Scalar field Restricted Boltzmann Machine as an ultraviolet regulator

TL;DR

This paper demonstrates that scalar field Restricted Boltzmann Machines (RBMs) can serve as ultraviolet regulators in lattice field theory, with the cutoff controlled by hidden nodes or a mass parameter, and explores their learning dynamics.

Contribution

It introduces the perspective of lattice field theory to analyze RBMs with scalar fields, revealing their role as ultraviolet regulators and examining their learning behavior.

Findings

RBMs act as ultraviolet regulators with cutoff set by hidden nodes or mass parameter.

Infrared modes are learned more quickly than ultraviolet modes.

Theoretical analysis verified with Gaussian fields and applied to MNIST data.

Abstract

Restricted Boltzmann Machines (RBMs) are well-known tools used in Machine Learning to learn probability distribution functions from data. We analyse RBMs with scalar fields on the nodes from the perspective of lattice field theory. Starting with the simplest case of Gaussian fields, we show that the RBM acts as an ultraviolet regulator, with the cutoff determined by either the number of hidden nodes or a model mass parameter. We verify these ideas in the scalar field case, where the target distribution is known, and explore implications for cases where it is not known using the MNIST data set. We also demonstrate that infrared modes are learnt quickest.