Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients

Sparse regression often uses `p norm priors (with p < 2). This paper demonstrates that the introduction of mixed-norms in such contexts allows one to go one step beyond in signal models, and promote some different, structured, forms of sparsity. It is shown that the particular case of the `1,2 and `2,1 norms leads to new group shrinkage operators. Mixed norm priors are shown to be particularly efficient in a generalized basis pursuit denoising approach, and are also used in a context of morphological component analysis. A suitable version of the Block Coordinate Relaxation algorithm is derived for the latter. The group-shrinkage operators are then modified to overcome some limitations of the mixed-norms. The proposed group shrinkage operators are tested on simulated signals in specific situations, to illustrate and compare their different behaviors. Results on real data are also used to illustrate the relevance of the approach.