# Tessellations of Semi-Discrete Flow Matching

arXiv: https://arxiv.org/abs/2605.07513 · HTML: https://mlsanity.com/paper/2605.07513/

## Key facts

- Cited by 3 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 15 of its 24 references that are in the map.
- First posted on arXiv 8 May 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, 15 of 15)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Convergence of a Newton algorithm for semi-discrete optimal transport](https://mlsanity.com/paper/1603.05579/) (2016)
- [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024)
- [Statistical Properties of Rectified Flow](https://mlsanity.com/paper/2511.03193/) (2025)
- [Gradient Variance Reveals Failure Modes in Flow-Based Generative Models](https://mlsanity.com/paper/2510.18118/) (2025)
- [One-Shot Generative Flows: Existence and Obstructions](https://mlsanity.com/paper/2604.15439/) (2026)

## Built on by (most cited first, 3 of 3)

- [Expected Batch Optimal Transport Plans and Consequences for Flow Matching](https://mlsanity.com/paper/2605.12174/) (2026)
- [Limit Points of Reflow with Minibatch Optimal Transport](https://mlsanity.com/paper/2608.07042/) (2026)
- [Beyond Transport Cost: Routing Differences between Flow Matching and Optimal Transport](https://mlsanity.com/paper/2610.05921/) (2026)

## Abstract (arXiv)

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.