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

arXiv: https://arxiv.org/abs/2609.14171 · HTML: https://mlsanity.com/paper/2609.14171/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 21 of its 23 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 21 of 21)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Expected Batch Optimal Transport Plans and Consequences for Flow Matching](https://mlsanity.com/paper/2605.12174/) (2026)
- [DisRFM: Polar Riemannian Flow Matching for Structure-Preserving Graph Domain Adaptation](https://mlsanity.com/paper/2602.00656/) (2026)
- [PhaseGen: A Diffusion-Based Approach for Complex-Valued MRI Data Generation](https://mlsanity.com/paper/2504.07560/) (2025)
- [UniverSR: Unified and Versatile Audio Super-Resolution via Vocoder-Free Flow Matching](https://mlsanity.com/paper/2510.00771/) (2025)
- [Generative Modeling of Complex-Valued Brain MRI Data](https://mlsanity.com/paper/2604.14800/) (2026)
- [Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation](https://mlsanity.com/paper/2605.05054/) (2026)
- [One-Sided Quantile Coupling for Flow Matching](https://mlsanity.com/paper/2608.00978/) (2026)

## Abstract (arXiv)

Complex-valued signals like MRI and audio spectrograms are typically modelled as flat two-channel Euclidean data. The inherited Euclidean metric dA2 + A2 dθ2 vanishes at the origin, leaving phase unpenalised exactly where the signal is weakest. We replace it with the decoupled product metric dA2 + dθ2 on the cylindrical closure [0, ∞) × S1, 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 ≈ 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).