Legendre Equivalences of Spherical 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.

arXiv commenttypos corrected

References in corpus (7)

• Equilibrium statistical mechanics of bipartite spin systems
• Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors
• The replica symmetric behavior of the analogical neural network
• Phase transitions in Restricted Boltzmann Machines with generic priors
• Free energy and complexity of spherical bipartite models
• A spherical Hopfield model
• Legendre Duality of Spherical and Gaussian Spin Glasses

Cited by in corpus (6)

• Restricted Boltzmann Machine, recent advances and mean-field theory
• Hopfield model with planted patterns: a teacher-student self-supervised learning model
• Dense Hopfield Networks in the Teacher-Student Setting
• Gaussian-Spherical Restricted Boltzmann Machines
• Modeling Structured Data Learning with Restricted Boltzmann Machines in the Teacher-Student Setting
• A remark on the spherical bipartite spin glass