A study on conditions for sparse solution recovery in compressive sensing

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It is well-known by now that tinder suitable conditions ℓ1 minimization can recover sparse solutions to under-determined linear systems of equations. More precisely, by solving the convex optimization problem min{∥ x∥1 : Ax = b}, where A is an m x n measurement matrix with m < n, one can obtain the sparsest solution x* to Ax = b provided that the measurement matrix A has certain properties and the sparsity level k of x* is sufficiently small. This fact has led to active research in the area of compressive sensing and other applications. The central question for this problem is the following. Given a type of measurements, a signal's length n and sparsity level k, what is the minimum measurement size m that ensures recovery? Or equivalently, given a type of measurements, a signal length n and a measurement size m, what is the maximum recoverable sparsity level k? The above fundamental question has been answered, with varying degrees of precision, by a number of researchers for a number of different random or semi-random measurement matrices. However, all the existing results still involve unknown constants of some kind and thus are unable to provide precise answers to specific situations. For example, let A be an m x n partial DCT matrix with n = 107 and m = 5 x 105 (n/m = 20). Can we provide a reasonably good estimate on the maximum recoverable sparsity k? In this research we attempt to provide a more precise answer to the central question raised above. By studying new sufficient conditions for exact recovery of sparse solutions, we propose a new technique to estimate recoverable sparsity for different kinds of deterministic, random and semi-random matrices. We will present empirical evidence to show the practical success of our approach, though further research is still needed to formally establish its effectiveness.

Doctor of Philosophy

Eydelzon, Anatoly. "A study on conditions for sparse solution recovery in compressive sensing." (2008) Diss., Rice University. https://hdl.handle.net/1911/22283.

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