@MISC{Berinde09advancesin, author = {Radu Berinde}, title = {Advances in sparse signal recovery methods}, year = {2009} }

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Abstract

The general problem of obtaining a useful succinct representation (sketch) of some piece of data is ubiquitous; it has applications in signal acquisition, data compression, sub-linear space algorithms, etc. In this thesis we focus on sparse recovery, where the goal is to recover sparse vectors exactly, and to approximately recover nearly-sparse vectors. More precisely, from the short representation of a vector x, we want to recover a vector x ∗ such that the approximation erorr ‖x − x ∗ ‖ is comparable to the “tail ” minx ′ ‖x − x ′ ‖ where x ′ ranges over all vectors with at most k terms. The sparse recovery problem has been subject to extensive research over the last few years, notably in areas such as data stream computing and compressed sensing. We consider two types of sketches: linear and non-linear. For the linear sketching case, where the compressed representation of x is Ax for a measurement matrix A, we introduce a class of binary sparse matrices as valid measurement matrices. We show that they can be used with the popular geometric “ℓ1 minimization ” recovery procedure. We also present two iterative recovery algorithms, Sparse Matching Pursuit and Sequential Sparse Matching Pursuit, that can be used with the same matrices. Thanks to the sparsity of the matrices, the resulting algorithms are