Optimal Covariance Inflation under Gaussian Tilts
Abstract
Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body K⊆ℝn, let μK,t (d x) ∝ e−t| x | 2 𝟙K(x) d x, and let Qn be the supremum of |Cov(μK,t)|op over all such K and all t>0. We prove the sharp bound Qn=Θ(n2/5), closing the gap between the known Ω(n1/3) lower bound and the O(√(nlog(en))) upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance Ω(n2/5).
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BibTeX
@misc{gao2026optimal,
title = {{Optimal Covariance Inflation under Gaussian Tilts}},
author = {Minbo Gao and Zhengfeng Ji and Chenghua Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08930},
eprint = {2609.08930},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08930},
}
RIS
TY - UNPB
TI - Optimal Covariance Inflation under Gaussian Tilts
AU - Minbo Gao
AU - Zhengfeng Ji
AU - Chenghua Liu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08930
UR - https://arxiv.org/abs/2609.08930
AB - Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body K⊆ℝ^n, let μ_K,t (d x) ∝ e^−t| x | ^2 𝟙_K(x) d x, and let Q_n be the supremum of |Cov(μ_K,t)|_op over all such K and all t>0. We prove the sharp bound Q_n=Θ(n^2/5), closing the gap between the known Ω(n^1/3) lower bound and the O(√(nlog(en))) upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance Ω(n^2/5).
ER -