Load--Reserve Wasserstein Propagation for Isotropic Diffusion Samplers

TL;DR

This paper introduces a profile-adapted propagation method for isotropic diffusion samplers that provides certified stability and tailored transportation costs, improving analysis of reverse-SDE processes.

Contribution

It develops a novel certified lower radial profile and an adapted cost framework that captures geometric features and stability in diffusion samplers.

Findings

Certified lower radial profiles improve stability analysis.

Adapted transportation costs reflect geometric and load features.

Error metrics are reported at terminal time using retained tail slope.

Abstract

Many Wasserstein analyses of diffusion samplers control reverse-time propagation by global stability summaries of the learned drift. These summaries can hide radial geometry: equal-height expansive regions of different width can yield different propagation costs. We give a profile-adapted propagation interface for scalar-isotropic reverse-SDE windows with certified learned-drift profiles. A certified lower radial profile is compiled into an affine-tail transportation cost: reflection coupling reduces stability to a one-dimensional slope budget, and Hardy capacity quantifies the load paid before a contractive tail reserve. The compiler yields an adapted cost, contraction rate, and retained tail slope. Score-modeling and solver residuals are treated as forcing inputs and propagate additively in the adapted Wasserstein distance. Quadratic Wasserstein error is reported only at terminal time, using the retained tail slope with tail, moment, or support information. Gaussian-smoothed denoising geometry supplies inverse-radius profiles for uniformly dissipative, bounded-amplitude, and common-covariance mixture windows. Fixed-height examples show that adverse height, even with eventual reserve, does not determine the certificate; barrier examples show that the load dependence is structural.