Operator-norm Sudakov minoration for Gaussian chaos of order two
Abstract
We prove that an operator-norm separated family of matrices satisfies 𝔼supA∈ T GTAG' ≥ calog |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.
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BibTeX
@misc{bednorz2026operatornorm,
title = {{Operator-norm Sudakov minoration for Gaussian chaos of order two}},
author = {Witold Bednorz and Rafał Martynek and Rafał Meller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23558},
eprint = {2609.23558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23558},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Operator-norm Sudakov minoration for Gaussian chaos of order two
AU - Witold Bednorz
AU - Rafał Martynek
AU - Rafał Meller
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.23558
UR - https://arxiv.org/abs/2609.23558
AB - We prove that an operator-norm separated family of matrices satisfies 𝔼sup_A∈ T G^TAG' ≥ calog |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.
ER -