Power-iterative strategy for ℓp−ℓ2 optimization for compressive sensing: Towards global solution

We study nonconvex relaxation of the combinatorial ℓ<inf>0</inf>-minimization for compressive sensing. In an ℓ<inf>p</inf>−ℓ<inf>2</inf> minimization setting with p &#60; 1, we propose an iterative algorithm with two distinct features: (i) use of a proximal-point (P-P) objective function composed of a convex quadratic term and an ℓ<inf>p</inf>-norm term, and a fast parallel-based solver for global minimization of the P-P function in each iteration; and (ii) a power-iterative strategy that begins by solving a convex ℓ<inf>1</inf>−ℓ<inf>2</inf> problem whose solution is then used to start next ℓ<inf>p</inf>−ℓ<inf>2</inf> problem with p close to but less than one. The process continues with gradually reduced p until a target power p<inf>t</inf> is reached. By simulations the algorithm is shown to offer considerable performance gain.

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