A simple tool for bounding the deviation of random matrices on geometric sets

Abstract

Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process (Formula presented) has sub-gaussian increments, that is, (Formula presented) for any x, y ∈ ℝn. Using this, we show that for any bounded set T ⊆ ℝn, the deviation of ||Ax||2 around its mean is uniformly bounded by the Gaussian complexity of T. We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, some of which are reviewed in this paper. In particular, we give a new result regarding model selection in the constrained linear model.