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Robust principal component analysis (RPCA) aims to recover a low-rank structure from noisy, partially observed data that is also corrupted by sparse, potentially large-magnitude outliers. Traditional RPCA models rely on convex relaxations, such as nuclear norm and $$ℓ_1$$ norm, to approximate the rank of a matrix and the ℓ0 functional (the number of non-zero elements) of another. In this work, we advocate a nonconvex regularization method, referred to as transformed $$ℓ_1$$ (TL1), to improve both approximations. The rationale is that by varying the internal parameter of TL1, its behavior asymptotically approaches either $$ℓ_0$$ or $$ℓ_1$$ . Since the rank is equal to the number of non-zero singular values and the nuclear norm is defined as their sum, applying TL1 to the singular values can approximate either the rank or the nuclear norm, depending on its internal parameter. We conduct a fine-grained theoretical analysis of statistical convergence rates, measured in the Frobenius norm, for both the low-rank and sparse components under general sampling schemes. These rates are comparable to those of the classical RPCA model based on the nuclear norm and $$ℓ_1$$ norm. Moreover, we establish constant-order upper bounds on the estimated rank of the low-rank component and the cardinality of the sparse component in the regime where TL1 behaves like $$ℓ_0,$$ assuming that the respective matrices are exactly low-rank and exactly sparse. Extensive numerical experiments on synthetic data and real-world applications demonstrate that the proposed approach achieves higher accuracy than the classic convex model, especially under non-uniform sampling schemes.
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e RPCA : Robust Principal Component Analysis for Exponential Family Distributions
Abstract Robust principal component analysis (RPCA) is a widely used method for recovering low‐rank structure from data matrices corrupted by significant and sparse outliers. These corruptions may arise from occlusions, malicious tampering, or other causes for anomalies, and the joint identification of such corruptions with low‐rank background is critical for process monitoring and diagnosis. However, existing RPCA methods and their extensions largely do not account for the underlying probabilistic distribution for the data matrices, which in many applications are known and can be highly non‐Gaussian. We thus propose a new method called RPCA for exponential family distributions (), which can perform the desired decomposition into low‐rank and sparse matrices when such a distribution falls within the exponential family. We present a novel alternating direction method of multiplier optimization algorithm for efficient decomposition, under either its natural or canonical parametrization. The effectiveness of is then demonstrated in two applications: the first for steel sheet defect detection and the second for crime activity monitoring in the Atlanta metropolitan area.
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- PAR ID:
- 10518507
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Statistical Analysis and Data Mining: The ASA Data Science Journal
- Volume:
- 17
- Issue:
- 2
- ISSN:
- 1932-1864
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1002/sam.11670
