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arXiv · 2101.09994

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

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

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Luca Corinzia, Paolo Penna, Wojciech Szpankowski, Joachim M. Buhmann. 2021-01-25. On maximum-likelihood estimation in the all-or-nothing regime. https://arxiv.org/abs/2101.09994

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