# Gaussian Flow-Matching Schedules: Implications for Sampling and Training

arXiv: https://arxiv.org/abs/2609.25839 · HTML: https://mlsanity.com/paper/2609.25839/

## 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 10 of its 12 references that are in the map.
- First posted on arXiv 22 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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [On Kinetic Optimal Probability Paths for Generative Models](https://mlsanity.com/paper/2306.06626/) (2023)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Optimal Scheduling of Dynamic Transport](https://mlsanity.com/paper/2504.14425/) (2025)
- [Optimizing Noise Schedules of Generative Models in High Dimensionss](https://mlsanity.com/paper/2501.00988/) (2025)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)

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

- [Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching](https://mlsanity.com/paper/2609.39488/) (2026)

## Abstract (arXiv)

Flow-matching schedules affect both sampling dynamics and the variance of the regression target. For centered commuting Gaussians, we show that a direction-dependent schedule decomposes into two independent design choices: a variance path, which fully determines the intermediate laws and probability flow, and a factorization, which leaves this flow unchanged while controlling irreducible regression variance. On the sampling side, we analyze finite-step Euler accuracy and derive a necessary drift bound for exact N -step sampling, connecting the geodesic and the logarithmic path. On the training side, for any fixed path, we derive closed-form factorizations that either minimize time-averaged regression variance or make it constant along the path.