Replica Symmetry Breaking in Bipartite Spin Glasses and Neural Networks
This work provides theoretical insights into neural network behavior for researchers in statistical physics and machine learning, but it is incremental as it extends known replica symmetry breaking techniques to a bipartite model.
The authors tackled the problem of understanding neural networks through the lens of disordered many-body physics by analyzing the bipartite Sherrington-Kirkpatrick spin glass model with replica symmetry breaking, finding it shares hierarchical ultrametric features with the original model and applying this to graph partitioning and showing that RBM outputs on MNIST data are more ultrametrically distributed than inputs.
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.