A Correspondence Between Deep Boltzmann Machines and p-Adic Statistical Field Theories

Abstract

There is a strong interest in studying the correspondence between Euclidean quantum fields and neural networks. This correspondence takes different forms depending on the type of networks considered. In this work, we study this correspondence in the case of deep Boltzmann machines (DBMs) having a tree-like topology. We use p-adic numbers to encode this type of topology. A p-adic continuous DBM is a statistical field theory (SFT) defined by an energy functional on the space of square-integrable functions defined on a p-adic N-dimensional ball. The energy functionals are non-local, meaning they depend on the interaction of all the neurons forming the network. Each energy functional defines a probability measure. A natural discretization process attaches to each probability measure a finite-dimensional Boltzmann distribution, which describes a hierarchical DBM. We provide a mathematically rigorous perturbative method for computing the correlation functions. A relevant novelty is that the general correlation functions cannot be calculated using the Wick-Isserlis theorem. We give a recursive formula for computing the correlation functions of an arbitrary number of points using certain 3-partitions of the sets of indices attached to the points.