Given a matrix of noisy data, principal component analysis (PCA) can be viewed as a "de-noising" technique that recovers the closest rank-one approximation. In certain matrix models, it is known that this procedure exhibits a phase transition: if the signal-to-noise ratio is below a critical value then PCA returns uninformative information. In this talk, we also consider a generalization of this problem to k-tensors (the matrix case corresponds to k=2). By studying the energy landscape of this model, we also find phase transitions akin to the matrix case. The proof of the results uses the Kac-Rice formula for the expected number of critical points of a random function and results about spiked random matrices.

Based on joint work with Gerard Ben Arous, Song Mei and Andrea Montanari.