Manifold Fitting by Successive Tangent-Space Projection
Abstract
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(σ2) 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(σ2) 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(σ2) 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.
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BibTeX
@misc{xia2026manifold,
title = {{Manifold Fitting by Successive Tangent-Space Projection}},
author = {Yuqing Xia and Bingjie Li and Zhigang Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09423},
eprint = {2610.09423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09423},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Manifold Fitting by Successive Tangent-Space Projection
AU - Yuqing Xia
AU - Bingjie Li
AU - Zhigang Yao
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09423
UR - https://arxiv.org/abs/2610.09423
AB - 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.
ER -