Statistical inference for empirical heat-kernel Sinkhorn metrics

Overview

Abstract

We estimate fixed smooth pullback entries of the inverse weighted-Laplacian metric of an unknown smooth strictly positive density on a flat torus. The estimator uses symmetric heat-kernel Sinkhorn normalization of the sample and prescribed score functions, without density evaluations. For every fixed dimension we establish a joint, population-centered root-n central limit theorem as the heat bandwidth tends to zero. The proof gives a signed-measure Taylor expansion on one Sobolev space, an explicit influence function including normalization and inverse response, and a consistent empirical covariance estimator using exact solves. A scalar Hessian identity with the moving integration measure improves the sufficient remainder budget. The population tensor has even finite asymptotic jets; fixed multiband extrapolation removes enough bias to retain the same limiting covariance. The influence is the canonical gradient, and the estimator attains the pointwise information bound under smooth local density submodels. Finally, an observed scaling, spectral-gap and inverse-residual certificate produces an actual finite first-hit output with negligible root-n metric error. Its certified-arithmetic realization requires effective convergent input enclosures. Bandwidth conditions are conservative; no practical runtime or optimal smoothing rate is asserted. A separate deterministic weighted-norm certificate controls inexact influence solves, residual-tilted laws and original-law covariance, including singular joint covariances. Its conditional gap and scale inputs do not give an atom-independent runtime.

Keywords Sinkhorn metric · diffusion maps · inverse weighted Laplacian · asymptotic linearity · semiparametric efficiency · observed stopping