Statistical Limits for Finite-Rank Tensor Estimation

TL;DR

This paper develops a comprehensive theoretical framework to analyze the fundamental limits of finite-rank tensor estimation problems, including complex scenarios with nonlinear observations and heteroskedastic noise.

Contribution

It introduces asymptotically exact formulas for mutual information and MMSE, and applies them to novel tensor estimation and permutation recovery problems.

Findings

Derived sharp statistical thresholds for tensor estimation under heteroskedastic noise.

Provided exact formulas for mutual information and MMSE in high-dimensional tensor models.

Characterized limits for permutation recovery from tensor data.

Abstract

This paper provides a unified framework for analyzing tensor estimation problems that allow for nonlinear observations, heteroskedastic noise, and covariate information. We study a general class of high-dimensional models where each observation depends on the interactions among a finite number of unknown parameters. Our main results provide asymptotically exact formulas for the mutual information (equivalently, the free energy) as well as the minimum mean-squared error in the Bayes-optimal setting. We then apply this framework to derive sharp characterizations of statistical thresholds for two novel scenarios: (1) tensor estimation in heteroskedastic noise that is independent but not identically distributed, and (2) higher-order assignment problems, where the goal is to recover an unknown permutation from tensor-valued observations.