Replica Symmetry Breaking in Bipartite Spin Glasses and Neural Networks

TL;DR

This paper applies replica symmetry breaking analysis to bipartite spin glass models, revealing complex hierarchical states similar to those in unipartite models, with implications for neural network understanding and graph partitioning.

Contribution

It extends the replica symmetry breaking framework to bipartite SK models, connecting them to neural networks and providing new insights into their state structure.

Findings

Bipartite SK model exhibits ultrametric pure states similar to unipartite models.

Ground state energy relates to optimal graph partitioning cost.

RBM outputs trained on MNIST show more ultrametric structure than input data.

Abstract

Some interesting recent advances in the theoretical understanding of neural networks have been informed by results from the physics of disordered many-body systems. Motivated by these findings, this work uses the replica technique to study the mathematically tractable bipartite Sherrington-Kirkpatrick (SK) spin glass model, which is formally similar to a Restricted Boltzmann Machine (RBM) neural network. The bipartite SK model has been previously studied assuming replica symmetry; here this assumption is relaxed and a replica symmetry breaking analysis is performed. The bipartite SK model is found to have many features in common with Parisi's solution of the original, unipartite SK model, including the existence of a multitude of pure states which are related in a hierarchical, ultrametric fashion. As an application of this analysis, the optimal cost for a graph partitioning problem is shown to be simply related to the ground state energy of the bipartite SK model. As a second application, empirical investigations reveal that the Gibbs sampled outputs of an RBM trained on the MNIST data set are more ultrametrically distributed than the input data itself.