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Minimax formula for the replica symmetric free energy of deep restricted Boltzmann machines


Genovese, Giuseppe (2023). Minimax formula for the replica symmetric free energy of deep restricted Boltzmann machines. Annals of Applied Probability, 33(3):2324-2341.

Abstract

We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming inde-pendent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for case.

Abstract

We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming inde-pendent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for case.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability Replica symmetry, restricted Boltzmann machines
Language:English
Date:1 June 2023
Deposited On:09 Jan 2024 11:27
Last Modified:30 Jun 2024 01:36
Publisher:Institute of Mathematical Statistics
ISSN:1050-5164
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/22-AAP1868
  • Content: Published Version
  • Language: English