Operator-norm Sudakov minoration for Gaussian chaos of order two

Abstract

We prove that an operator-norm separated family of matrices satisfies $\mathbb{E}{\sup }_{A\in T}{G}^{T}A{G}^{\prime }\ge ca\log |T|$, where G,G' are independent standard Gaussian vectors and a is the separation. The main information estimate concerns arbitrary separated coisometries: conditional entropy is bounded by a source-dependent operator energy times $\log |T|$, up to an additive quadratic term in the common row dimension. An adaptive Gaussian experiment proves this estimate by charging actual information increments to one weighted posterior-entropy potential. Convex separation and a Gaussian covering estimate then yield a bounded-radius result. To reach the general case, we first choose an operator scale preserving the Sudakov ratio, apply the known Hilbert-Schmidt minoration, and recompute a common Gaussian block compression at the retained entropy. This ordering preserves the normalization needed by the coisometry argument.

arXiv commentAI (GPT-6 Astra) was used in this research