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

2026-09-30

AI Summary

This study addresses the accumulation of numerical errors during finite-step sampling in flow matching models, which constrains generation quality. By leveraging the Wasserstein distance and an explicit midpoint sampling scheme, the authors systematically analyze error dependencies under Gaussian targets, revealing an error cancellation mechanism inherent to non-uniform step sizes. Based on this insight, a noise scheduling strategy is constructed to achieve exact Gaussian sampling. Theoretically, it is proven that the CondOT schedule eliminates the dominant error term, enabling precise calibration even with large step sizes while maintaining a convergence rate of 1/S. This work provides a rigorous theoretical foundation for improving few-step sampling accuracy in flow matching frameworks.

Abstract

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 $α_t=t$ and prove the existence of scalar noise schedules $β_t$ 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.

Problem

• Flow Matching
• Finite-Step Sampling
• Conditional Optimal Transport
• Sampling Error
• Noise Schedule

Innovation

• Flow Matching
• Conditional Optimal Transport
• Finite-Step Sampling
• Wasserstein Distance
• Noise Schedule