Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws
Loading original first page…
Abstract
Gaussian deconvolution is severely ill-posed: small differences between noisy observations can hide much larger differences between latent laws. For every fixed dimension and positive Gaussian noise level, we determine the sharp inverse modulus in quadratic Wasserstein distance around an isotropic reference with a twice continuously differentiable, uniformly convex potential of bounded Hessian. Over all isotropic log-concave inputs, the modulus has order (log(1/ε))⁻³ᐟ², whether the observed discrepancy is output W₁ distance or excess risk of the reference Bayes estimator. The input may have nonsmooth potential or compact support. A moment-recovery and covariance-alignment reduction removes input normalization and reference isotropy from both deterministic moduli. The upper bound uses log-concavity to control curvature at the resolution visible through Gaussian smoothing, then converts that control into a density-weighted transport field and pays the exterior correction. An explicit oscillatory perturbation preserves mass, mean and covariance and attains the matching order at every admissible reference. Uniform constants on fixed curvature intervals yield fixed-reference testing and minimax W₂ estimation at rate (log n)⁻³ᐟ²; the estimation lower bound holds when the curvature interval contains the Gaussian strictly. The Gaussian-reference case gives the sharp inverse modulus from the linear-MMSE gap at every fixed positive SNR.
Read the full paper in the original PDF.
Edition files
- Paper.pdf207.6 KiB · pdf
SHA-256
485991bc838f30b14da3a4f5ab56d31181c2c5096cca046a7db1c479e955a991 - Publication_Package.zip231.0 KiB · package
SHA-256
20b76ee3334940be31856f5c9eb80f2aea472095475489fc843ddc524ef20d54
Available editions
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 deterministic quadratic Wasserstein inverse moduli for log-concave inputs around C² references with positive lower and finite upper potential-Hessian bounds, at fixed dimension and positive Gaussian noise. Moment recovery and covariance alignment extend both output-W₁ and reference-estimator-regret moduli to unrestricted input moments and reference mean and covariance. Statistical testing, curvature-class estimation and Gaussian linear-MMSE conclusions use their stated parameter classes.
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 19 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 the files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep the included TeX files together. The paper is analytic and requires no empirical dataset.