噪声潜在结构下的流匹配：超越精确低维支撑
Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support
- Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对噪声潜在结构下的流匹配问题，提出空间正则ReLU速度类，证明样本复杂度由潜在维度主导，无需精确低维支撑即可获得Wasserstein收敛保证。
AI中文摘要：
流匹配（Flow Matching, FM）学习一个速度场，其常微分方程（ODE）将简单源分布输送到目标分布。现有的有限样本理论主要处理环境空间正则性或数据精确支撑在低维集合上的情况。我们研究线性流匹配在噪声潜在生成器模型下的表现，其中低维Hölder映射被非退化环境高斯噪声扰动，因此目标分布尽管具有潜在结构，却是全维的。我们构建了一个空间正则的ReLU速度类，并建立了非渐近的高概率逼近和估计界，其主导样本量指数由潜在维度而非环境维度决定，同时环境维度和噪声的影响保持显式。固定的正目标噪声使得插值在整个时间区间上保持非退化。相同的空间正则性将学习到的速度误差传播到传输ODE中，从而得到相应的Wasserstein收敛保证。这些结果表明，精确的低维支撑并非流匹配保持潜在维度统计行为所必需的。
英文摘要：
Flow Matching (FM) learns a velocity field whose ODE transports a simple source distribution to a target law. Existing finite-sample theory largely treats ambient-space regularity or data supported exactly on low-dimensional sets. We study linear FM under a noisy latent-generator model, where a low-dimensional Hölder map is perturbed by nondegenerate ambient Gaussian noise, so the target law is full-dimensional despite its latent structure. We construct a spatially regular ReLU velocity class and establish non-asymptotic high-probability approximation and estimation bounds whose leading sample-size exponent is governed by the latent dimension rather than the ambient dimension, with ambient and noise dependence kept explicit. Fixed positive target noise keeps the interpolation nondegenerate over the full time interval. The same spatial regularity propagates the learned velocity error through the transport ODE, yielding a corresponding Wasserstein convergence guarantee. These results show that exact low-dimensional support is not necessary for Flow Matching to retain latent-dimensional statistical behavior.
