All Research

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.