The dictionary approach for spherical deconvolution

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated `1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives very satisfying results in the numerical study when compared with other procedures.