Quantum field theories, Markov random fields and machine learning

TL;DR

This paper explores the connection between discretized Euclidean quantum field theories and Markov random fields, deriving neural network models from quantum theories to enhance probabilistic machine learning applications.

Contribution

It establishes the mathematical equivalence between lattice quantum field theories and Markov fields, and derives neural networks from quantum field theories for machine learning.

Findings

Discretized Euclidean field theories are equivalent to Markov fields.

Neural networks can be derived from quantum field theories.

Applications include minimizing Kullback-Leibler divergence in probability distributions.

Abstract

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning within quantum field theory. Here, we will discuss how discretized Euclidean field theories, such as the $\phi^{4}$ lattice field theory on a square lattice, are mathematically equivalent to Markov fields, a notable class of probabilistic graphical models with applications in a variety of research areas, including machine learning. The results are established based on the Hammersley-Clifford theorem. We will then derive neural networks from quantum field theories and discuss applications pertinent to the minimization of the Kullback-Leibler divergence for the probability distribution of the $\phi^{4}$ machine learning algorithms and other probability distributions.