The DOI matches its Crossref record; 20 of the 20 references it cites were found in a registry or an index.
On the Uniqueness of Nonnegative Sparse Solutions to Underdetermined Systems of Equations
Abstract
An underdetermined linear system of equationsAx=bwith nonnegativity constraintxges 0 is considered. It is shown that for matricesAwith a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required sparsity depends on a coherence property of the matrixA. This coherence measure can be improved by applying a conditioning stage onA, thereby strengthening the claimed result. The obtained uniqueness theorem relies on an extended theoretical analysis of the lscr0- lscr1equivalence developed here as well, considering a matrixAwith arbitrary column norms, and an arbitrary monotone element-wise concave penalty replacing the lscr1-norm objective function. Finally, from a numerical point of view, a greedy algorithm-a variant of the matching pursuit-is presented, such that it is guaranteed to find this sparse solution. It is further shown how this algorithm can benefit from well-designed conditioning ofA.
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Cites
Matching pursuits with time-frequency dictionaries
Mallat, S. G., & Zhang, Z. (1993) · IEEE Transactions on Signal Processing · doi:10.1109/78.258082 · Cited by 9,112
Atomic Decomposition by Basis Pursuit
Chen, S. S., Donoho, D. L., & Saunders, M. A. (2001) · SIAM Review · doi:10.1137/s003614450037906x · Cited by 5,085
Pati, Y. C., Rezaiifar, R., & Krishnaprasad, P. S. (2002) · Proceedings of 27th Asilomar Conference on Signals, Systems and Computers · doi:10.1109/acssc.1993.342465 · Cited by 4,323
Greed is Good: Algorithmic Results for Sparse Approximation
Tropp, J. A. (2004) · IEEE Transactions on Information Theory · doi:10.1109/tit.2004.834793 · Cited by 3,635
Entropy-based algorithms for best basis selection
Coifman, R. R., & Wickerhauser, M. V. (1992) · IEEE Transactions on Information Theory · doi:10.1109/18.119732 · Cited by 3,188
Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization
Donoho, D. L., & Elad, M. (2003) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0437847100 · Cited by 2,898
Self-citationStable recovery of sparse overcomplete representations in the presence of noise
Donoho, D. L., Elad, M., & Temlyakov, V. N. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2005.860430 · Cited by 2,183
Self-citationUncertainty principles and ideal atomic decomposition
Donoho, D. L., & Huo, X. (2001) · IEEE Transactions on Information Theory · doi:10.1109/18.959265 · Cited by 1,949
Just relax: convex programming methods for identifying sparse signals in noise
Tropp, J. A. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2005.864420 · Cited by 1,407
Uncertainty Principles and Signal Recovery
Donoho, D. L., & Stark, P. B. (1989) · SIAM Journal on Applied Mathematics · doi:10.1137/0149053 · Cited by 1,054
Cited by
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
Bioucas-Dias, J. M., Plaza, A., Dobigeon, N., Parente, M., Du, Q., Gader, P., & Chanussot, J. (2012) · IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing · doi:10.1109/jstars.2012.2194696 · Cited by 2,836
Comparing Measures of Sparsity
Hurley, N., & Rickard, S. (2009) · IEEE Transactions on Information Theory · doi:10.1109/tit.2009.2027527 · Cited by 757
Bioucas-Dias, J. M., & Figueiredo, M. A. T. (2010) · 2010 2nd Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing · doi:10.1109/whispers.2010.5594963 · Cited by 658
Hyperspectral Unmixing via $L_{1/2}$ Sparsity-Constrained Nonnegative Matrix Factorization
Qian, Y., Jia, S., Zhou, J., & Robles-Kelly, A. (2011) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2011.2144605 · Cited by 598
Collaborative Sparse Regression for Hyperspectral Unmixing
Iordache, M.-D., Bioucas-Dias, J. M., & Plaza, A. (2014) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2013.2240001 · Cited by 485
Metadata
- Authors
- Bruckstein, A. M., Elad, M., & Zibulevsky, M.
- Venue
- IEEE Transactions on Information Theory
- Published
- 2008-11
- Volume
- 54
- Issue
- 11
- Pages
- 4813-4820
- ISSN
- 0018-9448 (ISSN-L)
- OpenAlex
- W2135161976
- Cited by
- 285
- References
- 20
- FWCI
- 15.05
- Type
- journal article
- Language
- en
- Open access
- closed
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