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Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm
Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called reaper, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer US
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8549967/ https://www.ncbi.nlm.nih.gov/pubmed/34720419 http://dx.doi.org/10.1007/s10851-021-01019-1 |
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author | Beinert, Robert Steidl, Gabriele |
author_facet | Beinert, Robert Steidl, Gabriele |
author_sort | Beinert, Robert |
collection | PubMed |
description | Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called reaper, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must be determined in advance and the minimization is performed over symmetric positive semi-definite matrices having the size of the data, although the number of principal components is substantially smaller. This prohibits its use if the dimension of the data is large which is often the case in image processing. In this paper, we propose a regularized version of reaper which enforces the sparsity of the number of principal components by penalizing the nuclear norm of the corresponding orthogonal projector. If only an upper bound on the number of principal components is available, our approach can be combined with the L-curve method to reconstruct the appropriate subspace. Our second contribution is a matrix-free algorithm to find a minimizer of the regularized reaper which is also suited for high-dimensional data. The algorithm couples a primal-dual minimization approach with a thick-restarted Lanczos process. This appears to be the first efficient convex variational method for robust PCA that can handle high-dimensional data. As a side result, we discuss the topic of the bias in robust PCA. Numerical examples demonstrate the performance of our algorithm. |
format | Online Article Text |
id | pubmed-8549967 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer US |
record_format | MEDLINE/PubMed |
spelling | pubmed-85499672021-10-29 Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm Beinert, Robert Steidl, Gabriele J Math Imaging Vis Article Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called reaper, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must be determined in advance and the minimization is performed over symmetric positive semi-definite matrices having the size of the data, although the number of principal components is substantially smaller. This prohibits its use if the dimension of the data is large which is often the case in image processing. In this paper, we propose a regularized version of reaper which enforces the sparsity of the number of principal components by penalizing the nuclear norm of the corresponding orthogonal projector. If only an upper bound on the number of principal components is available, our approach can be combined with the L-curve method to reconstruct the appropriate subspace. Our second contribution is a matrix-free algorithm to find a minimizer of the regularized reaper which is also suited for high-dimensional data. The algorithm couples a primal-dual minimization approach with a thick-restarted Lanczos process. This appears to be the first efficient convex variational method for robust PCA that can handle high-dimensional data. As a side result, we discuss the topic of the bias in robust PCA. Numerical examples demonstrate the performance of our algorithm. Springer US 2021-02-15 2021 /pmc/articles/PMC8549967/ /pubmed/34720419 http://dx.doi.org/10.1007/s10851-021-01019-1 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Beinert, Robert Steidl, Gabriele Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title | Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title_full | Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title_fullStr | Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title_full_unstemmed | Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title_short | Robust PCA via Regularized Reaper with a Matrix-Free Proximal Algorithm |
title_sort | robust pca via regularized reaper with a matrix-free proximal algorithm |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8549967/ https://www.ncbi.nlm.nih.gov/pubmed/34720419 http://dx.doi.org/10.1007/s10851-021-01019-1 |
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