Correcting Source Mismatch in Flow Matching with Radial-Angular Transport

2026-04-05

AI Summary

This work addresses the performance degradation of flow matching when using Gaussian source distributions, which exhibit radial symmetry that mismatches the heavy-tailed or anisotropic structure of real-world data. To resolve this, the authors propose the Radial-Angular Flow Matching (RAFM) framework, which explicitly decouples and corrects this mismatch within standard simulation-free flow matching. RAFM constructs a source distribution aligned with the data’s radial component while employing a uniform angular conditional distribution on the sphere, thereby reducing the transport problem to angular alignment. The method introduces spherical geodesic interpolation to define conditional paths, combines Wasserstein and CDF-based metrics to estimate the radial law, and establishes a radial-angular KL decomposition with accompanying error stability theory. RAFM achieves state-of-the-art performance among non-Gaussian approaches on heavy-tailed and extreme-event datasets, while retaining lightweight, deterministic training.

Abstract

Flow Matching is typically built from Gaussian sources and Euclidean probability paths. For heavy-tailed or anisotropic data, however, a Gaussian source induces a structural mismatch already at the level of the radial distribution. We introduce \textit{Radial--Angular Flow Matching (RAFM)}, a framework that explicitly corrects this source mismatch within the standard simulation-free Flow Matching template. RAFM uses a source whose radial law matches that of the data and whose conditional angular distribution is uniform on the sphere, thereby removing the Gaussian radial mismatch by construction. This reduces the remaining transport problem to angular alignment, which leads naturally to conditional paths on scaled spheres defined by spherical geodesic interpolation. The resulting framework yields explicit Flow Matching targets tailored to radial--angular transport without modifying the underlying deterministic training pipeline. We establish the exact density of the matched-radial source, prove a radial--angular KL decomposition that isolates the Gaussian radial penalty, characterize the induced target vector field, and derive a stability result linking Flow Matching error to generation error. We further analyze empirical estimation of the radial law, for which Wasserstein and CDF metrics provide natural guarantees. Empirically, RAFM substantially improves over standard Gaussian Flow Matching and remains competitive with recent non-Gaussian alternatives while preserving a lightweight deterministic training procedure. Overall, RAFM provides a principled source-and-path design for Flow Matching on heavy-tailed and extreme-event data.

Problem

• Flow Matching
• source mismatch
• heavy-tailed data
• radial distribution
• anisotropic data

Innovation

• Radial-Angular Flow Matching
• source mismatch correction
• non-Gaussian source
• spherical geodesic interpolation
• flow matching