Preconditioned Flow Matching

TL;DR

This paper identifies a geometric bottleneck in flow matching due to ill-conditioned covariances and proposes preconditioned flow matching to improve training stability and sample quality across various datasets.

Contribution

It introduces a preconditioning framework that transforms target distributions to enhance the conditioning of flow matching, supported by theoretical analysis and empirical results.

Findings

Preconditioning improves flow matching convergence and sample quality.

The method enhances metrics like FID, MMD, precision, and recall.

Preconditioning benefits are not solely due to additional parameters, but improved geometry.

Abstract

Flow matching (FM) learns vector fields by regressing stochastic velocity targets along intermediate distributions $p_t$. We identify a geometric optimization bottleneck in this regression problem: when the covariance $\Sigma_t$ of $p_t$ is ill-conditioned, gradient-based training rapidly fits high-variance directions while making slow progress along low-variance ones. In an exactly solvable Gaussian setting, we prove that the excess risk is weighted by $\Sigma_t$, and that both gradient descent and stochastic gradient descent inherit condition-number-dependent convergence. We then extend the analysis to Gaussian mixtures, showing that multimodality does not average away this effect; instead, the slowest and worst-conditioned component can control optimization. Motivated by this analysis, we propose \emph{preconditioned flow matching}, a precondition-then-match framework that transforms the target distribution into a more isotropic representation, trains the main flow in the transformed space, and maps generated samples back through the inverse transformation. We show theoretically that preconditioning reshapes the intermediate FM path and improves its conditioning. Across controlled Gaussian and Gaussian-mixture experiments, latent MNIST and other high resolution image datasets up to $512{\times}512$ resolution, preconditioning improves path-conditioning diagnostics, low-eigenvalue recovery, FID, MMD, precision, and recall. Compute-matched baselines and preconditioner-quality ablations further show that the gains are not explained merely by additional preconditioner parameters, but by improved geometry of the downstream flow matching problem.