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

Let A be an isotropic, sub-gaussian m × n matrix. We prove that the process Zx := ‖Ax‖ 2 −√m ‖x‖ 2 has sub-gaussian increments, that is, ‖Zx−Zy‖ψ2 ≤ C‖x−y‖2 for any x, y ∈ R. Using this, we show that for any bounded set T ⊆ R, 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.