{"title":"尖峰协方差模型中特征值的最优收缩。","authors":"David L Donoho, Matan Gavish, Iain M Johnstone","doi":"10.1214/17-AOS1601","DOIUrl":null,"url":null,"abstract":"<p><p>We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker <i>η</i> that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker <i>η</i>* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues <i>and</i> eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.</p>","PeriodicalId":8032,"journal":{"name":"Annals of Statistics","volume":"46 4","pages":"1742-1778"},"PeriodicalIF":3.2000,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOS1601","citationCount":"181","resultStr":"{\"title\":\"Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model.\",\"authors\":\"David L Donoho, Matan Gavish, Iain M Johnstone\",\"doi\":\"10.1214/17-AOS1601\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker <i>η</i> that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker <i>η</i>* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues <i>and</i> eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.</p>\",\"PeriodicalId\":8032,\"journal\":{\"name\":\"Annals of Statistics\",\"volume\":\"46 4\",\"pages\":\"1742-1778\"},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2018-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1214/17-AOS1601\",\"citationCount\":\"181\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1214/17-AOS1601\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2018/6/27 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/17-AOS1601","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2018/6/27 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model.
We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.
期刊介绍:
The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.