Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization

TL;DR

This paper investigates the limits of PCA and other methods in detecting low-rank signals in spiked random matrices, revealing optimal thresholds, suboptimality of PCA in non-Gaussian cases, and gaps between statistical and computational feasibility.

Contribution

It provides a unified analysis of PCA's optimality in Gaussian models, introduces a pre-transform variant for non-Gaussian ensembles, and discusses the gap between statistical possibility and computational efficiency in detection.

Findings

PCA achieves optimal detection in Gaussian Wigner models with benign priors.

PCA is suboptimal in non-Gaussian Wigner models, but a pre-transform improves detection.

There is a conjectural gap between statistical detectability and efficient algorithms in certain models.

Abstract

A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, in which a prominent eigenvector is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and P\'ech\'e showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the signal strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise. However, not all the information about the spike is necessarily contained in the spectrum. We study the fundamental limitations of statistical methods, including non-spectral ones. Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike. We extend previous work on the spherically symmetric and i.i.d. Rademacher priors through an elementary, unified analysis. II) For any non-Gaussian Wigner ensemble, we show that PCA is always suboptimal for detection. However, a variant of PCA achieves the optimal threshold (for benign priors) by pre-transforming the matrix entries according to a carefully designed function. This approach has been stated before, and we give a rigorous and general analysis. III) For both the Gaussian Wishart ensemble and various synchronization problems over groups, we show that inefficient procedures can work below the threshold where PCA succeeds, whereas no known efficient algorithm achieves this. This conjectural gap between what is statistically possible and what can be done efficiently remains open.