{"title":"高维非凸 LASSO 型 M 估计器","authors":"Jad Beyhum , François Portier","doi":"10.1016/j.jmva.2024.105303","DOIUrl":null,"url":null,"abstract":"<div><p>A theory is developed to examine the convergence properties of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norm penalized high-dimensional <span><math><mi>M</mi></math></span>-estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>0</mn></mrow></msub><msqrt><mrow><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mi>d</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi></mrow></msqrt></mrow></math></span>, where <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the number of nonzero coefficients of the parameter of interest. Sufficient conditions for our main assumptions are then developed and finally used in several examples including robust linear regression, binary classification and nonlinear least squares.</p></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"High-dimensional nonconvex LASSO-type M-estimators\",\"authors\":\"Jad Beyhum , François Portier\",\"doi\":\"10.1016/j.jmva.2024.105303\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A theory is developed to examine the convergence properties of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norm penalized high-dimensional <span><math><mi>M</mi></math></span>-estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>0</mn></mrow></msub><msqrt><mrow><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mi>d</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi></mrow></msqrt></mrow></math></span>, where <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the number of nonzero coefficients of the parameter of interest. Sufficient conditions for our main assumptions are then developed and finally used in several examples including robust linear regression, binary classification and nonlinear least squares.</p></div>\",\"PeriodicalId\":16431,\"journal\":{\"name\":\"Journal of Multivariate Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-02-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Multivariate Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0047259X24000101\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Multivariate Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0047259X24000101","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
A theory is developed to examine the convergence properties of -norm penalized high-dimensional -estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence , where is the number of nonzero coefficients of the parameter of interest. Sufficient conditions for our main assumptions are then developed and finally used in several examples including robust linear regression, binary classification and nonlinear least squares.
期刊介绍：
Founded in 1971, the Journal of Multivariate Analysis (JMVA) is the central venue for the publication of new, relevant methodology and particularly innovative applications pertaining to the analysis and interpretation of multidimensional data.
The journal welcomes contributions to all aspects of multivariate data analysis and modeling, including cluster analysis, discriminant analysis, factor analysis, and multidimensional continuous or discrete distribution theory. Topics of current interest include, but are not limited to, inferential aspects of
Copula modeling
Functional data analysis
Graphical modeling
High-dimensional data analysis
Image analysis
Multivariate extreme-value theory
Sparse modeling
Spatial statistics.
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