Correcting Source Mismatch in Flow Matching with Radial-Angular Transport

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.

Figures (2)

• Figure 1: Conceptual comparison between standard Gaussian Flow Matching and Radial--Angular Flow Matching (RAFM). Left: intermediate marginals and radial diagnostics on the 2D toy example. Gaussian FM progressively corrects both radius and angle, whereas RAFM preserves the radial law and mainly reorganises mass angularly. Right: schematic illustration of the conditional interpolation geometries used in our comparison. RAFM follows a matched-radius geodesic path on the scaled sphere, whereas the Gaussian FM baseline uses a linear Euclidean interpolation.
• Figure 2: Radial source mismatch and generated radial fidelity on two representative hard regimes. The top row compares the test radial law with the Gaussian and empirical radial sources; the Gaussian source is strongly mismatched, whereas the empirical source closely follows the data. The bottom row shows the radial tail distributions of generated samples: Gaussian FM inherits poor radial fidelity, source-only already recovers much of the gap, and RAFM further improves the match to the target radial law. On Student-$t$ ($d=32$), MSGM remains slightly stronger on radial fidelity, whereas on PIV ($d=256$), RAFM outperforms MSGM while remaining substantially faster in wall-clock time.

Theorems & Definitions (40)

• Theorem 3.1: Radial KL decomposition
• Proposition 3.2: Radius preservation and tangency of the spherical path
• Proposition 3.3: Tangential flows preserve the radial law
• Theorem 3.4: Generation stability
• Proposition A.1: Density of the radial source
• proof
• Proposition A.2: Exact radial preservation
• proof
• Proposition A.3: Gaussian under-dispersion for regularly varying radial tails
• proof