Statistically and Computationally Optimal Estimation and Inference of Common Subspaces

Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this talk we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference is statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. Consequently, our results unveil a novel phenomenon: despite the SNR being ``above'' the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.

Spectral methods

Multilayer networks

Matrix analysis

Random matrix theory 

Section on Statistical Learning and Data Science