Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization
Amelia Perry111The first two authors contributed equally. Email: This work is supported in part by NSF CAREER Award CCF-1453261 and a grant from the MIT NEC Corporation. Department of Mathematics, Massachusetts Institute of Technology Alexander S. Wein Email: 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. Department of Mathematics, Massachusetts Institute of Technology Afonso S. Bandeira Email: 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. Department of Mathematics, Massachusetts Institute of Technology Department of Mathematics and Center for Data Science, Courant Institute of Mathematical Sciences, New York University Ankur Moitra Email: 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. Department of Mathematics, Massachusetts Institute of Technology Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology
Abstract
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.
原文 arXiv:1609.05573;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1609.05573v2