In this talk, we will investigate the fundamental limits of detecting the presence of a structured low-rank signal buried inside a large noise matrix. This setting serves among other things as a simple model for principal component analysis: Given a set of data points in Euclidean space, find out whether there exists a distinguished direction (a "spike") along which these data points align.
It is known from random matrix theory that the top eigenpair of the data matrix becomes indicative of the presence of the spike when and only when the strength of the spike is above a certain "spectral" threshold.
A natural question is then whether it is possible to identify the spike, or even tell if it's really present in the data below this spectral threshold?
I will show that the answer depends on the structure of this spike and then completely characterize the fundamental limits of its detection and estimation.
The analysis leading to this characterization relies on a connection with the mean-field theory of spin glasses, more specifically the study of the Sherrington-Kirkpatrick spin-glass model.
I will introduce the necessary tools and show how they can be used to obtain a precise control on the behavior of the posterior distribution of the spike as well as the fluctuations of the associated likelihood ratio process.