ESTIMATION OF LOW-RANK MATRICES VIA APPROXIMATE MESSAGE PASSING

• 成果类型: Article
• 署名作者: Montanari, Andrea; Venkataramanan, Ramji
• 署名单位: Stanford University; Stanford University; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1958
• 发表日期: 2021
• 页码: 321-345
• 关键词: Principal component analysis; LARGEST EIGENVALUE; phase-transitions; power method; Mutual information; algorithms; deformation

摘要

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise, a setting that is also known as spiked model or deformed random matrix. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper, we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation problems. We develop a new analysis of AMP that allows for spectral initializations, and builds on a decoupling between the outlier eigenvectors and the bulk in the spiked random matrix model. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related-via universality arguments-to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the Gaussian block model).