Statistical thresholds for Tensor PCA

TL;DR

This paper investigates the fundamental limits of detecting and estimating a rank-one signal in Gaussian tensors, identifying sharp thresholds and demonstrating the optimality of maximum likelihood estimators with a phase transition behavior.

Contribution

It precisely characterizes the statistical thresholds for tensor PCA and shows that maximum likelihood estimators achieve optimal correlation above these thresholds.

Findings

Sharp thresholds for hypothesis testing and estimation are computed.

Maximum likelihood estimator achieves maximal correlation above the threshold.

Discontinuous phase transition in estimator performance similar to BBP transition.

Abstract

We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that the maximum likelihood estimator achieves the maximal correlation with the planted vector among measurable estimators above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.