Concentration inequalities under sub-Gaussian and sub-exponential conditions

We prove analogues of the popular bounded difference inequality (also called McDiarmid’s inequality) for functions of independent random variables under sub-Gaussian and sub-exponential conditions. Applied to vector-valued concentration and the method of Rademacher complexities these inequalities allow an easy extension of uniform convergence results for PCA and linear regression to the case of potentially unbounded input- and output variables

On the Concentration of the Minimizers of Empirical Risks

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters

Weak Continuity and Compactness for Nonlinear Partial Differential Equations

We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. We then analyze the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.Comment: 29 page