On Talagrand's functional and generic chaining

TL;DR

This paper extends Talagrand's chaining functionals to general distributions and uses generic chaining to bound the moments of stochastic process suprema, with applications to Johnson-Lindenstrauss lemma, Gaussian chaos, and convex signal recovery.

Contribution

It introduces Talagrand's functionals for general distributions and applies generic chaining to derive bounds on stochastic process moments in broader settings.

Findings

Derived upper bounds for all p-th moments of process suprema.

Extended Talagrand's functionals to non-identically distributed cases.

Applied results to Johnson-Lindenstrauss lemma and Gaussian chaos.

Abstract

In the study of the supremum of stochastic processes, Talagrand's chaining functionals and his generic chaining method are heavily related to the distribution of stochastic processes. In the present paper, we construct Talagrand's type functionals in the general distribution case and obtain the upper bound for the suprema of all $p$-th moments of the stochastic process using the generic chaining method. As applications, we obtained the Johnson-Lindenstrauss lemma, the upper bound for the supremum of all $p$-th moment of order 2 Gaussian chaos, and convex signal recovery in our setting.