Scale-Dependent Radial Geometry and Metric Mismatch in Wasserstein Propagation for Reverse Diffusion

2026-03-20

AI Summary

This work addresses the mismatch between Wasserstein-2 error propagation under Euclidean geometry and early-stage radial contraction dynamics in reverse diffusion processes. To resolve this, the authors propose a scale-dependent radial geometric structure and a single-switch routing strategy: initially employing reflection coupling under a concave transport metric tailored to radial contours to achieve effective contraction, then switching to the Euclidean Wasserstein-2 metric for short-horizon error propagation. By establishing a dual-limit characterization—termed “radial contraction reserve” and “Euclidean load”—they formalize the metric-switching mechanism, derive a structural upper bound on the transition exponent, and obtain an optimal switching-time criterion. Under standard assumptions, the framework yields a non-asymptotic end-to-end Wasserstein-2 error bound, an optimizable scalar switching objective, and sharp limits on the transition exponent within the class of affine-tail concave distributions.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Existing analyses of reverse diffusion often propagate sampling error in the Euclidean geometry underlying \(\Wtwo\) along the entire reverse trajectory. Under weak log-concavity, however, Gaussian smoothing can create contraction first at large separations while short separations remain non-dissipative. The first usable contraction is therefore radial rather than Euclidean, creating a metric mismatch between the geometry that contracts early and the geometry in which the terminal error is measured. We formalize this mismatch through an explicit radial lower profile for the learned reverse drift. Its far-field limit gives a contraction reserve, its near-field limit gives the Euclidean load governing direct \(\Wtwo\) propagation, and admissible switch times are characterized by positivity of the reserve on the remaining smoothing window. We exploit this structure with a one-switch routing argument. Before the switch, reflection coupling yields contraction in a concave transport metric adapted to the radial profile. At the switch, we convert once from this metric back to \(\Wtwo\) under a \(p\)-moment budget, and then propagate the converted discrepancy over the remaining short window in Euclidean geometry. For discretizations of the learned reverse SDE under \(L^2\) score-error control, a one-sided Lipschitz condition of score error, and standard well-posedness and coupling hypotheses, we obtain explicit non-asymptotic end-to-end \(\Wtwo\) guarantees, a scalar switch-selection objective, and a sharp structural limit on the conversion exponent within the affine-tail concave class.

Problem

• reverse diffusion
• Wasserstein propagation
• metric mismatch
• radial geometry
• scale-dependent contraction

Innovation

• radial geometry
• metric mismatch
• Wasserstein propagation
• reverse diffusion
• one-switch coupling