Nonlinear estimation in anisotropic multi-index denoising. Sparse case

In dimension one, it has long been observed that the minimax rates of convergences in the scale of Besov spaces present essentially two regimes ( and a boundary): dense and the sparse zones. In this paper, we consider the problem of denoising a function depending on a multidimensional variable ( for instance, an image), with anisotropic constraints of regularity ( especially providing a possible disparity of the inhomogeneous aspect in different directions). The case of the dense zone has been investigated in the former paper [ G. Kerkyacharian, O. Lepski, and D. Picard, Probab. Theory Related Fields, 121 ( 2001), pp. 137 - 170]. Here, our aim is to investigate the case of the sparse region. This case is more delicate in some aspects. For instance, it was an open question to decide whether this sparse case, in the d- dimensional context, has to be split into different regions corresponding to different minimax rates. We will see here that the answer is negative: we still observe a sparse region but with a unique minimax behavior, except, as usual, on the boundary. It is worthwhile to notice that our estimation procedure admits the choice of its parameters under which it is adaptive up to logarithmic factor in the dense case [ G. Kerkyacharian, O. Lepski, and D. Picard, Probab. Theory Related Fields, 121 ( 2001), pp. 137 - 170] and minimax adaptive in the sparse case. It is also interesting to observe that in the sparse case the embedding properties of the spaces are fundamental.