Sampling subspaces using determinantal point processes.

Summary Determinantal point processes are probabilistic models of repulsion. These models have been studied in different fields: random matrices, quantum optics, spatial statistics, image processing, machine learning and recently quadratures.In this thesis, we study the sampling of subspaces using determinantal point processes. This problem is at the intersection of three branches of approximation theory: sub-selection in discrete sets, kernel quadrature and kernel interpolation. We study these classical questions through a new interpretation of these random models: a determinantal point process is a natural way to define a random subspace. In addition to giving a unified analysis of numerical integration and interpolation under PLRs, this new approach allows us to develop the theoretical guarantees of several PLR-based algorithms, and even to prove their optimality for certain problems.