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

A central problem of random matrix theory is to understand the eigenvalues of ‘spiked’ or ‘deformed’ random matrix models, in which a prominent eigenvector (or ‘spike’) 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 Péché [2005] 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. Subsequently, sharp spectral phase transitions have been proven in many other random matrix models. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise, and how well it can estimate it. 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: • 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. • 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, based on a linearization of approximate message passing, and we give a rigorous and general analysis. • Finally, for both the Gaussian Wishart ensemble and various synchronization problems over groups, we show that computationally 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 an interesting open question. Our results are based on several new tools for establishing that two matrix distributions are contiguous. In some cases, we establish non-asymptotic bounds for hypothesis testing, and also transfer our results to the corresponding estimation problems. ∗The first two authors contributed equally. †Email: [email protected]. This work is supported in part by NSF CAREER Award CCF-1453261 and a grant from the MIT NEC Corporation. ‡Email: [email protected]. This research was conducted with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. §Email: [email protected]. A.S.B. was supported by NSF Grant DMS-1317308. Part of this work was done while A.S.B. was with the Department of Mathematics at the Massachusetts Institute of Technology. ¶Email: [email protected]. This work is supported in part by NSF CAREER Award CCF-1453261, NSF Large CCF-1565235, a grant from the MIT NEC Corporation and a Google Faculty Research Award. 1 ar X iv :1 60 9. 05 57 3v 1 [ m at h. ST ] 1 9 Se p 20 16