Match the Radius, Learn the Angles: Radial-Angular Flow Matching

Flow Matching commonly relies on Gaussian sources and Euclidean conditional paths. For data with non-Gaussian amplitude statistics, this introduces a source-induced radial mismatch: the learned transport must reshape the distribution of norms while simultaneously modeling the directional structure of the data. We propose Radial-Angular Flow Matching (RAFM), which matches the target radial law at the source and preserves it along spherical conditional paths, separating radial estimation from directional transport. We further show that, although this construction removes radial motion from the path, conventional velocity regression still inherits the scale of the data. This motivates a scale-free angular formulation whose targets are uniformly bounded independently of the radial distribution, while still recovering the corresponding Flow Matching velocity field after radial rescaling. Our analysis characterizes the Gaussian radial mismatch removed by source matching and shows that, unlike velocity targets, angular targets require no finite radial moments. Across controlled heavy-tailed benchmarks and real vector, image, and audio data, ablations isolate the contributions of source matching, path geometry, and angular regression. The results show that these components provide complementary benefits while retaining the simulation-free training framework of Flow Matching.