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

Complex-valued signals like MRI and audio spectrograms are typically modelled as flat two-channel Euclidean data. The inherited Euclidean metric $dA^2 + A^2 dθ^2$ vanishes at the origin, leaving phase unpenalised exactly where the signal is weakest. We replace it with the decoupled product metric $dA^2 + dθ^2$ on the cylindrical closure $[0, \infty) \times S^1$, which stays non-degenerate at $A = 0$. We measure what this substitution costs and buys. Exact analytical bridges across synthetic fields, fastMRI knee data, and LibriSpeech spectrograms show Cartesian paths induce a heavy-tailed angular velocity distribution (Pareto index $\approx 1$). Under independent coupling, 43%-49% of signal energy falls on paths turning faster than $π$ rad per unit time. Cylindrical paths never reach this speed. We formulate Cylindrical Flow Matching (CyFM) to strictly bound the angular regression target, coupling noise and data via exact minibatch Optimal Transport jointly over whole fields. This coupling reduces few-step generation error by 3%-60%. CyFM achieves lower generative error than the best Cartesian baseline at every step up to $k = 8$ on synthetic fields and speech spectrograms, with all seeds separated. On knee MRI, the single-step advantage is 1.8x. At convergence ($k = 100$), the two geometries show no significant difference. Finally, a prior-only control exposes the cost of flat parametrisation: on synthetic fields, a single Cartesian Euler step performs worse than the unintegrated noise prior (0.376 vs. 0.150).