An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds

TL;DR

This paper introduces an explicit, training-free surrogate for Gaussian mixture flow matching that simplifies computation of transport costs, with theoretical bounds and practical insights on its accuracy.

Contribution

It develops a closed-form surrogate for Gaussian mixture transport cost, analyzes its approximation quality, and proposes a path-splitting strategy for nonlocal regimes.

Findings

The surrogate provides a second-order approximation in local commuting regimes.

A cubic error bound is derived for the surrogate in local commuting regimes.

The practical regime map indicates when the surrogate or exact methods are preferable.

Abstract

We study training-free flow matching between two Gaussian mixture models (GMMs) using explicit velocity fields that transport one mixture into the other over time. Our baseline approach constructs component-wise Gaussian paths with affine velocity fields satisfying the continuity equation, which yields to a closed-form surrogate for the pairwise kinetic transport cost. In contrast to the exact Gaussian Wasserstein cost, which relies on matrix square-root computations, the surrogate admits a simple analytic expression derived from the kinetic energy of the induced flow. We then analyze how closely this surrogate approximates the exact cost. We prove second-order agreement in a local commuting regime and derive an explicit cubic error bound in the local commuting regime. To handle nonlocal regimes, we introduce a path-splitting strategy that localizes the covariance evolution and enables piecewise application of the bound. We finally compare the surrogate with an exact construction based on the Gaussian Wasserstein geodesic and summarize the results in a practical regime map showing when the surrogate is accurate and the exact method is preferable.