Quantum field theories, Markov random fields and machine learning

To compare the probability distribution $p(\phi;\theta)$ of the Markov field with another probability distribution $q(\phi)$, we define the Kullback-Leibler divergence, a nonnegative quantity, as

We now consider a target Boltzmann probability distribution $q(\phi)=\exp[-\mathcal{A}]/Z_{\mathcal{A}}$ that describes an arbitrary statistical system and substitute the two probability distributions in the Kullback-Leibler divergence to obtain:

where $\mathcal{F}$ is the variational free energy, $F_{\mathcal{A}}=-\ln Z_{\mathcal{A}}$, and $\langle O\rangle_{p(\phi;\theta)}$ denotes the expectation value of an observable $O$ under the probability distribution $p(\phi;\theta)$. By minimizing this quantity the two probability distributions $p(\phi;\theta)$ and $q(\phi)$ will become equal and we can therefore use the distribution of the $\phi^{4}$ theory to draw samples from the target probability distribution $q(\phi)$.

Based on a gradient-based approach, we are able to minimize Eq. 8 via

and the coupling constants $\theta$ are updated at each epoch $t$ as:

where $\eta$ is the learning rate and $\mathcal{L}=\partial\mathcal{F}/\partial\theta^{(t)}$.

We now consider a variation of the $\phi^{4}$ theory which includes next-nearest neighbor interactions $nnn$, and with a complex action $\mathcal{A}$ defined as

where $i$ denotes the imaginary unit. The coupling constants can have arbitrary values but for this example we consider $g_{1}=g_{4}=-1$, $g_{2}=1.52425$, $g_{3}=0.175$ and $g_{5}=0.15$, see Ref. [3].

We now utilize the $\phi^{4}$ Markov field of action $S$, to approximate a probability distribution which is described by an action $\mathcal{A}_{\{4\}}=\sum_{k=1}^{4}g_{k}\mathcal{A}^{(k)}$. To investigate how accurate the equivalence between the two probability distributions is, we will implement reweighting based on the probability distribution of the $\phi^{4}$ Markov field to calculate expectation values of the full complex action $\mathcal{A}$. The reweighting relation is given by

details of which can be found in Refs. [3, 4, 5, 6].

We consider $j=4$ and calculate the expectation values of the action $\mathcal{A}$ and the magnetization $m$ by extrapolating with reweighting in the range $g_{4}^{\prime}\in[-1.15,-0.85]$. The results, shown in Fig. 2, overlap within statistical uncertainty with independent calculations, therefore verifying that observables which would correspond to probability distribution of the full action $\mathcal{A}$ can be accurately calculated based on reweighting from the inhomogeneous action $S$.

A different class of machine learning applications considers the case where the form of the target probability distribution is unknown but there exists a set of available data in which an empirical probability distribution is encoded. To explore this type of applications we consider the following expression of the Kullback-Leibler divergence:

By substituting and taking the derivative in terms of the variational parameters $\theta$ we obtain:

and the update rule of the parameters $\theta$ at each epoch $t$ is given based on Eq. 10, where $\mathcal{L}=-\partial\ln p(\phi;\theta^{(t)})/{\partial\theta^{(t)}}$.

As an example of a distribution to be learned by the machine learning algorithm, we consider the simple case of a Gaussian distribution with $\mu=-0.5$ and $\sigma=0.05$. Since the lattice action is invariant under the $Z_{2}$ symmetry we expect that the symmetric values of the dataset are equiprobable in being reproduced. This invariance can be removed via the inclusion of a term $\sum_{i}r_{i}\phi_{i}$ which breaks the symmetry of the action explicitly. The results are shown in Fig. 3 (left) where the anticipated behaviour is observed.

Finally, we illustrate the approach in an image from the CIFAR-10 dataset. The thermalization of the trained Markov field is depicted in Fig. 3 (right), where the image emerges within the equilibrium probability distribution. Since the machine learning algorithm learns the correct values of coupling constants in the action that solve the considered problem, extensions towards learning the appropriate coupling constants which describe renormalized systems can potentially be explored [7].

To increase the expressivity of the machine learning algorithm, we will introduce a new set of latent or hidden variables $h_{j}$ within the graph structure. In addition, we will restrict the interactions to be exclusively between the visible $\phi_{i}$ and the hidden $h_{j}$ variables, giving rise to the lattice action

The $\phi^{4}$ neural network implemented above is a generalization of other neural network architectures: if $b_{i}=n_{j}=0$ the $\phi^{4}$ neural network reduces to a Gaussian-Gaussian restricted Boltzmann machine and if $b_{i}=n_{j}=m_{j}=0$ and $h_{j}\in\{-1,1\}$ then one obtains a Gaussian-Bernoulli restricted Boltzmann machine, see Refs [8, 9]. Restricted Boltzmann machines were inspired by Ising models and since the $\phi^{4}$ theory can become equivalent to an Ising model [10], novel physical insights into how these machine learning algorithms work might emerge from the perspective of quantum field theory. We now train the $\phi^{4}$ neural network on the Olivetti faces dataset. Some representative features, which resemble face structures, are shown in Fig. 4.