Taibleson Lecture: "Bayes adaptive procedures associated with operators"

Abstract: 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, or data defined on some specific manifolds such as graphs, trees, or matrices), or data with a probabilistic structure involving an operator but also in the so-called linear inverse problems. In most cases the geometry is well described by a linear operator (the Laplacian for example).

The spectral decomposition of this operator plays an important role, generally translating a deep and intrinsic structure which leads to a natural choice of estimates (covariance kernel, spectral clustering, inverse problems). Moreover, 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 bayesian functional estimation considering Gaussian processes as a-priori measures. In particular, 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 also investigate the problem from the more explicit angle of an a priori measure on ‘manifold-wavelet’ coefficients.

We extend the results of Ghosal, Ghosh and van der Vaart [2], 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.

Host: Todd Kuffner

Tea @3:45pm in Cupples I, room 200