
Basis Pursuit and Orthogonal Matching Pursuit for Subspacepreserving Recovery: Theoretical Analysis
Given an overcomplete dictionary A and a signal b = Ac^* for some sparse...
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Greedy Signal Space Recovery Algorithm with Overcomplete Dictionaries in Compressive Sensing
Compressive Sensing (CS) is a new paradigm for the efficient acquisition...
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BlockSparse Recovery via Convex Optimization
Given a dictionary that consists of multiple blocks and a signal that li...
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Exact Reconstruction Conditions for Regularized Modified Basis Pursuit
In this correspondence, we obtain exact recovery conditions for regulari...
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Sparse Representation Classification Beyond L1 Minimization and the Subspace Assumption
The sparse representation classifier (SRC) proposed in Wright et al. (20...
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Sparse Signal Subspace Decomposition Based on Adaptive Overcomplete Dictionary
This paper proposes a subspace decomposition method based on an overcom...
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The Use of Mutual Coherence to Prove ℓ^1/ℓ^0Equivalence in Classification Problems
We consider the decomposition of a signal over an overcomplete set of ve...
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SubspaceSparse Representation
Given an overcomplete dictionary A and a signal b that is a linear combination of a few linearly independent columns of A, classical sparse recovery theory deals with the problem of recovering the unique sparse representation x such that b = A x. It is known that under certain conditions on A, x can be recovered by the Basis Pursuit (BP) and the Orthogonal Matching Pursuit (OMP) algorithms. In this work, we consider the more general case where b lies in a lowdimensional subspace spanned by some columns of A, which are possibly linearly dependent. In this case, the sparsest solution x is generally not unique, and we study the problem that the representation x identifies the subspace, i.e. the nonzero entries of x correspond to dictionary atoms that are in the subspace. Such a representation x is called subspacesparse. We present sufficient conditions for guaranteeing subspacesparse recovery, which have clear geometric interpretations and explain properties of subspacesparse recovery. We also show that the sufficient conditions can be satisfied under a randomized model. Our results are applicable to the traditional sparse recovery problem and we get conditions for sparse recovery that are less restrictive than the canonical mutual coherent condition. We also use the results to analyze the sparse representation based classification (SRC) method, for which we get conditions to show its correctness.
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