Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws
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Abstract
Gaussian smoothing can conceal large differences between latent probability laws. We determine the sharp inverse modulus in quadratic Wasserstein distance around every reference whose twice continuously differentiable potential has positive lower and finite upper Hessian bounds. In every fixed dimension and at every fixed positive scalar Gaussian noise level, the modulus over all log-concave inputs has order (log(1/ε))⁻³ᐟ², for either output W₁ discrepancy or excess risk of the reference Bayes estimator. Inputs may be nonsmooth or compactly supported, and their moments need not be prescribed. The upper proof combines log-concave curvature control at the observable resolution with a density-weighted transport field and an exterior flux correction. Matching oscillatory alternatives preserve the reference's mass, mean and covariance exactly. Uniform upper constants give fixed-reference testing on the stated isotropic class and minimax estimation with unknown moments on two-sided potential-curvature classes. The estimation rate is (log n)⁻³ᐟ² in expected W₂ loss, also for known Gaussian noise covariance of positive rank; mean anchoring controls exceptional samples. The Gaussian-reference specialization gives the corresponding sharp modulus from the linear-MMSE gap.
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Research details
- Subjects
- Mathematics · Geometry and Optimal Transport · Statistics and Data Science · Statistical Theory
- Keywords
- Gaussian deconvolution, log-concavity, Wasserstein distance, inverse stability, minimax estimation
- Manuscript date
- 2026-10-23
- Reader reuse
- none
Scope
Sharp quadratic Wasserstein inverse moduli around C² references with positive lower and finite upper potential-Hessian bounds, for arbitrary log-concave inputs in fixed dimension and fixed positive scalar Gaussian noise. Statistical testing uses the stated isotropic alternatives; minimax estimation on two-sided curvature classes allows unrestricted latent moments and known positive-rank Gaussian noise covariance.
Disclosures
OpenAI ChatGPT/Codex assisted with literature discovery, proof exploration and drafting. The author reviewed the resulting manuscript and takes responsibility for its mathematical statements and references.
Reproduction
The manuscript has 20 pages. The source package contains 9 files: lower.tex, main.tex, normalization.tex, statistics.tex, upper.tex, main.pdf, README.txt, CITATION.cff, MANIFEST.json. Extract all files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep included TeX files together. The mathematical results are analytic.