Title

Sparse Approximation Property And Stable Recovery Of Sparse Signals From Noisy Measurements

Keywords

Additive noise; approximation methods; compressed sensing; signal reconstruction

Abstract

In this correspondence, we introduce a sparse approximation property of order s for a measurement matrix A: ∥xs∥2≤D∥Ax∥2+β(σ s(x))/√s for all x, where xs is the best s -sparse approximation of the vector x in ℓ2, σs(x) is the s-sparse approximation error of the vector x in ℓ1 , and D and β are positive constants. The sparse approximation property for a measurement matrix can be thought of as a weaker version of its restricted isometry property and a stronger version of its null space property. In this correspondence, we show that the sparse approximation property is an appropriate condition on a measurement matrix to consider stable recovery of any compressible signal from its noisy measurements. In particular, we show that any compressible signal can be stably recovered from its noisy measurements via solving an ℓ1-minimization problem if the measurement matrix has the sparse approximation property with β∈(0,1), and conversely the measurement matrix has the sparse approximation property with β∈(0,∞) if any compressible signal can be stably recovered from its noisy measurements via solving an ℓ1 -minimization problem. © 2011 IEEE.

Publication Date

10-1-2011

Publication Title

IEEE Transactions on Signal Processing

Volume

59

Issue

10

Number of Pages

5086-5090

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1109/TSP.2011.2161470

Socpus ID

80052879453 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/80052879453

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