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On Generalized Sub-Gaussian Canonical Processes and Their Applications

Yiming Chen, Yuxuan Wang, Kefan Zhu

Abstract

We obtain the tail probability of generalized sub-Gaussian canonical processes. It can be viewed as a variant of the Bernstein-type inequality in the i.i.d case, and we further get a tighter bound of concentration inequality through uniformly randomized techniques. A concentration inequality for general functions involving independent random variables is also derived as an extension. As for applications, we derive convergence results for principal component analysis and the Rademacher complexities method.

On Generalized Sub-Gaussian Canonical Processes and Their Applications

Abstract

We obtain the tail probability of generalized sub-Gaussian canonical processes. It can be viewed as a variant of the Bernstein-type inequality in the i.i.d case, and we further get a tighter bound of concentration inequality through uniformly randomized techniques. A concentration inequality for general functions involving independent random variables is also derived as an extension. As for applications, we derive convergence results for principal component analysis and the Rademacher complexities method.
Paper Structure (6 sections, 12 theorems, 64 equations)

This paper contains 6 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Main Results
  4. Applications
  5. Uniform bound for PCA
  6. Application in Rademacher complexities

Key Result

Lemma 1

(Buldygin and Kozachenko BK) For any Orlicz $N$-function $\varphi$, the following results hold:

Theorems & Definitions (29)

  • definition 1
  • Lemma 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • Theorem 1
  • proof : Proof of Theorem
  • remark 1
  • Corollary 1
  • ...and 19 more