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Abstract In many applications, we seek to recover signals from linear measurements far fewer than the ambient dimension, given the signals have exploitable structures such as sparse vectors or low rank matrices. In this paper, we work in a general setting where signals are approximately sparse in a so-called atomic set. We provide general recovery results stating that a convex programming can stably and robustly recover signals if the null space of the sensing map satisfies certain properties. Moreover, we argue that such null space property can be satisfied with high probability if each measurement is sub-Gaussian even when the number of measurements are very few. Some new results for recovering signals sparse in a frame, and recovering low rank matrices are also derived as a result.
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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
