Research Article · Preprint Statistics and Data Science
A sharp logarithmic inverse bound for Gaussian estimation
- Primary subject
- Statistics and Data Science
- Subject subfields
- Statistical Theory · Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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.
Keywords Statistical Theory · Geometry and Optimal Transport · Gaussian estimation · inverse stability
Scope of the work
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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Scott Schubr. (2026). A sharp logarithmic inverse bound for Gaussian estimation (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/a-sharp-logarithmic-inverse-bound-for-gaussian-estimation/v/1
BibTeX
@techreport{luminary_a_sharp_logarithmic_inverse_bound_for_gaussian_estimation_v1,
title = {A sharp logarithmic inverse bound for Gaussian estimation},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/a-sharp-logarithmic-inverse-bound-for-gaussian-estimation/v/1}
}Loading discussion…