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

Complex-valued MRI and audio signals are commonly modeled as two Cartesian channels, although their phase is circular. In Cartesian flow matching, straight paths between nearly opposite phases can pass close to the origin and induce arbitrarily large angular velocities. We formulate Cylindrical Flow Matching (CyFM) on amplitude and phase coordinates $[0,\infty )\times S^1$ with the decoupled metric $d{A}^2+d{\theta }^2$. Shortest-arc phase interpolation bounds the angular training target by $\pi$, while exact minibatch Optimal Transport (OT) pairs noise and data jointly over whole fields. For equal-amplitude Cartesian endpoints with uniform phase differences, we derive an index-1 Pareto tail in peak angular velocity. Across exact bridges on synthetic copula fields, LibriSpeech spectrograms, and fastMRI knee wavefields, 42.6%-49.4% of target signal energy lies on independently coupled paths whose peak angular velocity exceeds $\pi$; no cylindrical target exceeds this bound. Under matched training budgets, CyFM with joint OT achieves lower pooled sliced $W_2$ than the best Cartesian baseline at every tested solver step $k\le 8$ on synthetic fields at three resolutions and on speech spectrograms. On $64\times 64$ knee MRI, its pooled error is 1.8x lower at one step; amplitude texture favors CyFM from four steps, and phase coherence from eight steps. Joint OT reduces synthetic few-step error by 3%-60%, whereas factorizing the coupling disrupts joint dependencies. At full $320\times 320$ MRI resolution, CyFM preserves a 2.1x single-step advantage, demonstrating that cylindrical geometry and joint transport provide an effective inductive bias for rapid, few-step generation of complex-valued physical fields.