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

Luca Corinzia, Paolo Penna, Wojciech Szpankowski, Joachim M. Buhmann

Abstract: We study the problem of estimating a rank-1additive deformation of a Gaussian tensor according to the 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 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. A full version of this paper is accessible at: https://arxiv.org/pdf/2101.09994.pdf