The solution of the deep Boltzmann machine on the Nishimori line

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.

arXiv comment30 pages, 1 figure. To appear in Communications in Mathematical Physics

References in corpus (11)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• The Thermodynamic Limit in Mean Field Spin Glass Models
• The free energy in a multi-species Sherrington-Kirkpatrick model
• Equilibrium statistical mechanics of bipartite spin systems
• Free energy and complexity of spherical bipartite models
• Griffiths inequalities for the Gaussian spin glass
• Annealing and replica-symmetry in Deep Boltzmann Machines
• Deep Boltzmann machines: rigorous results at arbitrary depth
• Replica symmetry breaking in multi-species Sherrington-Kirkpatrick model
• Surface terms on the Nishimori line of the Gaussian Edwards-Anderson model
• Nonconvex interactions in mean-field spin glasses

Cited by in corpus (9)

• Statistical limits of dictionary learning: random matrix theory and the spectral replica method
• Free energy in multi-species mixed $p$-spin spherical models
• An inference problem in a mismatched setting: a spin-glass model with Mattis interaction
• Hopfield model with planted patterns: a teacher-student self-supervised learning model
• Estimating rank-one matrices with mismatched prior and noise: universality and large deviations
• Statistical-mechanical study of deep Boltzmann machine given weight parameters after training by singular value decomposition
• Gibbs-Bogoliubov inequality on Nishimori line
• Analyticity of the energy in an Ising spin glass with correlated disorder
• Toward mean-field bound for critical temperature on Nishimori line