Sparse and low-rank matrix decompositions
Venkat Chandrasekaran,Sujay Sanghavi,Pablo A. Parrilo,Alan S. Willsky +3 more
- pp 962-967
TLDR
The uncertainty principle is a quantification of the notion that a matrix cannot be sparse while having diffuse row/column spaces and forms the basis for the decomposition method and its analysis.Abstract:
We consider the following fundamental problem: given a matrix that is the sum of an unknown sparse matrix and an unknown low-rank matrix, is it possible to exactly recover the two components? Such a capability enables a considerable number of applications, but the goal is both ill-posed and NP-hard in general. In this paper we develop (a) a new uncertainty principle for matrices, and (b) a simple method for exact decomposition based on convex optimization. Our uncertainty principle is a quantification of the notion that a matrix cannot be sparse while having diffuse row/column spaces. It characterizes when the decomposition problem is ill-posed, and forms the basis for our decomposition method and its analysis. We provide deterministic conditions — on the sparse and low-rank components — under which our method guarantees exact recovery.read more
Citations
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Book ChapterDOI
Robust photometric stereo via low-rank matrix completion and recovery
TL;DR: This work presents a new approach to robustly solve photometric stereo problems by using advanced convex optimization techniques that are guaranteed to find the correct low-rank matrix by simultaneously fixing its missing and erroneous entries.
Journal ArticleDOI
Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization
Yuan Shen,Zaiwen Wen,Yin Zhang +2 more
TL;DR: Numerical studies indicate that the effectiveness of the proposed model is limited to problems where the sparse matrix does not dominate the low-rank one in magnitude, but results show that the proposed method in general has a much faster solution speed than nuclear-norm minimization algorithms and often provides better recoverability.
Journal ArticleDOI
Bayesian Robust Principal Component Analysis
TL;DR: The Bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low-rank and sparse-outlier contributions; the model is robust to a broad range of noise levels, without having to change model hyperparameter settings.
Proceedings Article
SpaRCS: Recovering low-rank and sparse matrices from compressive measurements
TL;DR: This work proposes a natural optimization problem for signal recovery under this model and develops a new greedy algorithm called SpaRCS to solve it, which inherits a number of desirable properties from the state-of-the-art CoSaMP and ADMiRA algorithms.
Journal ArticleDOI
Low-Rank Matrix Recovery From Errors and Erasures
TL;DR: In this article, the authors considered the recovery of a low-rank matrix from an observed version that simultaneously contains both erasures, most entries are not observed, and errors, values at a constant fraction of (unknown) locations are arbitrarily corrupted.
References
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