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

Elham Rostami, Taous-Meriem Laleg-Kirati, Hamidou Tembine

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.

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

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.

Paper Structure

This paper contains 23 sections, 6 theorems, 66 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Method A
  3. Component-wise Gaussian path and the affine field
  4. Closed-form kinetic cost for a pair $(i,j)$
  5. Global coupling and mixture flow
  6. How Close Is $C$ to the Wasserstein Cost?
  7. Explicit Cubic Gap Bound Under $\hat{\rho}<1$
  8. Piecewise-Local Transport via Path Splitting
  9. Method B: Exact Gaussian Wasserstein Transport
  10. Numerical Experiments
  11. Accuracy Across Gaussian Regimes
  12. Computational Complexity and Runtime Scaling
  13. PRACTICAL REGIME MAP
  14. Conclusion
  15. Proof of Lemma \ref{['lem:unique-affine']}
  16. ...and 8 more sections

Key Result

Lemma 1

Let where $m(\cdot)$ and $\Sigma(\cdot)$ are differentiable in $t$. Consider affine vector fields of the form $v(t,x)=a(t)+B(t)\bigl(x-m(t)\bigr).$ Then the continuity equation $\partial_t \rho_t+\nabla\!\cdot(\rho_t v)=0$ admits an affine solution, namely Equivalently,

Figures (1)

  • Figure 1: Runtime scaling of the transport-construction pipeline for the surrogate and exact Wasserstein methods across three Gaussian regimes.

Theorems & Definitions (13)

  • Lemma 1: Affine continuity-equation field
  • proof
  • Lemma 2: Closed-form surrogate cost (Gaussian pair)
  • proof
  • Theorem 1: Local agreement of $C_{ij}$ and $W_{2,ij}^2$
  • proof
  • Remark 1
  • Theorem 2: Explicit cubic gap bound
  • proof
  • Proposition 1: Local validity under subdivision
  • ...and 3 more