  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.
