Papers › Fast L1-L2 minimization via a proximal operator

Fast L1-L2 minimization via a proximal operator

29 Sep 2016arXiv:1609.09530links table onlyarchive 2025-07-28

Yifei Lou, Ming Yan

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This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as L₁ minimization do not work well. Recently, the difference of the L₁ and L₂ norms, denoted as L₁-L₂, is shown to have superior performance over the classic L₁ method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the L₁-L₂ metric, and it makes some fast L₁ solvers such as forward-backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for L₁-L₂. We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of L₁-L₂ based on a difference-of-convex approach in the numerical experiments.

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