Correcting Source Mismatch in Flow Matching with Radial-Angular Transport

Fouad Oubari; Mathilde Mougeot

cs.LGApr 5, 2026

Correcting Source Mismatch in Flow Matching with Radial-Angular Transport

Authors: Fouad Oubari, Mathilde Mougeot

Organizations: Université Paris-Saclay, CNRS, ENS Paris-Saclay, Centre Borelli · ENSIIE

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 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 & tables

Panel of Figure 1 — 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.

\frac{1}{B}\sum_{b=1}^{B}\bigl\|v_{\theta}(t_{b},x_{t_{b}}^{(b)})-\dot{x}_{t_{b}}^{(b)}\bigr\|_{2}^{2}

Algorithm 1 RAFM training

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.

Dataset	Method	Radial W1 $\downarrow$	KS $\downarrow$	Sliced W1 $\downarrow$	Train time
Student- $t$ ( $d=16$ )	Gaussian FM	$3.3415\scriptstyle{\pm 0.9074}$	$0.1745\scriptstyle{\pm 0.0627}$	$0.7595\scriptstyle{\pm 0.2315}$	$18.3\scriptstyle{\pm 0.2}$ s
	Source-only	$0.3986\scriptstyle{\pm 0.0444}$	$0.0207\scriptstyle{\pm 0.0055}$	$0.4379\scriptstyle{\pm 0.0644}$	$19.0\scriptstyle{\pm 0.3}$ s
	RAFM	$\mathbf{0.2264}\scriptstyle{\pm 0.0476}$	$\mathbf{0.0119}\scriptstyle{\pm 0.0029}$	$\mathbf{0.3316}\scriptstyle{\pm 0.0298}$	$35.2\scriptstyle{\pm 0.1}$ s
	MSGM	$0.3760\scriptstyle{\pm 0.0650}$	$0.0142\scriptstyle{\pm 0.0036}$	$0.4864\scriptstyle{\pm 0.0279}$	$49.3\scriptstyle{\pm 0.3}$ min
Student- $t$ ( $d=32$ )	Gaussian FM	$8.3012\scriptstyle{\pm 0.5585}$	$0.2960\scriptstyle{\pm 0.0208}$	$1.5186\scriptstyle{\pm 0.0261}$	$18.4\scriptstyle{\pm 0.2}$ s
	Source-only	$1.4295\scriptstyle{\pm 0.2491}$	$0.0369\scriptstyle{\pm 0.0090}$	$0.7513\scriptstyle{\pm 0.2040}$	$19.2\scriptstyle{\pm 0.0}$ s

Table 1: Main benchmark results under a harmonized setting with a common batch size of 256 and 10,000 training steps for all methods. RAFM substantially improves over Gaussian FM and source-only baselines. Compared with MSGM, RAFM remains competitive across the main regimes while being dramatically faster in wall-clock time, and achieves its strongest result on PIV (

d=256

), where it outperforms MSGM on all reported metrics.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	Source-only		RAFM
Dataset	Empirical	Oracle	Empirical	Oracle
Student- $t$ ( $d=16$ ), Radial W1	$0.3986\pm 0.0444$	$0.5083\pm 0.1319$	$0.2264\pm 0.0476$	$0.2377\pm 0.0562$
Student- $t$ ( $d=16$ ), KS	$0.0207\pm 0.0055$	$0.0271\pm 0.0048$	$0.0119\pm 0.0029$	$0.0133\pm 0.0037$
Student- $t$ ( $d=16$ ), Sliced W1	$0.4379\pm 0.0644$	$0.4332\pm 0.0595$	$0.3316\pm 0.0298$	$0.3195\pm 0.0242$
Student- $t$ ( $d=32$ ), Radial W1	$1.4295\pm 0.2491$	$1.4800\pm 0.4106$	$0.6162\pm 0.0076$	$0.4652\pm 0.0185$
Student- $t$ ( $d=32$ ), KS	$0.0369\pm 0.0090$	$0.0376\pm 0.0076$	$0.0120\pm 0.0008$	$0.0110\pm 0.0021$

Appendix

Table 2: Empirical versus oracle radial distributions on the synthetic Student-

t

benchmarks. The empirical variants remain of the same order as the oracle ones, supporting the practical viability of radial estimation from training data.

Method	Radial W1 $\downarrow$	KS $\downarrow$	Sliced W1 $\downarrow$	Angular SW $\downarrow$
Gaussian FM	$0.0821\pm 0.0426$	$0.0563\pm 0.0068$	$0.0800\pm 0.0134$	$0.0502\pm 0.0051$
Source-only	$0.0688\pm 0.0245$	$0.0330\pm 0.0090$	$0.0889\pm 0.0241$	$0.0980\pm 0.0375$
RAFM	$0.0233\pm 0.0049$	$0.0291\pm 0.0292$	$0.3371\pm 0.0421$	$0.4695\pm 0.0988$
MSGM	$0.0488\pm 0.0087$	$0.0203\pm 0.0026$	$0.0449\pm 0.0038$	$0.0347\pm 0.0011$

Appendix

Table 3: Failure mode on the 2D radial–angular toy. RAFM still improves radial fidelity, but its current spherical angular transport remains clearly weaker on global and angular geometry in this low-dimensional near-origin regime.

Dataset	RAFM variant	Radial W1 $\downarrow$	KS $\downarrow$	Sliced W1 $\downarrow$
Gaussian aniso. ( $d=16$ )	w/ tangential projection	$\mathbf{0.0607}\scriptstyle{\pm 0.0030}$	$\mathbf{0.0127}\scriptstyle{\pm 0.0011}$	$0.1568\scriptstyle{\pm 0.0107}$
Gaussian aniso. ( $d=16$ )	w/o tangential projection	$0.1909\scriptstyle{\pm 0.0355}$	$0.0200\scriptstyle{\pm 0.0026}$	$\mathbf{0.1444}\scriptstyle{\pm 0.0150}$
PIV ( $d=16$ )	w/ tangential projection	$\mathbf{0.0111}\scriptstyle{\pm 0.0005}$	$\mathbf{0.0824}\scriptstyle{\pm 0.0050}$	$0.0114\scriptstyle{\pm 0.0011}$
PIV ( $d=16$ )	w/o tangential projection	$0.0164\scriptstyle{\pm 0.0013}$	$0.0848\scriptstyle{\pm 0.0156}$	$\mathbf{0.0113}\scriptstyle{\pm 0.0017}$
PIV ( $d=64$ )	w/ tangential projection	$\mathbf{0.0482}\scriptstyle{\pm 0.0019}$	$\mathbf{0.0469}\scriptstyle{\pm 0.0026}$	$\mathbf{0.0273}\scriptstyle{\pm 0.0029}$
PIV ( $d=64$ )	w/o tangential projection	$0.1656\scriptstyle{\pm 0.0119}$	$0.1304\scriptstyle{\pm 0.0177}$	$0.0289\scriptstyle{\pm 0.0016}$

Appendix

Table 4: Full tangential-projection ablation for RAFM. Tangential projection is not uniformly beneficial on the easiest control regimes, and can even be slightly worse on sliced Wasserstein in the mild anisotropic Gaussian control and on PIV (

d=16

). However, it becomes increasingly important as radial mismatch and dimensionality grow, substantially improving radial fidelity on Student-

t

(

d=16,32

), PIV (

d=64

), and especially PIV (

d=256

). On the 2D toy, removing projection leads to severe degradation and numerical instability.

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