发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对函数流匹配的离散化问题，证明了有限秩重构下的强L²收敛性，推导了Wasserstein界与超额风险项，给出了端到端收敛速率的显式结果。
AI 中文摘要
函数流匹配是针对函数分布提出的，但通过有限个系数或点值实现。在分散或自适应细化下，所得条件σ代数未必嵌套，故鞅收敛无法证明传感器极限的合理性。我们对每一个强一致的有限秩重构序列，证明了有限条件速度目标的强L²收敛性，给出了正交投影的定量界，以及通过正则性空间的点传感器扩展。对于学习到的流，直接耦合到总体叠加路径可得到端到端Wasserstein界，无需假设总体有限维常微分方程的唯一性。我们验证了归一化求积神经算子的传感器无关常数，包括通过显式幅度递推得到的全局Lipschitz激活函数。非交换迹类高斯例子在投影限制下给出边界乘子0，在精确条件下给出0.72。空间正则性-求积证书封闭了算子实现项，伯恩斯坦论证给出了固定模型维度和包络下的O~(n⁻¹)超额风险项，而一个可精确实现的截断高斯缩放特例则得到了显式端到端速率。
英文摘要
Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier $0$ under projected restriction and $0.72$ under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a $\widetilde{O}(n^{-1})$ excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.
Comments31 pages, 2 tables
