# Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization

arXiv: https://arxiv.org/abs/2510.02308 · HTML: https://mlsanity.com/paper/2510.02308/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 83 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 21 of 21)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Strong Consistency, Graph Laplacians, and the Stochastic Block Model](https://mlsanity.com/paper/2004.09780/) (2020)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Spectral Convergence of the connection Laplacian from random samples](https://mlsanity.com/paper/1306.1587/) (2013)
- [Global registration of multiple point clouds using semidefinite programming](https://mlsanity.com/paper/1306.5226/) (2013)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise](https://mlsanity.com/paper/2006.00402/) (2020)
- [Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud](https://mlsanity.com/paper/2011.10725/) (2020)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Selecting the independent coordinates of manifolds with large aspect ratios](https://mlsanity.com/paper/1907.01651/) (2019)
- [Large sample spectral analysis of graph-based multi-manifold clustering](https://mlsanity.com/paper/2107.13610/) (2021)
- [Low-dimensional embeddings of high-dimensional data](https://mlsanity.com/paper/2508.15929/) (2025)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling](https://mlsanity.com/paper/2209.08004/) (2022)
- [Spectral Embedding Norm: Looking Deep into the Spectrum of the Graph Laplacian](https://mlsanity.com/paper/1810.10695/) (2018)
- [On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs](https://mlsanity.com/paper/2312.10295/) (2023)
- [Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds](https://mlsanity.com/paper/2506.00171/) (2025)
- [Non-degenerate Rigid Alignment in a Patch Framework](https://mlsanity.com/paper/2303.11620/) (2023)

## Abstract (arXiv)

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise setting due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.