Sparse Approximation Property and Stable Recovery of Sparse Signals From Noisy Measurements

    Authors

    Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    IEEE Trans. Signal Process.

    Keywords

    Additive noise; approximation methods; compressed sensing; signal; reconstruction; L(1) MINIMIZATION; RECONSTRUCTION; Engineering, Electrical & Electronic

    Abstract

    In this correspondence, we introduce a sparse approximation property of order for a measurement matrix A : parallel to x(s)parallel to(2) < = D parallel to Ax parallel to(2) + beta(sigma(s)(x))/root s for all x, where x(s) is the best s-sparse approximation of the vector x in l(5), sigma(s)(x) is the s-sparse approximation error of the vector x in l(1), and D and beta 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 l(1)-minimization problem if the measurement matrix has the sparse approximation property with beta is an element of (0, 1), and conversely the measurement matrix has the sparse approximation property with beta is an element of (0, infinity) if any compressible signal can be stably recovered from its noisy measurements via solving an l(1)-minimization problem.

    Journal Title

    Ieee Transactions on Signal Processing

    Volume

    59

    Issue/Number

    10

    Publication Date

    1-1-2011

    Document Type

    Article

    Language

    English

    First Page

    5086

    Last Page

    5090

    WOS Identifier

    WOS:000297111500048

    ISSN

    1053-587X

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