Exact sampling method improves efficiency in Gaussian flow matching
Correcting CondOT: Exact Finite-Step Sampling in Gaussian Flow Matching
Machine Learning
Summary
Sampling methods often transform random noise into data step by step, but using only a few steps can cause errors. This paper studies how to reduce those errors when working with Gaussian distributions by choosing better schedules for sampling steps. The authors show that a certain way of scheduling, called conditional optimal transport, can cancel out the main errors and even achieve exact sampling when enough steps are used. Experiments confirm their predictions and show faster convergence to the true distribution.
What this means in practice
- •For machine learning engineers: Design more accurate and efficient Gaussian flow sampling for generative models with fewer steps while maintaining sample quality.
- •For signal processing teams: Improve numerical precision in simulations and transformations involving Gaussian noise through optimized sampling schedules.
Authors
Ron Levy, Michael Elad
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.