A sharp logarithmic inverse bound for Gaussian estimation
- Primary subject
- Statistics and Data Science
- Subject subfields
- Statistical Theory · Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
At fixed dimension and positive Gaussian noise level, fix a centered isotropic reference whose twice continuously differentiable potential has Hessian between two positive constant multiples of the identity. Uniformly over isotropic log-concave inputs, we prove a sharp inverse-logarithmic Wasserstein modulus of order 3/2 from either noisy-output W1 distance or excess quadratic risk of the reference Bayes estimator. Explicit narrow-Gaussian perturbations with zero mass, mean and covariance corrections give the matching lower order at every permitted reference. The Gaussian-reference case yields a sharp total linear-MMSE gap modulus at every fixed positive SNR. Curvature-uniform constants yield sharp minimax W2 estimation rates of order (log n)^(-3/2) on two-sided curvature classes and fixed-reference testing at the same separation order. Inputs may have compact support or nonsmooth convex potentials.