Legendre Equivalences of Spherical Boltzmann Machines

TL;DR

This paper establishes a mathematical relationship between the free energies of spherical and Gaussian Boltzmann machines using Legendre transformations, providing new insights and a variational derivation for spherical models.

Contribution

It introduces a novel Legendre equivalence between spherical and Gaussian Boltzmann machines and offers a new variational approach to compute free energies.

Findings

Free energies of spherical and Gaussian models are related by a Legendre transform.

Provides a new variational derivation of the free energy for spherical models.

Enhances understanding of the mathematical structure of Boltzmann machines.

Abstract

We study either fully visible and restricted Boltzmann machines with sub-Gaussian random weights and spherical or Gaussian priors. We prove the free energies of the spherical and Gaussian models are related by a Legendre transformation. Incidentally our analysis brings also a new purely variational derivation of the free energy of the spherical models.