Title:CyFM: Cylindrical Optimal Transport for Few-Step Complex-Valued Flow Matching

Abstract:Complex-valued signals, such as Magnetic Resonance Imaging (MRI) and audio spectrograms, are almost always modelled as flat two-channel Euclidean data. For nonzero values the amplitude-phase chart $z \mapsto (|z|, z/|z|)$ identifies the signal domain with the cylinder $(0, \infty) \times S^1$, on which we deliberately replace the inherited metric $dA^2 + A^2 d\theta^2$ by the decoupled product metric $dA^2 + d\theta^2$. In this empirical study we measure what that substitution costs and what it buys. By computing exact analytical bridges, we demonstrate that Cartesian paths induce a heavy-tailed distribution of angular velocity (power law index $\approx 1.0$), with nearly half of the probability paths exceeding an angular speed of $\pi$ under independent coupling, a rate no cylindrical path ever exceeds. To resolve this, we analyze Cylindrical Flow Matching (CyFM), which strictly bounds the regression target, and couple noise and data by exact minibatch Optimal Transport computed jointly over whole fields in the cylindrical metric. Although the transport-cost reduction of this coupling collapses with field dimension (from 86% for scalar pairs to 3% for $64\times64$ fields), its benefit to few-step generation does not: it lowers the few-step error of the cylindrical model by 3-60% at every evaluated resolution. With this coupling, CyFM has a lower error than the best Cartesian baseline at every step count up to $k = 8$ and every evaluated resolution, with all five seeds separated and without distillation, and at convergence we detect no significant difference between the two geometries. Finally, we expose the "Factorized Coupling Trap," showing that dimension-wise or patch-wise transport factorizations silently destroy the joint distribution of the data. All experiments are on synthetic complex fields.

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