Abstract
Exact sparse support recovery from noisy deterministic Fourier measurements is studied in this paper. The ideal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -regularized least squares (LS) program is known to have strict local minimizers for multiple sparse supports. By applying a specific non-convex surrogate of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -regularizer, we show that, under certain conditions, the set of local minimizers with particularly separated supports is a singleton. Thus, any converging algorithms that can enforce the corresponding separation constraint will lead to the true support as the unique solution. In this regard, we construct a novel optimal mapping operator that can fulfill the separation condition, and several empirically effective algorithms are developed. The simulation results demonstrate the theoretical claims made in this paper.
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