Sparse Signal Approximation via Nonseparable Regularization

The calculation of a sparse approximate solution to a linear system of equations is often performed using either L1-norm regularization and convex optimization or nonconvex regularization and nonconvex optimization. Combining these principles, this paper describes a type of nonconvex regularization that maintains the convexity of the objective function, thereby allowing the calculation of a sparse approximate solution via convex optimization. The preservation of convexity is viable in the proposed approach because it uses a regularizer that is nonseparable. The proposed method is motivated and demonstrated by the calculation of sparse signal approximation using tight frames. Examples of denoising demonstrate improvement relative to L1 norm regularization.