Recovering sparse signals with non-convex penalties and DC programming

This paper considers the problem of recovering a sparse signal representation according to a signal dictionary. This problem is usually formalized as a penalized least-squares problem in which sparsity is usually induced by a l1-norm penalty on the coefficient. Such an approach known as the Lasso or Basis Pursuit Denoising has been shown to perform reasonably well in some situations. However, it has also been proved that non-convex penalties like lq-norm with q < 1 or SCAD penalty are able to recover sparsity in a more efficient way than the Lasso. Several algorithms have been proposed for solving the resulting non-convex least-squares problem. This paper proposes a generic algorithm to address such a sparsity recovery problem with non-convex penalty. The main contribution is that our methodology is based on an iterative algorithm which solves at each iteration a convex weighted Lasso problem. It relies on the decomposition of the non-convex penalty into a difference of convex functions. This allows us to apply difference of convex functions programming which is a generic and principled way for solving non-smooth and non-convex optimization problem. We also show that several algorithms in the literature which solve such a problem are particular cases of our algorithm. Experimental results then demonstrate that our method performs better than previously proposed algorithms.