Translation-invariant shrinkage/thresholding of group sparse signals

• We present an algorithm for denoising signals with group-sparse behavior.
• We use fully overlapping groups for shift-invariance and to avoid blocking artifacts.
• The algorithm, ‘overlapping group shrinkage’ (OGS), converges fast and robustly.
• For speech enhancement, the algorithm yields speech relatively free of musical noise.
• A simple method for setting the regularization parameter is presented.

This paper addresses signal denoising when large-amplitude coefficients form clusters (groups). The L1-norm and other separable sparsity models do not capture the tendency of coefficients to cluster (group sparsity). This work develops an algorithm, called ‘overlapping group shrinkage’ (OGS), based on the minimization of a convex cost function involving a group-sparsity promoting penalty function. The groups are fully overlapping so the denoising method is translation-invariant and blocking artifacts are avoided. Based on the principle of majorization–minimization (MM), we derive a simple iterative minimization algorithm that reduces the cost function monotonically. A procedure for setting the regularization parameter, based on attenuating the noise to a specified level, is also described. The proposed approach is illustrated on speech enhancement, wherein the OGS approach is applied in the short-time Fourier transform (STFT) domain. The OGS algorithm produces denoised speech that is relatively free of musical noise.