发表机构
Zhejiang University of Finance and Economics; Huazhong University of Science and Technology; National University of Singapore（浙江财经大学; 华中科技大学; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出连续切空间投影（STSP）迭代流形拟合法，在降噪同时保持几何覆盖，并给出理论保证及多尺度扩展MS-STSP，实验验证其平衡精度与覆盖的优势。
AI 中文摘要
流形拟合旨在从含噪声的环境观测中恢复其背后的几何结构。我们提出了连续切空间投影（STSP）方法，这是一种迭代式的流形拟合方法，能够在减少法向噪声的同时保留切向变化。前者提高了拟合精度，而后者有助于保持几何覆盖范围。我们将STSP的邻近总体不动点集刻画为一个流形拟合对象。在从具有正reach值的紧致光滑流形上均匀采样并添加各向同性高斯噪声的条件下，该集合位于底层流形的$O(\sigma^2)$范围内，且邻近总体轨道几何收敛于该集合。在有限样本层面，以高概率，局部管内的不动点位于底层流形的$O(\sigma^2)$范围内（直至采样误差），并且从固定参考样本生成的、在该管内初始化的经验轨道会进入并停留在同一邻域内。我们进一步开发了多尺度扩展方法MS-STSP，旨在减少曲率偏差，同时在总体和有限样本层面保持STSP的$O(\sigma^2)$几何定位阶数。数值实验表明，STSP在平衡拟合精度和几何覆盖范围方面优于竞争方法，且MS-STSP能减少曲率引起的收缩，尤其是在高噪声条件下。
英文摘要
Manifold fitting seeks to recover the geometric structure underlying noisy ambient observations. We propose Successive Tangent Space Projection (STSP), an iterative manifold-fitting method that reduces normal noise while preserving tangential variation. The former improves fitting accuracy, whereas the latter helps retain geometric coverage. We characterise the \rev{nearby population fixed-point set} of STSP as a manifold-fitting object. Under uniform sampling from a compact smooth manifold with positive reach and isotropic Gaussian noise, this set lies within $O(σ^2)$ of the underlying manifold, and \rev{nearby population orbits} converge geometrically to it. At the finite-sample level, with high probability, fixed points in the local tube lie within $O(σ^2)$ of the underlying manifold up to sampling error, and empirical orbits \rev{initialised within that tube} and generated from a fixed reference sample enter and remain in the same neighbourhood. We further develop a multi-scale extension, MS-STSP, designed to reduce curvature bias while preserving the $O(σ^2)$ geometric localization order of STSP at both the population and finite-sample levels. Numerical experiments show that STSP compares favourably with competing methods in balancing fitting accuracy and geometric coverage, and that MS-STSP reduces curvature-induced shrinkage, particularly under high noise.
Comments51 pages, 22 figures
