# Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching

arXiv: https://arxiv.org/abs/2609.39488 · HTML: https://mlsanity.com/paper/2609.39488/

## 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 10 of its 10 references that are in the map.
- First posted on arXiv 30 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, 10 of 10)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting](https://mlsanity.com/paper/2503.11615/) (2025)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)
- [Mixture-of-Gaussians-Guided Schedule Design for Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2607.03517/) (2026)
- [Gaussian Flow-Matching Schedules: Implications for Sampling and Training](https://mlsanity.com/paper/2609.25839/) (2026)

## Abstract (arXiv)

Flow matching generates samples by gradually transforming noise into data. In practice, using a finite number of sampling steps introduces a numerical error that depends on the chosen schedule. We study this dependence for Gaussian targets and the explicit midpoint sampling method, using the exact flow field. We measure sampling error by the squared Wasserstein distance between the target distribution and the final distribution produced by the midpoint sampler. We show that the standard conditional optimal transport (CondOT) schedule cancels the leading midpoint error and improves the general convergence bound, even when the sampling steps are unequally spaced. On a uniform grid of S sampling steps, we fix the signal schedule at αt=t and prove the existence of scalar noise schedules βt that approach the CondOT noise schedule 1-t at rate 1/S and yield exact Gaussian sampling for every sufficiently large S. Controlled Gaussian experiments illustrate the convergence rates and exact calibration.