  Tensor models play an increasingly prominent role in many fields, notably in
machine learning. In several applications, such as community detection, topic
modeling and Gaussian mixture learning, one must estimate a low-rank signal
from a noisy tensor. Hence, understanding the fundamental limits of estimators
of that signal inevitably calls for the study of random tensors. Substantial
progress has been recently achieved on this subject in the large-dimensional
limit. Yet, some of the most significant among these results--in particular, a
precise characterization of the abrupt phase transition (with respect to
signal-to-noise ratio) that governs the performance of the maximum likelihood
(ML) estimator of a symmetric rank-one model with Gaussian noise--were derived
based of mean-field spin glass theory, which is not easily accessible to
non-experts. In this work, we develop a sharply distinct and more elementary
approach, relying on standard but powerful tools brought by years of advances
in random matrix theory. The key idea is to study the spectra of random
matrices arising from contractions of a given random tensor. We show how this
gives access to spectral properties of the random tensor itself. For the
aforementioned rank-one model, our technique yields a hitherto unknown
fixed-point equation whose solution precisely matches the asymptotic
performance of the ML estimator above the phase transition threshold in the
third-order case. A numerical verification provides evidence that the same
holds for orders 4 and 5, leading us to conjecture that, for any order, our
fixed-point equation is equivalent to the known characterization of the ML
estimation performance that had been obtained by relying on spin glasses.
Moreover, our approach sheds light on certain properties of the ML problem
landscape in large dimensions and can be extended to other models, such as
asymmetric and non-Gaussian.

Machine Learning;Machine Learning;Probability