Hamilton-Jacobi equations for nonsymmetric matrix inference

TL;DR

This paper analyzes the high-dimensional free energy in nonsymmetric matrix inference problems using Hamilton-Jacobi equations, providing bounds and convergence rates applicable to various vector distributions.

Contribution

It introduces a novel approach linking free energy limits to Hamilton-Jacobi equations and provides bounds and convergence rates for nonsymmetric matrix inference.

Findings

Bound the difference between free energy and Hamilton-Jacobi solution.

Apply the approach to i.i.d. and spherical cases.

Identify limits and convergence rates for different distributions.

Abstract

We study the high-dimensional limit of the free energy associated with the inference problem of a rank-one nonsymmetric matrix. The matrix is expressed as the outer product of two vectors, not necessarily independent. The distributions of the two vectors are only assumed to have scaled bounded supports. We bound the difference between the free energy and the solution to a suitable Hamilton-Jacobi equation in terms of two much simpler quantities: concentration rate of this free energy, and the convergence rate of a simpler free energy in a decoupled system. To demonstrate the versatility of this approach, we apply our result to the i.i.d. case and the spherical case. By plugging in estimates of the two simpler quantities, we identify the limits and obtain convergence rates.