Hamilton-Jacobi equations for mean-field disordered systems

TL;DR

This paper demonstrates that Hamilton-Jacobi equations are effective for analyzing the large-scale behavior of mean-field disordered systems, exemplified through the inference of a rank-one matrix, by deriving the free energy limit.

Contribution

It introduces a Hamilton-Jacobi framework for mean-field disordered systems and computes the free energy limit via an approximate PDE approach.

Findings

Derived the large-scale limit of free energy using Hamilton-Jacobi equations

Showed the approach applies to rank-one matrix inference

Validated the method with asymptotic analysis

Abstract

We argue that Hamilton-Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton-Jacobi equation with asymptotically vanishing viscosity parameter and error term.