Model selection and Geometry

The idea of selecting a model via penalizing a log-likelihood type criterion is an old idea in statistics. It goes back to the early seventies with the pioneering works of Mallows and Akaike.

One can find many consistency results in the literature for such criteria. These results rely on limit theorems from probability theory and are asymptotic in the sense that one deals with a given number of models and the number of observations tends to infinity. This setting turns out to be irrelevant for many applications (especially for high-dimensional data). Taking as a central illustrative example the Gaussian white noise framework, we shall show that it is possible to develop a non asymptotic theory for model selection. This theory for model selection deeply relies on the concentration of mesure phenomenon which is known to be of a geometric nature. Conversely some important issues of geometrical inference can be solved by using the model selection approach. We shall illustrate this point by focusing on a specific example: the use of model section for simplicial approximation.