Sparsity amplified

Abstract

The L1 norm is often used as a penalty function to obtain a sparse approximate solution to a system of linear equations, but it often underestimates the true values. This paper proposes a different type of penalty that (1) estimates sparse solutions more accurately and (2) maintains the convexity of the cost function. The new penalty is a multivariate generalization of the minimax-concave (MC) penalty. To define the generalized MC (GMC) penalty we first define a multivariate generalized Huber function. The resulting cost function can be minimized by proximal algorithms comprising simple computations. The effectiveness of the GMC penalty is illustrated in a denoising example.