Abstract. In this paper we provide an algorithm allowing to solve the variational Bayesian issue as a functional optimization problem. The main contribution of this paper is to transpose a classical iterative algorithm of optimization in the metric space of probability densities involved in the Bayesian methodology. The main advantage of this methodology is that it allows to address large dimensional inverse problems by unsupervised algorithms. The interest of our algorithm is enhanced by its application to large dimensional linear inverse problems involving sparse objects. Finally, we provide simulation results. First we show the good numerical performances of our method by comparing it with classical ones on a small tomographic problem. On a second time we treat a large dimensional dictionary learning problem and compare our method with a wavelet based one.

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.

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We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli-Generalized Gaussian distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a p-th order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bitrate and their slope factors at low bitrate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed.