The DOI matches its Crossref record; 34 of the 34 references it cites were found in a registry or an index.
Dense Error Correction Via $\ell^1$-Minimization
Abstract
This paper studies the problem of recovering a sparse signal x ∈ ℝnfrom highly corrupted linear measurements y = Ax + e ∈ ℝm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any sufficiently sparse signal x can be recovered by solving an ℓ1-minimization problem min ||x||1 + ||e||1 subject to y = Ax + e. More precisely, if the fraction of the support of the error e is bounded away from one and the support of a: is a very small fraction of the dimension m, then as m becomes large the above ℓ1-minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100% of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of independent identically distributed (i.i.d.) Gaussian vectors with nonzero mean and small variance, dubbed the "cross-and-bouquet" (CAB) model. Simulations and experiments corroborate the findings, and suggest extensions to the result.
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Cites
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Robust Face Recognition via Sparse Representation
Wright, J., Yang, A. Y., Ganesh, A., Sastry, S. S., & Ma, Y. (2009) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2008.79 · Cited by 9,388
Self-citationDecoding by Linear Programming
Candes, E. J., & Tao, T. (2005) · IEEE Transactions on Information Theory · doi:10.1109/tit.2005.858979 · Cited by 7,121
Robust principal component analysis?
Candès, E. J., Li, X., Ma, Y., & Wright, J. (2011) · Journal of the ACM · doi:10.1145/1970392.1970395 · Cited by 6,921
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Greed is Good: Algorithmic Results for Sparse Approximation
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Cited by
Fast ℓ1-minimization algorithms and an application in robust face recognition: A review
Yang, A. Y., Sastry, S. S., Ganesh, A., & Ma, Y. (2010) · 2010 IEEE International Conference on Image Processing · doi:10.1109/icip.2010.5651522 · Cited by 414
Self-citationHalf-Quadratic-Based Iterative Minimization for Robust Sparse Representation
He, R., Zheng, W.-S., Tan, T., & Sun, Z. (2014) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2013.102 · Cited by 342
Fast $\ell_{1}$-Minimization Algorithms for Robust Face Recognition
Yang, A. Y., Zhou, Z., Balasubramanian, A. G., Sastry, S. S., & Ma, Y. (2013) · IEEE Transactions on Image Processing · doi:10.1109/tip.2013.2262292 · Cited by 341
Self-citationGeometric median and robust estimation in Banach spaces
Minsker, S. (2015) · Bernoulli · doi:10.3150/14-bej645 · Cited by 261
Metadata
- Authors
- Wright, J., & Ma, Y.
- Venue
- IEEE Transactions on Information Theory
- Published
- 2010-07
- Volume
- 56
- Issue
- 7
- Pages
- 3540-3560
- ISSN
- 0018-9448 (ISSN-L)
- 1557-9654
- OpenAlex
- W2096586829
- Cited by
- 305
- References
- 34
- FWCI
- 28.74
- Type
- journal article
- Language
- en
- Open access
- closed
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