Exact Reconstruction of Sparse Signals via Nonconvex Minimization

Abstract

Several authors have shown recently that It is possible to reconstruct exactly a sparse signal from fewer linear measurements than would be expected from traditional sampling theory. The methods used involve computing the signal of minimum lscr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> norm among those having the given measurements. We show that by replacing the lscr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> norm with the lscr <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> norm with p < 1, exact reconstruction is possible with substantially fewer measurements. We give a theorem in this direction, and many numerical examples, both in one complex dimension, and larger-scale examples in two real dimensions.

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