Hamilton-Jacobi equations for inference of matrix tensor products

TL;DR

This paper investigates the high-dimensional limit of free energy in matrix tensor product inference problems, using Hamilton-Jacobi equations to characterize the limit and providing explicit results for specific models.

Contribution

It introduces a Hamilton-Jacobi framework to analyze the asymptotic free energy of matrix tensor inference, identifying the limit under certain conditions and applying it to specific tensor models.

Findings

Bounded the limit of free energy using Hamilton-Jacobi equations.

Identified the limit explicitly for models with i.i.d. entries and symmetric interactions.

Provided convergence rate estimates for first and even order tensor products.

Abstract

We study the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. In general, we bound the limit from above by the unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on the nonlinearity in the equation which is determined explicitly by the model, we identify the limit with the solution. Two notions of solutions, weak solutions and viscosity solutions, are considered, each of which has its own advantages and requires different treatments. For concreteness, we apply our results to a model with i.i.d. entries and symmetric interactions. In particular, for the first order and even order tensor products, we identify the limit and obtain estimates on convergence rates; for other odd orders, upper bounds are obtained.