Statistical inference of finite-rank tensors

TL;DR

This paper develops a theoretical framework for understanding the statistical inference of finite-rank tensor models, deriving the limit free energy through a variational approach and Hamilton-Jacobi equations.

Contribution

It introduces a general method to compute the limit free energy for finite-rank tensor inference models using viscosity solutions of Hamilton-Jacobi equations.

Findings

Derived the limit free energy as a variational formula.

Connected free energy to viscosity solutions of Hamilton-Jacobi equations.

Applicable to any interaction structure and tensor order.

Abstract

We consider a general statistical inference model of finite-rank tensor products. For any interaction structure and any order of tensor products, we identify the limit free energy of the model in terms of a variational formula. Our approach consists of showing first that the limit free energy must be the viscosity solution to a certain Hamilton-Jacobi equation.