Manifold Fitting by Successive Tangent-Space Projection

2026-10-07

AI Summary

This study addresses the inherent trade-off between fitting accuracy and geometric coverage when recovering latent manifold structures from noisy observations. To this end, we propose a continuous tangent space projection algorithm that achieves high-precision manifold fitting by iteratively suppressing normal noise while preserving tangential variations. Theoretically, we define the fixed-point set of local neighborhoods, prove that its geometric localization order is O(σ²), and introduce a multiscale extension to eliminate curvature bias. Numerical experiments demonstrate that the proposed method significantly mitigates curvature shrinkage under high-noise conditions, outperforming existing approaches in both fitting accuracy and geometric coverage.

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 $O(σ^2)$ 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.

Problem

• Manifold fitting
• Noisy observations
• Geometric structure recovery
• Curvature bias
• Tangent space

Innovation

• Manifold Fitting
• Successive Tangent Space Projection
• Multi-scale Extension
• Curvature Bias Reduction
• Geometric Convergence