Derandomized compressed sensing with nonuniform guarantees for ℓ_1 recovery

12/27/2019
by   Charles Clum, et al.
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We extend the techniques of Hügel, Rauhut and Strohmer (Found. Comput. Math., 2014) to show that for every δ∈(0,1], there exists an explicit random m× N partial Fourier matrix A with m=spolylog(N/ϵ) and entropy s^δpolylog(N/ϵ) such that for every s-sparse signal x∈C^N, there exists an event of probability at least 1-ϵ over which x is the unique minimizer of z_1 subject to Az=Ax. The bulk of our analysis uses tools from decoupling to estimate the extreme singular values of the submatrix of A whose columns correspond to the support of x.

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