Solve-Select-Scale: A Three Step Process For Sparse Signal Estimation
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In the theory of compressed sensing (CS), the sparsity x_0 of the unknown signal x∈R^n is of prime importance and the focus of reconstruction algorithms has mainly been either x_0 or its convex relaxation (via x_1). However, it is typically unknown in practice and has remained a challenge when nothing about the size of the support is known. As pointed recently, x_0 might not be the best metric to minimize directly, both due to its inherent complexity as well as its noise performance. Recently a novel stable measure of sparsity s(x) := x_1^2/x_2^2 has been investigated by Lopes Lopes2012, which is a sharp lower bound on x_0. The estimation procedure for this measure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. The usage of the quantity s(x) in sparse signal estimation problems has not received much importance yet. We develop the idea of incorporating s(x) into the signal estimation framework. We also provide a three step algorithm to solve problems of the form Ax=b with no additional assumptions on the original signal x.
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