  We study the statistical limits of testing and estimation for a rank one
deformation of a Gaussian random tensor. We compute the sharp thresholds for
hypothesis testing and estimation by maximum likelihood and show that they are
the same. Furthermore, we find that the maximum likelihood estimator achieves
the maximal correlation with the planted vector among measurable estimators
above the estimation threshold. In this setting, the maximum likelihood
estimator exhibits a discontinuous BBP-type transition: below the critical
threshold the estimator is orthogonal to the planted vector, but above the
critical threshold, it achieves positive correlation which is uniformly bounded
away from zero.
