Pattern reconstruction with restricted Boltzmann machines

TL;DR

This paper analyzes how the tail behavior of the hidden layer prior in restricted Boltzmann machines influences their ability to reconstruct patterns, revealing that tail properties critically determine retrieval performance.

Contribution

It provides a theoretical characterization of the energy landscape and shows how the tail behavior of the hidden prior affects pattern reconstruction capabilities.

Findings

Super-Gaussian tails lead to logarithmic loss in retrieval

Sub-Gaussian tails significantly hinder pattern retrieval

Gaussian tails' retrieval depends on the number of hidden units

Abstract

Restricted Boltzmann machines are energy models made of a visible and a hidden layer. We identify an effective energy function describing the zero-temperature landscape on the visible units and depending only on the tail behaviour of the hidden layer prior distribution. Studying the location of the local minima of such an energy function, we show that the ability of a restricted Boltzmann machine to reconstruct a random pattern depends indeed only on the tail of the hidden prior distribution. We find that hidden priors with strictly super-Gaussian tails give only a logarithmic loss in pattern retrieval, while an efficient retrieval is much harder with hidden units with strictly sub-Gaussian tails; if the hidden prior has Gaussian tails, the retrieval capability is determined by the number of hidden units (as in the Hopfield model).