Sparse Coding Algorithm with Negentropy and Weighted ℓ1-Norm for Signal Reconstruction

Compressive sensing theory has attracted widespread attention in recent years and sparse signal reconstruction has been widely used in signal processing and communication. This paper addresses the problem of sparse signal recovery especially with non-Gaussian noise. The main contribution of this paper is the proposal of an algorithm where the negentropy and reweighted schemes represent the core of an approach to the solution of the problem. The signal reconstruction problem is formalized as a constrained minimization problem, where the objective function is the sum of a measurement of error statistical characteristic term, the negentropy, and a sparse regularization term, `p-norm, for 0 < p < 1. The `p-norm, however, leads to a non-convex optimization problem which is difficult to solve efficiently. Herein we treat the `p-norm as a serious of weighted `1-norms so that the sub-problems become convex. We propose an optimized algorithm that combines forward-backward splitting. The algorithm is fast and succeeds in exactly recovering sparse signals with Gaussian and non-Gaussian noise. Several numerical experiments and comparisons demonstrate the superiority of the proposed algorithm.