On maximum-likelihood estimation in the all-or-nothing regime
This work provides theoretical insights into estimation methods for sparse signals, but it is incremental as it extends known results from MMSE to MLE.
The paper tackles the problem of estimating a rank-1 additive deformation of a Gaussian tensor using the maximum-likelihood estimator (MLE) in a sparse setting, showing that the MLE undergoes an all-or-nothing phase transition similar to the minimum mean-square-error estimator (MMSE).
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.