{"title":"Bayesian Sparse Spiked Covariance Model with a Continuous Matrix Shrinkage Prior","authors":"Fangzheng Xie, J. Cape, C. Priebe, Yanxun Xu","doi":"10.1214/21-ba1292","DOIUrl":null,"url":null,"abstract":". We propose a Bayesian methodology for estimating spiked covariance matrices with a jointly sparse structure in high dimensions. The spiked covariance matrix is reparameterized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the spectral norm as well as that for the principal subspace with respect to the projection spectral norm loss. We also study the posterior contraction rate of the principal subspace with respect to the two-to-infinity norm loss, a novel loss function measuring the distance between subspaces that is able to capture entrywise eigenvector perturbations. We show that the posterior contraction rate with respect to the two-to-infinity norm loss is tighter than that with respect to the routinely used projection spectral norm loss under certain low-rank and bounded coherence conditions. In addition, a point estimator for the principal subspace is proposed with the rate-optimal risk bound with respect to the projection spectral norm loss. The numerical performance of the proposed methodology is assessed through synthetic examples and the analysis of a real-world face data example. 62H25, 62C10; secondary 62H12.","PeriodicalId":55398,"journal":{"name":"Bayesian Analysis","volume":" ","pages":""},"PeriodicalIF":4.9000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bayesian Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/21-ba1292","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 3
Abstract
. We propose a Bayesian methodology for estimating spiked covariance matrices with a jointly sparse structure in high dimensions. The spiked covariance matrix is reparameterized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the spectral norm as well as that for the principal subspace with respect to the projection spectral norm loss. We also study the posterior contraction rate of the principal subspace with respect to the two-to-infinity norm loss, a novel loss function measuring the distance between subspaces that is able to capture entrywise eigenvector perturbations. We show that the posterior contraction rate with respect to the two-to-infinity norm loss is tighter than that with respect to the routinely used projection spectral norm loss under certain low-rank and bounded coherence conditions. In addition, a point estimator for the principal subspace is proposed with the rate-optimal risk bound with respect to the projection spectral norm loss. The numerical performance of the proposed methodology is assessed through synthetic examples and the analysis of a real-world face data example. 62H25, 62C10; secondary 62H12.
期刊介绍:
Bayesian Analysis is an electronic journal of the International Society for Bayesian Analysis. It seeks to publish a wide range of articles that demonstrate or discuss Bayesian methods in some theoretical or applied context. The journal welcomes submissions involving presentation of new computational and statistical methods; critical reviews and discussions of existing approaches; historical perspectives; description of important scientific or policy application areas; case studies; and methods for experimental design, data collection, data sharing, or data mining.
Evaluation of submissions is based on importance of content and effectiveness of communication. Discussion papers are typically chosen by the Editor in Chief, or suggested by an Editor, among the regular submissions. In addition, the Journal encourages individual authors to submit manuscripts for consideration as discussion papers.