Analogy between Boltzmann machines and Feynman path integrals

TL;DR

This paper explores the deep connection between Boltzmann machines in machine learning and Feynman path integrals in quantum mechanics, offering new interpretations and models linking the two fields.

Contribution

It establishes a formal equivalence between Boltzmann machines and Feynman path integrals, enabling quantum circuit models and insights into interpretable hidden layers.

Findings

Boltzmann machine hidden layers are discrete path elements.

Path weights in networks capture the $x ightarrow y$ mapping.

Quantum circuit models applicable to both frameworks.

Abstract

We provide a detailed exposition of the connections between Boltzmann machines commonly utilized in machine learning problems and the ideas already well known in quantum statistical mechanics through Feynman's description of the same. We find that this equivalence allows the interpretation that the hidden layers in Boltzmann machines and other neural network formalisms are in fact discrete versions of path elements that are present within the Feynman path-integral formalism. Since Feynman paths are the natural and elegant depiction of interference phenomena germane to quantum mechanics, it appears that in machine learning, the goal is to find an appropriate combination of ``paths'', along with accumulated path-weights, through a network that cumulatively capture the correct $x \rightarrow y$ map for a given mathematical problem. As a direct consequence of this analysis, we are able to provide general quantum circuit models that are applicable to both Boltzmann machines and to Feynman path integral descriptions. Connections are also made to inverse quantum scattering problems which allow a robust way to define ``interpretable'' hidden layers.