Legendre equivalences of spherical Boltzmann machines
Giuseppe Genovese, Daniele Tantari · Journal of Physics A Mathematical and Theoretical · 2020
Abstract We study either fully visible and restricted Boltzmann machines with sub-Gaussian random weights and spherical or Gaussian priors. We prove that 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.