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Sharp Wasserstein Stability for Gaussian Deconvolution of Log-Concave Laws

Scott Schubr ORCID

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Abstract

Fix a dimension, a positive Gaussian noise level, and an isotropic reference whose C² potential has two-sided positive Hessian bounds. We prove a sharp inverse-logarithmic modulus of order 3/2 for the latent quadratic Wasserstein distance, uniformly over isotropic log-concave inputs, including nonsmooth and compactly supported densities. The observation may be either the W₁ distance between noisy outputs or the excess quadratic risk of the reference Bayes estimator. An explicit narrow-Gaussian perturbation preserves mass, mean and covariance exactly and gives the matching order at every permitted reference. The Gaussian-reference specialization yields the sharp modulus from the linear-MMSE gap at every fixed positive SNR. The upper proof combines bandlimited comparison, conditional curvature measures and a density-weighted divergence correction. Its constants are uniform over references in a fixed curvature interval. Consequently, known Gaussian noise gives fixed-reference testing against all isotropic log-concave alternatives and minimax W₂ estimation on the stated two-sided curvature classes at order (log n)^(-3/2). The matching estimation lower bound assumes 0<c<1<C; constants depend on the fixed dimension, noise and curvature bounds.

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Available editions

  1. Edition 4 · 2026-11-01 · public
  2. Edition 3 · 2026-10-31 · public
  3. Edition 2 · 2026-10-27 · public
  4. Edition 1 · 2026-10-24 · public

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
No additional permission granted

Scope

Fixed-dimension, fixed-positive-noise sharp latent W₂ inverse modulus of logarithmic order 3/2, for all isotropic log-concave inputs around each isotropic C² reference with two-sided positive Hessian bounds. Observations are noisy-output W₁ discrepancy or reference Bayes-estimator regret. Matching exact-moment alternatives at every reference; uniform upper bounds on fixed curvature classes; stated testing and minimax estimation consequences.

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 16-page manuscript includes complete analytic upper, matching lower and statistical proofs. The source package includes all four TeX files, bibliography, build instructions and fixed-timestamp build.py. No numerical experiment or computer-generated certificate is a premise of the mathematical proof.

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