流形稳定的流匹配

中文摘要 AI 辅助

流匹配（FM）通过速度回归学习生成动力学。几何流匹配变体通常假设先验分布支持在数据流形上，这需要通常不可用的几何知识。在没有此类知识的情况下，仅靠低回归误差并不能保证流形一致性。一致性使生成的样本保持在有效配置内，并且经验上与更好的任务性能相关。我们引入了流形稳定的流匹配（MSFM），它可以从任意的环境先验开始，不必支持在流形上。利用非线性动力学工具，即收缩理论，MSFM将学习的切向传输与规定的法向收缩相结合。该构造对已知流形使用解析投影算子，对未知数据几何使用通过主成分分析估计的局部仿射代理。通过在已知和未知流形两种情况下实现收缩理论，我们保证了流形不变性和在期望时间窗口（例如，一秒）内横向收敛到流形。我们推导了一族兼容的概率路径，并将训练损失分解为可学习的切向项和法向残差。一个椭圆实验达到了数量级为$10^{-6}$的平均终端离流形误差。在Push-T机器人实验中，MSFM将成功率从$74\%$提高到$82\%$。在Robomimic Square任务中，成功率从$60\%$增加到$72\%$，而旋转流形偏差从数量级$10^{-2}$减少到$10^{-7}$。MSFM终端几何误差由所选的数值容差控制。这些结果表明更强的几何一致性和更高的观察任务性能，支持规定的法向收缩作为学习生成传输的补充。

英文摘要

Flow matching (FM) learns generative dynamics through velocity regression. Geometric FM variants commonly assume a prior supported on the data manifold, requiring geometric knowledge that is often unavailable. Without such knowledge, low regression error alone does not guarantee manifold adherence. Adherence keeps generated samples within valid configurations and is empirically associated with better task performance. We introduce manifold-stable flow matching (MSFM), which can start from an arbitrary ambient prior, not necessarily supported on the manifold. Using tools from nonlinear dynamics, namely contraction theory, MSFM combines learned tangential transport with prescribed normal contraction. The construction uses analytical projectors for known manifolds and local affine proxies estimated by principal component analysis for unknown data geometry. By implementing contraction theory in both cases of known and unknown manifolds, we guarantee manifold invariance and transverse convergence to the manifold within a desired time window (e.g., one second). We derive a family of compatible probability paths and decompose the training loss into a learnable tangential term and a normal residual. An ellipse experiment attains a mean terminal off-manifold error of order $10^{-6}$. In Push-T robotic experiments, MSFM raises success from $74\%$ to $82\%$. In the Robomimic Square task, success increases from $60\%$ to $72\%$, while rotation-manifold deviation decreases from order $10^{-2}$ to $10^{-7}$. The MSFM terminal geometric errors are controlled by the chosen numerical tolerance. These results demonstrate stronger geometric adherence and higher observed task performance, supporting prescribed normal contraction as a complement to learned generative transport.

发表机构

• Colorado State University（科罗拉多州立大学）

机构由 AI 辅助整理，请以论文原文为准。