Discrepancy, chaining and subgaussian processes

Abstract

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs inf_(εi) sup_f∈F|∑_i=1^k ε_if(X_i)| is asymptotically smaller than the expectation over signs as a function of the dimension k, if the canonical Gaussian process indexed by F is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of ℝ^k using properties of the canonical Gaussian process the set indexes, and then obtain quantitative structural information on a typical coordinate projection of a subgaussian class.