# Coupled Flow Matching

arXiv: https://arxiv.org/abs/2510.23015 · HTML: https://mlsanity.com/paper/2510.23015/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 29 of its 45 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## 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, 29 of 29)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Multimodal Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/1804.04732/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Autoencoders: Toward a Meaningful and Decodable Representation](https://mlsanity.com/paper/2111.15640/) (2021)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)
- [Multimodal Generative Models for Scalable Weakly-Supervised Learning](https://mlsanity.com/paper/1802.05335/) (2018)
- [One Transformer Fits All Distributions in Multi-Modal Diffusion at Scale](https://mlsanity.com/paper/2303.06555/) (2023)
- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Dual Diffusion for Unified Image Generation and Understanding](https://mlsanity.com/paper/2501.00289/) (2024)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [InfoDiffusion: Representation Learning Using Information Maximizing Diffusion Models](https://mlsanity.com/paper/2306.08757/) (2023)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [Latent Bayesian Optimization via Autoregressive Normalizing Flows](https://mlsanity.com/paper/2504.14889/) (2025)
- [EasyGen: Easing Multimodal Generation with BiDiffuser and LLMs](https://mlsanity.com/paper/2310.08949/) (2023)
- [Invertible Kernel PCA with Random Fourier Features](https://mlsanity.com/paper/2303.05043/) (2023)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024)
- [Quadratic-form Optimal Transport](https://mlsanity.com/paper/2501.04658/) (2025)

## Built on by (most cited first, 1 of 1)

- [Representation Learning in Diffusion and Flow-based Model: An Application Aspect](https://mlsanity.com/paper/2608.24068/) (2026)

## Abstract (arXiv)

We introduce Coupled Flow Matching (CPFM), a framework that integrates controllable dimensionality reduction and high-fidelity reconstruction. CPFM learns coupled continuous flows for both the high-dimensional data x and the low-dimensional embedding y, which enables sampling p(y|x) via a latent-space flow and p(x|y) via a data-space flow. Unlike classical dimension-reduction methods, where information discarded during compression is often difficult to recover, CPFM preserves the knowledge of residual information within the weights of a flow network. This design provides bespoke controllability: users may decide which semantic factors to retain explicitly in the latent space, while the complementary information remains recoverable through the flow network. Coupled flow matching builds on two components: (i) an extended Gromov-Wasserstein optimal transport objective that establishes a probabilistic correspondence between data and embeddings, and (ii) a dual-conditional flow-matching network that extrapolates the correspondence to the underlying space. Experiments on multiple benchmarks show that CPFM yields semantically rich embeddings and reconstructs data with higher fidelity than existing baselines.