Theory of linear and integer programming
Theory of linear and integer programming
Sparse Approximate Solutions to Linear Systems
SIAM Journal on Computing
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Dictionary learning algorithms for sparse representation
Neural Computation
Analysis of sparse representation and blind source separation
Neural Computation
Blind Source Separation by Sparse Decomposition in a Signal Dictionary
Neural Computation
Underdetermined blind source separation based on sparse representation
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
IEEE Transactions on Information Theory
IEEE Transactions on Neural Networks
IEEE Transactions on Neural Networks
Simple Method for High-Performance Digit Recognition Based on Sparse Coding
IEEE Transactions on Neural Networks
Equivalence Probability and Sparsity of Two Sparse Solutions in Sparse Representation
IEEE Transactions on Neural Networks
Learning Bimodal Structure in Audio–Visual Data
IEEE Transactions on Neural Networks
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In sparse representation, two important sparse solutions, the 0-norm and 1-norm solutions, have been receiving much of attention. The 0-norm solution is the sparsest, however it is not easy to obtain. Although the 1-norm solution may not be the sparsest, it can be easily obtained by the linear programming method. In many cases, the 0-norm solution can be obtained through finding the 1-norm solution. Many discussions exist on the equivalence of the two sparse solutions. This paper analyzes two conditions for the equivalence of the two sparse solutions. The first condition is necessary and sufficient, however, difficult to verify. AIthough, the second is necessary but is not sufficient, it is easy to verify. In this paper, we analyze the second condition within the stochastic framework and propose a variant. We then prove that the equivalence of the two sparse solutions holds with high probability under the variant of the second condition. Furthermore, in the limit case where the 0- norm solution is extremely sparse, the second condition is also a sufficient condition with probability 1.