The solution of the deep Boltzmann machine on the Nishimori line

TL;DR

This paper provides an exact solution for deep Boltzmann machines on the Nishimori line, revealing phase transitions in the order parameter without magnetic fields through a variational approach.

Contribution

It introduces a theorem that exactly solves deep Boltzmann machines on the Nishimori line using a finite-dimensional variational problem.

Findings

Exact solution for deep Boltzmann machines on the Nishimori line

Identification of phase transition in the order parameter without magnetic fields

Use of a min-max variational problem to characterize the pressure

Abstract

The deep Boltzmann machine on the Nishimori line with a finite number of layers is exactly solved by a theorem that expresses its pressure through a finite dimensional variational problem of min-max type. In the absence of magnetic fields the order parameter is shown to exhibit a phase transition.