用显式速度场构建高斯混合模型间流匹配的近似代价，计算更快且有误差保证。
An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds
- 设计分量级仿射速度场，导出可解析计算的近似传输代价
- 在局部交换条件下误差为三阶，提供显式误差上界
- 提出路径分割策略处理非局部情形，实用性强
我们研究无需训练的高斯混合模型（GMM）间流匹配，通过显式速度场将一个混合模型随时间变换至另一个。基准方法采用分量级高斯路径与满足连续性方程的仿射速度场，得到配对动能传输代价的闭式近似。相比依赖矩阵平方根计算的精确高斯沃瑟斯坦代价，该近似仅需简单解析表达式。我们分析其逼近精度，在局部交换条件下证明二阶一致性，并推导出三阶误差上界。针对非局部情形，引入路径分割策略以局部化协方差演化，支持分段应用该上界。最终，将该近似与基于高斯沃瑟斯坦测地线的精确构造进行对比，给出实用的性能比较图谱，明确指示何时使用近似更优、何时需精确方法。
原文摘要 · Abstract (English)
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.
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