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

@article{Liaw2016AST,
  title={A simple tool for bounding the deviation of random matrices on geometric sets},
  author={Christopher Liaw and Abbas Mehrabian and Yaniv Plan and Roman Vershynin},
  journal={CoRR},
  year={2016},
  volume={abs/1603.00897}
}
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… CONTINUE READING
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