Admissible Sequences for Talagrand's $γ_2$-functional
Simona Diaconu
TL;DR
The work tackles Talagrand's $\gamma_2$-functional for Gaussian processes by constructing near-optimal admissible sequences that bound the majorizing measure expression more explicitly. It introduces a structured block-decomposition strategy for Gaussian matrices, separating off-diagonal and diagonal contributions, and then patches admissible sequences across blocks to obtain a global bound on $\sum_{h\ge0} 2^{h/2} d(t,\mathcal{A}_h)$. A mixture of deterministic constructions (for off-diagonal parts) and randomized bounding (for diagonal blocks) yields a universal bound of the form $\sup_{t\in\mathbb{S}^{d-1}}\sum_{h\ge0} 2^{h}\, (d(t,\mathcal{A}_h))^2 \le c_0 \cdot C^2$, with $c_0=121{,}931{,}520$ and $C$ capturing per-row variance scales, thus advancing understanding of Gaussian suprema via $\gamma_2$. The approach offers potential localization-delocalization insights for leading eigenvectors and suggests avenues to extend to subgaussian settings, though the full $\gamma_2$-sum presents ongoing technical challenges requiring further probabilistic tools.
Abstract
Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in T}{X_t}]$ is comparable to the $γ_2$-functional of $T,$ a quantity that depends solely on the space $(T,d),$ where $d$ denotes the pseudometric $d(u,v)=\sqrt{\mathbb{E}[(X_u-X_v)^2]}.$ Despite the explicit definition of this functional, an infimum over admissible sequences, this tool tends to be used exclusively as a means to bound the expectation of the supremum of a random process by that of another. This work considers the $γ_2$-functional as a proxy for the quantity of interest by constructing admissible sequences that are close to being optimal, and aims to provide a promising avenue towards understanding expectations of suprema of Gaussian random processes.