A Simple Tool for Bounding the Deviation of Random Matrices on Geometric Sets

Let A be an isotropic, sub-gaussian m x n matrix. We prove that the process Z(x) := parallel to Ax parallel to(2) - root m parallel to x parallel to(2) has sub-gaussian increments, that is parallel to Z(x) - Z(y)parallel to psi(2) <= C parallel to x - y parallel to(2) for any x, y is an element of R-n. Using this, we show that for any bounded set T subset of R-n, the deviation of parallel to Ax parallel to(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.