Statistical Limits in Random Tensors with Multiple Correlated Spikes

We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank-$r$ deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank-$r$ tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator.