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Article: An ℓ∞ eigenvector perturbation bound and its application to robust covariance estimation
Title | An ℓ<inf>∞</inf> eigenvector perturbation bound and its application to robust covariance estimation |
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Authors | |
Keywords | Approximate factor model Sparsity Matrix perturbation theory Incoherence Low-rank matrices |
Issue Date | 2018 |
Citation | Journal of Machine Learning Research, 2018, v. 18, article no. 207 How to Cite? |
Abstract | In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan sin θ theorem is often used to bound the difference between the eigenvectors of a matrix A and those of a perturbed matrix A˜ = A + E, in terms of ℓ2 norm. In this paper, we prove that when A is a low-rank and incoherent matrix, the ℓ∞ norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of √d1 or √d2 for left and right vectors, where d1 and d2 are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments. |
Persistent Identifier | http://hdl.handle.net/10722/303565 |
ISSN | 2023 Impact Factor: 4.3 2023 SCImago Journal Rankings: 2.796 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Fan, Jianqing | - |
dc.contributor.author | Wang, Weichen | - |
dc.contributor.author | Zhong, Yiqiao | - |
dc.date.accessioned | 2021-09-15T08:25:34Z | - |
dc.date.available | 2021-09-15T08:25:34Z | - |
dc.date.issued | 2018 | - |
dc.identifier.citation | Journal of Machine Learning Research, 2018, v. 18, article no. 207 | - |
dc.identifier.issn | 1532-4435 | - |
dc.identifier.uri | http://hdl.handle.net/10722/303565 | - |
dc.description.abstract | In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan sin θ theorem is often used to bound the difference between the eigenvectors of a matrix A and those of a perturbed matrix A˜ = A + E, in terms of ℓ2 norm. In this paper, we prove that when A is a low-rank and incoherent matrix, the ℓ∞ norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of √d1 or √d2 for left and right vectors, where d1 and d2 are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments. | - |
dc.language | eng | - |
dc.relation.ispartof | Journal of Machine Learning Research | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject | Approximate factor model | - |
dc.subject | Sparsity | - |
dc.subject | Matrix perturbation theory | - |
dc.subject | Incoherence | - |
dc.subject | Low-rank matrices | - |
dc.title | An ℓ<inf>∞</inf> eigenvector perturbation bound and its application to robust covariance estimation | - |
dc.type | Article | - |
dc.description.nature | published_or_final_version | - |
dc.identifier.scopus | eid_2-s2.0-85048935983 | - |
dc.identifier.volume | 18 | - |
dc.identifier.spage | article no. 207 | - |
dc.identifier.epage | article no. 207 | - |
dc.identifier.eissn | 1533-7928 | - |
dc.identifier.isi | WOS:000435627900001 | - |