A sharp logarithmic inverse bound for Gaussian estimation
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Abstract
At fixed dimension and positive Gaussian noise level, fix a centered isotropic reference whose twice continuously differentiable potential has Hessian between two positive constant multiples of the identity. Uniformly over isotropic log-concave inputs, we prove a sharp inverse-logarithmic Wasserstein modulus of order 3/2 from either noisy-output W1 distance or excess quadratic risk of the reference Bayes estimator. Explicit narrow-Gaussian perturbations with zero mass, mean and covariance corrections give the matching lower order at every permitted reference. The Gaussian-reference case yields a sharp total linear-MMSE gap modulus at every fixed positive SNR. Curvature-uniform constants yield sharp minimax W2 estimation rates of order (log n)^(-3/2) on two-sided curvature classes and fixed-reference testing at the same separation order. Inputs may have compact support or nonsmooth convex potentials.
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Research details
- Subjects
- Mathematics · Geometry and Optimal Transport · Statistics and Data Science · Statistical Theory
- Keywords
- Statistical Theory, Geometry and Optimal Transport, Gaussian estimation, inverse stability
- Manuscript date
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
At fixed dimension and positive Gaussian noise level, fix a centered isotropic reference whose twice continuously differentiable potential has Hessian between two positive constant multiples of the identity. Uniformly over isotropic log-concave inputs, we prove a sharp inverse-logarithmic Wasserstein modulus of order 3/2 from either noisy-output W1 distance or excess quadratic risk of the reference Bayes estimator. Explicit narrow-Gaussian perturbations with zero mass, mean and covariance corrections give the matching lower order at every permitted reference. The Gaussian-reference case yields a sharp total linear-MMSE gap modulus at every fixed positive SNR. Curvature-uniform constants yield sharp minimax W2 estimation rates of order (log n)^(-3/2) on two-sided curvature classes and fixed-reference testing at the same separation order. Inputs may have compact support or nonsmooth convex potentials. Constants depend on dimension, fixed curvature interval and noise; the matching estimation lower bound assumes 0 < c < 1 < C.
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