Year: 2021
Type: article
Abstract: 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, calledreaper, 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 ofreaperwhich 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 theL-curve method to reconstruct the appropriate subspace. Our second contribution is a matrix-free algorithm to find a minimizer of the regularizedreaperwhich 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.
Authors: Robert Beinert, Gabriele Steidl
Institution: Technische Universität Berlin
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,780
OpenAlex ID: W3024545334