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Greed is Good: Algorithmic Results for Sparse Approximation
Abstract
This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.
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Citations
Cited by
Donoho, D. L. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.871582 · Cited by 23,994
Candes, E. J., Romberg, J., & Tao, T. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2005.862083 · Cited by 15,639
A sparse signal reconstruction perspective for source localization with sensor arrays
Malioutov, D., Cetin, M., & Willsky, A. S. (2005) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2005.850882 · Cited by 2,638
Sparse Representation for Color Image Restoration
Mairal, J., Elad, M., & Sapiro, G. (2008) · IEEE Transactions on Image Processing · doi:10.1109/tip.2007.911828 · Cited by 1,716
Ouyang, R., Curtarolo, S., Ahmetcik, E., Scheffler, M., & Ghiringhelli, L. M. (2018) · Physical Review Materials · doi:10.1103/physrevmaterials.2.083802 · Cited by 944
Inferring biological networks by sparse identification of nonlinear dynamics
Mangan, N. M., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · IEEE Transactions on Molecular, Biological and Multi-Scale Communications · doi:10.1109/tmbmc.2016.2633265 · Cited by 436
Liang, H., & Li, Q. (2016) · Remote Sensing · doi:10.3390/rs8020099 · Cited by 238
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Large Margin Local Estimate With Applications to Medical Image Classification
Song, Y., Cai, W., Huang, H., Zhou, Y., Feng, D. D., Wang, Y., Fulham, M. J., & Chen, M. (2015) · IEEE Transactions on Medical Imaging · doi:10.1109/tmi.2015.2393954 · Cited by 87
Hernandez, E. M. (2014) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2013.12.009 · Cited by 74
Energy-Efficient Multi-Mode Compressed Sensing System for Implantable Neural Recordings
Suo, Y., Zhang, J., Xiong, T., Chin, P. S., Etienne-Cummings, R., & Tran, T. D. (2014) · IEEE Transactions on Biomedical Circuits and Systems · doi:10.1109/tbcas.2014.2359180 · Cited by 47
Multi-Focus Image Fusion: Algorithms, Evaluation, and a Library
Zafar, R., Farid, M. S., & Khan, M. H. (2020) · Journal of Imaging · doi:10.3390/jimaging6070060 · Cited by 29
Sparsity-Aware Learning and Compressed Sensing: An Overview
Theodoridis, S., Kopsinis, Y., & Slavakis, K. (2014) · Academic Press Library in Signal Processing · doi:10.1016/b978-0-12-396502-8.00023-1 · Cited by 14
İLERİ-GERİ TAKİP ALGORİTMASI TABANLI SEYREK AŞIRI ÖĞRENME MAKİNESİ
ALÇİN, Ö. F., ŞENGÜR, A., & İNCE, M. C. (2015) · Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi · doi:10.17341/gummfd.91022 · Cited by 2
Metadata
- Authors
- Tropp, J. A.
- Venue
- IEEE Transactions on Information Theory
- Published
- 2004-10
- Volume
- 50
- Issue
- 10
- Pages
- 2231-2242
- ISSN
- 0018-9448 (ISSN-L)
- OpenAlex
- W2116148865
- Cited by
- 3,635
- References
- 0
- FWCI
- 75.86
- Type
- journal article
- Language
- en
- Open access
- closed
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