A new regularization for sparse optimization

Abstract

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Several numerical studies have shown that non-convex sparsity-induced regularization can outperform the convex ℓ1-penalty. In this article, we introduce a new non-convex and non-smooth regularization. This new regularization is a continuous and separable function which provides a tighter approximation to the cardinality function than any ℓq-penalty (0 < q < 1). We then apply the Proximal Gradient Method to solve a regularized optimization problem with the new regularization. The convergence analysis shows that the algorithm converges to a critical point and we also provide a pseudo-code for fast implementation. In addition, we conduct a simple numerical experiment with a regularized least square problem to illustrate the performance of the new regularization.