Improved Bounds on Restricted Isometry Constants for Gaussian Matrices
SIAM Journal on Matrix Analysis and Applications
Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics
Mathematics of Operations Research
An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices
Journal of Mathematical Imaging and Vision
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We discuss necessary and sufficient conditions for a sensing matrix to be “s-good”—to allow for exact ℓ 1-recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect ℓ 1-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the ℓ 1-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ 1-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.