Residual Certificates and Empirical Sharpness for Entropic Self-Transport Scores

Overview

Abstract

We study how numerical errors in symmetric kernel scaling affect the spatial score obtained from its smooth kernel extension. A full square-root-mass residual gives a sharp kernel-energy bound for every positive candidate scaling, including singular positive semidefinite Gram matrices. For Gaussian kernels this yields computable, scale-invariant pointwise score certificates. For iid samples from a strictly positive C³ density on the d-dimensional torus, we prove a simultaneous nonlinear score bound and determine the exact local modulus in a shrinking-bandwidth regime, with leading constant C_d/√f(x). Under the stated normalization and sampling assumptions, residual o(1/√n) suffices for negligible normalized score error; positive sample-dependent candidates with residual asymptotic to c/√n attain nonzero normalized error. The sufficient sampling regime is not claimed optimal. A reproducible solver experiment illustrates the certificate on actual iterates. These numerical-accuracy results do not assert a new statistical central limit theorem or an iteration lower bound.

Keywords entropic self-transport · matrix scaling · residual error bounds · score estimation · empirical sharpness