On maximum-likelihood estimation in the all-or-nothing regime

TL;DR

This paper analyzes the maximum-likelihood estimator for a sparse rank-1 Gaussian tensor deformation, revealing an all-or-nothing phase transition similar to the MMSE, with technical insights linking MLE and MMSE.

Contribution

It establishes the all-or-nothing phase transition for MLE in sparse tensor estimation and connects MLE behavior to MMSE using novel first and second-moment methods.

Findings

MLE exhibits an all-or-nothing phase transition in sparse tensor estimation.

The phase transition for MLE matches that of MMSE in the same setting.

A new recovery regime for MMSE is identified, stricter than standard error vanishing.

Abstract

We study the problem of estimating a rank-1 additive deformation of a Gaussian tensor according to the \emph{maximum-likelihood estimator} (MLE). The analysis is carried out in the sparse setting, where the underlying signal has a support that scales sublinearly with the total number of dimensions. We show that for Bernoulli distributed signals, the MLE undergoes an \emph{all-or-nothing} (AoN) phase transition, already established for the minimum mean-square-error estimator (MMSE) in the same problem. The result follows from two main technical points: (i) the connection established between the MLE and the MMSE, using the first and second-moment methods in the constrained signal space, (ii) a recovery regime for the MMSE stricter than the simple error vanishing characterization given in the standard AoN, that is here proved as a general result.