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

Christopher Liaw; Abbas Mehrabian; Yaniv Plan; Roman Vershynin

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arXiv · 1603.00897

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

Christopher Liaw·Abbas Mehrabian·Yaniv Plan·Roman Vershynin

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Abstract

Let $A$ be an isotropic, sub-gaussian $m \times n$ matrix. We prove that the process $Z_x := \|Ax\|_2 - \sqrt m \|x\|_2$ has sub-gaussian increments. Using this, we show that for any bounded set $T \subseteq \mathbb{R}^n$, 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.

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BibTeX RIS

Christopher Liaw, Abbas Mehrabian, Yaniv Plan, Roman Vershynin. 2016-03-02. A simple tool for bounding the deviation of random matrices on geometric sets. https://arxiv.org/abs/1603.00897

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