Statistical Thresholds for Tensor PCA

Recent work in theoretical physics has produced numerous conjectures linking problems in high- dimensional statistics to the behavior of disordered magnetic materials known as spin glasses. In this talk, I will discuss results from joint work with Aukosh Jagannath and Léo Miolane that rigorously verify some of these predictions for the spiked tensor model. This model consists of a low-rank tensor corrupted by Gaussian noise, and is useful for studying a variety of questions involving denoising high-dimensional tensor data. Specifically, we consider (i) estimating the planted tensor by maximum likelihood and (ii) testing against the null of a purely Gaussian tensor.
In the rank one case, we show there is a sharp threshold in the norm of the planted vector below which both estimation and testing are uninformative. Further, we find that the maximum likelihood estimator exhibits a discontinuous phase transition. Below the critical threshold, the estimator is orthogonal to the planted vector, but above this threshold, it achieves positive correlation, which is uniformly bounded away from zero. I will conclude by discussing the higher-rank case and future research directions.

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