Variable selection in high-dimensional additive models based on norms of projections

We consider the problem of variable selection in high-dimensional sparse additive models. We focus on the case that the components belong to nonparametric classes of functions. The proposed method is motivated by geometric considerations in Hilbert spaces and consists of comparing the norms of the projections of the data onto various additive subspaces. Under minimal geometric assumptions, we prove concentration inequalities which lead to new conditions under which consistent variable selection is possible. As an application, we establish conditions under which a single component can be estimated with the rate of convergence corresponding to the situation in which the other components are known.Comment: 27 page

Lower bounds for invariant statistical models with applications to principal component analysis

This paper develops nonasymptotic information inequalities for the estimation of the eigenspaces of a covariance operator. These results generalize previous lower bounds for the spiked covariance model, and they show that recent upper bounds for models with decaying eigenvalues are sharp. The proof relies on lower bound techniques based on group invariance arguments which can also deal with a variety of other statistical models.Comment: 42 pages, to appear in Annales de l'Institut Henri Poincar\'e Probabilit\'es et Statistique

From mapping class groups to automorphism groups of free groups

We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.Comment: to appear in J. Lond. Math. So