Research Article · Preprint Statistics and Data Science
Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws
- Primary subject
- Statistics and Data Science
- Subject subfields
- Statistical Theory · Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
Fix a dimension, a positive Gaussian noise level, and a centered isotropic reference density whose twice continuously differentiable potential has Hessian between two positive constant multiples of the identity. We prove that, uniformly over all isotropic log-concave inputs, the input's quadratic Wasserstein distance from the reference has the sharp inverse-logarithmic modulus of order 3/2 from either the W_1 distance of the noisy outputs or the excess quadratic risk of the reference Bayes estimator. An explicit narrow-Gaussian perturbation has exactly zero mass, mean and covariance corrections and supplies the matching lower order at every permitted reference. The Gaussian-reference case yields the corresponding sharp modulus from the total linear-MMSE gap at every fixed positive SNR. The upper proof uses a positive bandlimited core comparison, conditional curvature measures, a convex gradient-oscillation estimate and a weighted divergence correction. Inputs may have compact support or nonsmooth convex potentials. The upper constants are uniform when the reference Hessian lies in a fixed positive curvature interval. This yields sharp minimax W_2 estimation rates of order (log n)^(-3/2) on isotropic two-sided curvature classes, and fixed-reference goodness-of-fit testing at the same separation order from noisy independent samples. Constants depend on dimension, curvature bounds and noise level; the matching estimation lower bound assumes 0<c<1<C.
Keywords Statistical Theory · Geometry and Optimal Transport · Gaussian estimation · inverse stability
Scope of the work
Sharp inverse-logarithmic order-3/2 Wasserstein deconvolution moduli hold for isotropic log-concave inputs at fixed dimension and Gaussian noise, relative to smooth uniformly convex isotropic references, with the stated MMSE, minimax and testing consequences.
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Scott Schubr. (2026). Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/a-sharp-logarithmic-inverse-bound-for-gaussian-estimation/v/2
BibTeX
@techreport{luminary_a_sharp_logarithmic_inverse_bound_for_gaussian_estimation_v2,
title = {Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/a-sharp-logarithmic-inverse-bound-for-gaussian-estimation/v/2}
}Loading discussion…