In lots of practical problems, the statistical issue is very much connected to a genuine geometry. It is obviously the case for data observed on geometrical objects: directional data, data defined on some specific manifolds, on graphs, trees, or matrices... In most cases the geometry is well described by a linear operator -the Laplacian in most cases-.
The spectral decomposition of this operator plays an important role translating an intrinsic structure which leads to a natural choice of estimates Morover, the operator often induces a genuine regularization, leading to a regularity definition adapted to the structure of the data, which is fundamental in various situations : denoising, semi-supervised learning, classification...
We illustrate this problem by the example of density estimation using a heat kernel littlewood Paley decomposition, as well as bayesian functional estimation considering Gaussian processes as a-priori measures. In this particular later case, the problem of adaptation shows the need for fitting the a priori distribution to an harmonic analysis of the structure of the data, and in particular we associate the choice of the Gaussian measure with the Laplacian of the structure. We extend the results of Ghosal, Ghosh and van der Vaart, on the concentration a posteriori measures, for the case of geometrical data.
We also relate this construction to a characterization of the regularity of Gaussian processes on geometric objects.