Sparse Approximate Solutions to Linear Systems
SIAM Journal on Computing
Greedy algorithms and M-term approximation with regard to redundant dictionaries
Journal of Approximation Theory
On the Optimality of the Backward Greedy Algorithm for the Subset Selection Problem
SIAM Journal on Matrix Analysis and Applications
Atomic Decomposition by Basis Pursuit
SIAM Review
Approximation of functions over redundant dictionaries using coherence
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Sparse signal reconstruction from limited data using FOCUSS: are-weighted minimum norm algorithm
IEEE Transactions on Signal Processing
Uncertainty principles and ideal atomic decomposition
IEEE Transactions on Information Theory
A generalized uncertainty principle and sparse representation in pairs of bases
IEEE Transactions on Information Theory
Sparse representations in unions of bases
IEEE Transactions on Information Theory
On sparse representations in arbitrary redundant bases
IEEE Transactions on Information Theory
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
Stable recovery of sparse overcomplete representations in the presence of noise
IEEE Transactions on Information Theory
High-resolution radar via compressed sensing
IEEE Transactions on Signal Processing
A plurality of sparse representations is better than the sparsest one alone
IEEE Transactions on Information Theory
Distributed recognition of human actions using wearable motion sensor networks
Journal of Ambient Intelligence and Smart Environments
Sparse representations and approximation theory
Journal of Approximation Theory
Two-dimensional random projection
Signal Processing
Distributed recognition of human actions using wearable motion sensor networks
Journal of Ambient Intelligence and Smart Environments
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Given a signal S ∈ RN and a full-rank matrix D ∈ RN × L with N , we define the signal's overcomplete representation as α ∈ RL satisfying S= Dα. Among the infinitely many solutions of this under-determined linear system of equations, we have special interest in the sparsest representation, i.e., the one minimizing ||α||0. This problem has a combinatorial flavor to it, and its direct solution is impossible even for moderate L. Approximation algorithms are thus required, and one such appealing technique is the basis pursuit (BP) algorithm. This algorithm has been the focus of recent theoretical research effort. It was found that if indeed the representation is sparse enough, BP finds it accurately.When an error is permitted in the composition of the signal, we no longer require exact equality S=Dα. The BP has been extended to treat this case, leading to a denoizing algorithm. The natural question to pose is how the above-mentioned theoretical results generalize to this more practical mode of operation. In this paper we propose such a generalization. The behavior of the basis pursuit in the presence of noise has been the subject of two independent very wide contributions released for publication very recently. This paper is another contribution in this direction, but as opposed to the others mentioned, this paper aims to present a somewhat simplified picture of the topic, and thus could be referred to as a primer to this field. Specifically, we establish here the stability of the BP in the presence of noise for sparse enough representations. We study both the case of a general dictionary D, and a special case where D is built as a union of orthonormal bases. This work is a direct generalization of noiseless BP study, and indeed, when the noise power is reduced to zero, we obtain the known results of the noiseless BP.