach, combined with the theory for Markov chains can be summarized as follows:
+A Markov chain Monte Carlo method for the simulation of a distribution \( w \) is any method producing an
+ergodic Markov chain of events \( x \) whose stationary distribution is \( w \). The Metropolis algorithm can be phrased as
+
+
+
Generate an initial value \( x^{(i)} \).
+
Generate a trial value \( y_t \) with probability \( T(y_t|x^{(i)}) \). The latter quantity represents the probability of generating \( y_t \) given \( x^{(i)} \).
+Why use a generative model rather than the more well known discriminative deep neural networks (DNN)?
+
+
+
Discriminitave methods have several limitations: They are mainly supervised learning methods, thus requiring labeled data. And there are tasks they cannot accomplish, like drawing new examples from an unknown probability distribution.
+
A generative model can learn to represent and sample from a probability distribution. The core idea is to learn a parametric model of the probability distribution from which the training data was drawn. As an example
+
+
+
A model for images could learn to draw new examples of cats and dogs, given a training dataset of images of cats and dogs.
+
Generate a sample of an ordered or disordered Ising model phase, having been given samples of such phases.
+
Model the trial function for Monte Carlo calculations
Both use gradient-descent based learning procedures for minimizing cost functions
+
Energy based models don't use backpropagation and automatic differentiation for computing gradients, instead turning to Markov Chain Monte Carlo methods.
+
DNNs often have several hidden layers. A restricted Boltzmann machine has only one hidden layer, however several RBMs can be stacked to make up Deep Belief Networks, of which they constitute the building blocks.
+
+
+History: The RBM was developed by amongst others Geoffrey Hinton, called by some the "Godfather of Deep Learning", working with the University of Toronto and Google.
+
+
A function \( \mathbf{x} \) that represents the visible layer, a vector of \( M \) elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be the pixels of an image, the spin values of the Ising model, or coefficients representing speech.
+
The function \( \mathbf{h} \) represents the hidden, or latent, layer. A vector of \( N \) elements (nodes). Also called "feature detectors".
+The goal of the hidden layer is to increase the model's expressive power. We encode complex interactions between visible variables by introducing additional, hidden variables that interact with visible degrees of freedom in a simple manner, yet still reproduce the complex correlations between visible degrees in the data once marginalized over (integrated out).
+
+
+Examples of this trick being employed in physics:
+
+
+
The Hubbard-Stratonovich transformation
+
The introduction of ghost fields in gauge theory
+
Shadow wave functions in Quantum Monte Carlo simulations
+
+
+The network parameters, to be optimized/learned:
+
+
+
\( \mathbf{a} \) represents the visible bias, a vector of same length as \( \mathbf{x} \).
+
\( \mathbf{b} \) represents the hidden bias, a vector of same lenght as \( \mathbf{h} \).
+
\( W \) represents the interaction weights, a matrix of size \( M\times N \).
+The restricted Boltzmann machine is described by a Bolztmann distribution
+$$
+\begin{align}
+ P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})},
+\tag{5}
+\end{align}
+$$
+
+where \( Z \) is the normalization constant or partition function, defined as
+$$
+\begin{align}
+ Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}.
+\tag{6}
+\end{align}
+$$
+
+It is common to ignore \( T_0 \) by setting it to one.
+
+
+The function \( E(\mathbf{x},\mathbf{h}) \) gives the energy of a
+configuration (pair of vectors) \( (\mathbf{x}, \mathbf{h}) \). The lower
+the energy of a configuration, the higher the probability of it. This
+function also depends on the parameters \( \mathbf{a} \), \( \mathbf{b} \) and
+\( W \). Thus, when we adjust them during the learning procedure, we are
+adjusting the energy function to best fit our problem.
+
+
+There are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function \( E(\mathbf{x},\mathbf{h}) \).
+
+
+
+
+
+
+
+RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:
+$$
+\begin{align}
+ E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j,
+\tag{7}
+\end{align}
+$$
+
+where the binary values taken on by the nodes are most commonly 0 and 1.
+
+
+
+
+
+
+
+
+Another varient is the RBM where the visible units are Gaussian while the hidden units remain binary:
+$$
+\begin{align}
+ E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}.
+\tag{8}
+\end{align}
+$$
+
+In order to sample from the RBM probability distribution it is common to use Markov Chain Monte Carlo (MCMC) algorithms such as Metropolis-Hastings or Gibbs sampling.
+
+
+Metropolis sampling starts by suggesting a new configuration \( \boldsymbol{x}^{k+1} \). In the brute force method this is done by some random change of the visible units. The new configuration is then accepted with the acceptance probability
+$$
+\begin{align}
+ A(\boldsymbol{x}^k \rightarrow \boldsymbol{x}^{k+1}) = \text{min} (1, \frac{P(\boldsymbol{x}^{k+1})}{P(\boldsymbol{x}^k)}),
+\tag{9}
+\end{align}
+$$
+
+where we need the marginalized probability
+$$
+\begin{align}
+ P(\boldsymbol{x}) &= \sum_\mathbf{h} P_{rbm}(\mathbf{x}, \mathbf{h})
+\tag{10}\\
+ &= \frac{1}{Z}\sum_\mathbf{h} e^{-E(\mathbf{x}, \mathbf{h})}.
+\tag{11}
+\end{align}
+$$
+
+
+In this method we sample from the joint probability \( P_{rbm} (\mathbf{x}, \mathbf{h}) \) by way of a two step sampling process. We alternately update the visible and hidden units.
+New samples are generated according to the conditional probabilities \( P(x_i|\mathbf{h}) \) and \( P(h_j|\mathbf{x}) \) respectively and accepted with the probability of \( 1 \). While the the visible nodes are dependent on the hidden nodes and vice versa, the nodes are independent of other nodes within the same layer. This is due to there being no intra layer interactions in the restricted Boltzmann machine.
+
+
+The conditional probabilities are often referred to as the activitation functions in the neural networks context due to their role in determining the node outputs. For the binary-binary RBM they are
+$$
+\begin{align}
+ P(h_j = 1 | \boldsymbol{x}) &= \frac{1}{1 + e^{-b_j - \sum_i x_i w_{ij}}}
+\tag{12}\\
+ P(x_i = 1 | \boldsymbol{h}) &= \frac{1}{1 + e^{-a_j - \sum_j h_j w_{ij}}},
+\tag{13}
+\end{align}
+$$
+
+where we recognize the logistic sigmoid function \( \sigma (x) = 1/(1+exp(-x)) \).
+
+
+For the Gaussian-Binary RBM the conditional probabilities are
+$$
+\begin{align}
+ P(x_i|\mathbf{h}) &= \mathcal{N}(x_i; a_i+ \sum_j h_j w_{ij}, \sigma^2)
+\tag{14}\\
+ P(h_j=1|\mathbf{x}) &= \frac{1}{1+e^{-b_j-\frac{1}{\sigma^2} \sum_i x_i w_{ij}}},
+\tag{15}
+\end{align}
+$$
+
+while the visible units now follow a normal distribution, we see the hidden units again follow the logistic sigmoid function.
+
+
+When working with a training dataset, the most common training approach is maximizing the log-likelihood of the training data. The log likelihood characterizes the log-probability of generating the observed data using our generative model. Using this method our cost function is chosen as the negative log-likelihood. The learning then consists of trying to find parameters that maximize the probability of the dataset, and is known as Maximum Likelihood Estimation (MLE).
+Denoting the parameters as \( \boldsymbol{\theta} = a_1,...,a_M,b_1,...,b_N,w_{11},...,w_{MN} \), the log-likelihood is given by
+$$
+\begin{align}
+ \mathcal{L}(\{ \theta_i \}) &= \langle \text{log} P_\theta(\boldsymbol{x}) \rangle_{data}
+\tag{16}\\
+ &= - \langle E(\boldsymbol{x}; \{ \theta_i\}) \rangle_{data} - \text{log} Z(\{ \theta_i\}),
+\tag{17}
+\end{align}
+$$
+
+where we used that the normalization constant does not depend on the data, \( \langle \text{log} Z(\{ \theta_i\}) \rangle = \text{log} Z(\{ \theta_i\}) \)
+Our cost function is the negative log-likelihood, \( \mathcal{C}(\{ \theta_i \}) = - \mathcal{L}(\{ \theta_i \}) \)
+
+
+The training procedure of choice often is Stochastic Gradient Descent (SGD). It consists of a series of iterations where we update the parameters according to the equation
+$$
+\begin{align}
+ \boldsymbol{\theta}_{k+1} = \boldsymbol{\theta}_k - \eta \nabla \mathcal{C} (\boldsymbol{\theta}_k)
+\tag{18}
+\end{align}
+$$
+
+at each \( k \)-th iteration. There are a range of variants of the algorithm which aim at making the learning rate \( \eta \) more adaptive so the method might be more efficient while remaining stable.
+
+
+We now need the gradient of the cost function in order to minimize it. We find that
+$$
+\begin{align}
+ \frac{\partial \mathcal{C}(\{ \theta_i\})}{\partial \theta_i}
+ &= \langle \frac{\partial E(\boldsymbol{x}; \theta_i)}{\partial \theta_i} \rangle_{data}
+ + \frac{\partial \text{log} Z(\{ \theta_i\})}{\partial \theta_i}
+\tag{19}\\
+ &= \langle O_i(\boldsymbol{x}) \rangle_{data} - \langle O_i(\boldsymbol{x}) \rangle_{model},
+\tag{20}
+\end{align}
+$$
+
+where in order to simplify notation we defined the "operator"
+$$
+\begin{align}
+ O_i(\boldsymbol{x}) = \frac{\partial E(\boldsymbol{x}; \theta_i)}{\partial \theta_i},
+\tag{21}
+\end{align}
+$$
+
+and used the statistical mechanics relationship between expectation values and the log-partition function:
+$$
+\begin{align}
+ \langle O_i(\boldsymbol{x}) \rangle_{model} = \text{Tr} P_\theta(\boldsymbol{x})O_i(\boldsymbol{x}) = - \frac{\partial \text{log} Z(\{ \theta_i\})}{\partial \theta_i}.
+\tag{22}
+\end{align}
+$$
+
+
+The data-dependent term in the gradient is known as the positive phase of the gradient, while the model-dependent term is known as the negative phase of the gradient. The aim of the training is to lower the energy of configurations that are near observed data points (increasing their probability), and raising the energy of configurations that are far from observed data points (decreasing their probability).
+
+
+The gradient of the negative log-likelihood cost function of a Binary-Binary RBM is then
+$$
+\begin{align}
+ \frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial w_{ij}} =& \langle x_i h_j \rangle_{data} - \langle x_i h_j \rangle_{model}
+\tag{23}\\
+ \frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial a_{ij}} =& \langle x_i \rangle_{data} - \langle x_i \rangle_{model}
+\tag{24}\\
+ \frac{\partial \mathcal{C} (w_{ij}, a_i, b_j)}{\partial b_{ij}} =& \langle h_i \rangle_{data} - \langle h_i \rangle_{model}.
+\tag{25}\\
+\tag{26}
+\end{align}
+$$
+
+To get the expecation values with respect to the data, we set the visible units to each of the observed samples in the training data, then update the hidden units according to the conditional probability found before. We then average over all samples in the training data to calculate expectation values with respect to the data.
+
+
+To get the expectation values with respect to the model, we use Gibbs sampling. We can either initialize the \( \boldsymbol{x} \) randomly or with a training sample. While we ideally want a large number of Gibbs iterations \( n\rightarrow n \), one might decide to truncate it earlier for efficiency. Doing this while having intialized \( \boldsymbol{x} \) with a training data vector is referred to as contrastive divergence (CD), because one is then closer to approximating the gradient of this function than the negative log-likelihood. The contrastive divergence function is the difference between two Kullback-Leibler divergences (also called relative entropy), which measure how one probability distribution diverges from a second, expected probability distribution (in this case the estimated one from the ground truth one).
+
+
+The idea of applying RBMs to quantum many body problems was presented by G. Carleo and M. Troyer, working with ETH Zurich and Microsoft Research.
+
+
+Some of their motivation included
+
+
+
"The wave function \( \Psi \) is a monolithic mathematical quantity that contains all the information on a quantum state, be it a single particle or a complex molecule. In principle, an exponential amount of information is needed to fully encode a generic many-body quantum state."
+
There are still interesting open problems, including fundamental questions ranging from the dynamical properties of high-dimensional systems to the exact ground-state properties of strongly interacting fermions.
+
The difficulty lies in finding a general strategy to reduce the exponential complexity of the full many-body wave function down to its most essential features. That is
+
+
+
\( \rightarrow \) Dimensional reduction
+
\( \rightarrow \) Feature extraction
+
+
+
Among the most successful techniques to attack these challenges, artifical neural networks play a prominent role.
+
Want to understand whether an artifical neural network may adapt to describe a quantum system.
+Carleo and Troyer applied the RBM to the quantum mechanical spin lattice systems of the Ising model and Heisenberg model, with encouraging results. Our goal is to test the method on systems of moving particles. For the spin lattice systems it was natural to use a binary-binary RBM, with the nodes taking values of 1 and -1. For moving particles, on the other hand, we want the visible nodes to be continuous, representing position coordinates. Thus, we start by choosing a Gaussian-binary RBM, where the visible nodes are continuous and hidden nodes take on values of 0 or 1. If eventually we would like the hidden nodes to be continuous as well the rectified linear units seem like the most relevant choice.
+
+
+The wavefunction should be a probability amplitude depending on \( \boldsymbol{x} \). The RBM model is given by the joint distribution of \( \boldsymbol{x} \) and \( \boldsymbol{h} \)
+$$
+\begin{align}
+ F_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})}.
+\tag{27}
+\end{align}
+$$
+
+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)
+$$
+\begin{align}
+ \Psi (\mathbf{X}) &= F_{rbm}(\mathbf{x})
+\tag{30}\\
+ &= \frac{1}{Z}\sum_{\boldsymbol{h}} e^{-E(\mathbf{x}, \mathbf{h})}
+\tag{31}\\
+ &= \frac{1}{Z} \sum_{\{h_j\}} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2} + \sum_j^N b_j h_j + \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma^2}}
+\tag{32}\\
+ &= \frac{1}{Z} e^{-\sum_i^M \frac{(x_i - a_i)^2}{2\sigma^2}} \prod_j^N (1 + e^{b_j + \sum_i^M \frac{x_i w_{ij}}{\sigma^2}}).
+\tag{33}\\
+\tag{34}
+\end{align}
+$$
+
+
+Now we don't necessarily have training data (unless we generate it by using some other method). However, what we do have is the variational principle which allows us to obtain the ground state wave function by minimizing the expectation value of the energy of a trial wavefunction (corresponding to the untrained NQS). Similarly to the traditional variational Monte Carlo method then, it is the local energy we wish to minimize. The gradient to use for the stochastic gradient descent procedure is
+$$
+\begin{align}
+ G_i = \frac{\partial \langle E_L \rangle}{\partial \theta_i}
+ = 2(\langle E_L \frac{1}{\Psi}\frac{\partial \Psi}{\partial \theta_i} \rangle - \langle E_L \rangle \langle \frac{1}{\Psi}\frac{\partial \Psi}{\partial \theta_i} \rangle ),
+\tag{35}
+\end{align}
+$$
+
+where the local energy is given by
+$$
+\begin{align}
+ E_L = \frac{1}{\Psi} \hat{\mathbf{H}} \Psi.
+\tag{36}
+\end{align}
+$$
+
+
+The trial wave function is based on the product of a Slater determinant with Gaussian orbitals, a simple Jastrow factor \( \exp{(r_{ij})} \) and the reduced Boltzmann machines.
+
+
+The Broyden-Fletcher-Goldfarb-Shanno algorithm was used to perform the minimization. We used \( 14 \) hidden nodes in the calculations below.
+
+
Anti-symmetry dealt with multiplying the trail wave function with an optimized Slater determinant.
+
To come: Analysis of wave function from ML and compare with diffusion and Variational Monte Carlo calculations as well as the analytical results of Taut for the two-electron case.
+
Extend to more fermions. How do we deal with the antisymmetry of the multi-fermion wave function?
+
+
+
Here we used standard Hartree-Fock theory to define an optimal Slater determinant. Takes care of the antisymmetry. What about constructing an anti-symmetrized network function?
+
Use thereafter ML to determine the correlated part of the wafe function (including a standard Jastrow factor).
+
Test this for multi-fermion systems and compare with other many-body methods.
+
+
+
Can we use ML to find out which correlations are relevant and thereby diminish the dimensionality problem in say CC or SRG theories?
