Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization

• URL: http://arxiv.org/abs/2510.02308v1
• Date: Thu, 02 Oct 2025 17:59:45 GMT
• Title: Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization
• Authors: Dhruv Kohli, Sawyer J. Robertson, Gal Mishne, Alexander Cloninger,
• Abstract summary: Estimating the tangent spaces of a data manifold is a fundamental problem in data analysis.<n>We propose a method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation.
• Score: 48.25304391127552
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Estimating the tangent spaces of a data manifold is a fundamental problem in data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise settings 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 a tubular neighborhood of a manifold shows that gradients of the low-frequency Laplacian 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. Through comprehensive experiments, we 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.

Related papers

• A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting [0.0]
  This work presents a novel and effective method for fitting multidimensional ellipsoids to scattered data in the contamination of noise and outliers. We incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain. We apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks.
  arXiv  Detail & Related papers  (2024-07-27T14:31:51Z)
• Boundary Detection Algorithm Inspired by Locally Linear Embedding [8.259071011958254]
  We propose a method for detecting boundary points inspired by the widely used locally linear embedding algorithm.<n>In the presence of high-dimensional noise, we propose a framework aimed at enhancing boundary detection in noisy data.
  arXiv  Detail & Related papers  (2024-06-26T16:05:57Z)
• Neural Gradient Learning and Optimization for Oriented Point Normal Estimation [53.611206368815125]
  We propose a deep learning approach to learn gradient vectors with consistent orientation from 3D point clouds for normal estimation. We learn an angular distance field based on local plane geometry to refine the coarse gradient vectors. Our method efficiently conducts global gradient approximation while achieving better accuracy and ability generalization of local feature description.
  arXiv  Detail & Related papers  (2023-09-17T08:35:11Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Random Smoothing Regularization in Kernel Gradient Descent Learning [24.383121157277007]
  We present a framework for random smoothing regularization that can adaptively learn a wide range of ground truth functions belonging to the classical Sobolev spaces. Our estimator can adapt to the structural assumptions of the underlying data and avoid the curse of dimensionality.
  arXiv  Detail & Related papers  (2023-05-05T13:37:34Z)
• Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling [8.271859911016719]
  We develop tools for robust inference under high-dimensional noise. We show that our approach is robust to variability in technical noise levels across cell types.
  arXiv  Detail & Related papers  (2022-09-16T15:39:11Z)
• Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach [5.975670441166475]
  We propose a kernel-spectral embedding algorithm for learning low-dimensional nonlinear structures from high-dimensional and noisy observations. The algorithm employs an adaptive bandwidth selection procedure which does not rely on prior knowledge of the underlying manifold. The obtained low-dimensional embeddings can be further utilized for downstream purposes such as data visualization, clustering and prediction.
  arXiv  Detail & Related papers  (2022-02-28T22:46:34Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• Tensor Laplacian Regularized Low-Rank Representation for Non-uniformly Distributed Data Subspace Clustering [2.578242050187029]
  Low-Rank Representation (LRR) suffers from discarding the locality information of data points in subspace clustering. We propose a hypergraph model to facilitate having a variable number of adjacent nodes and incorporating the locality information of the data. Experiments on artificial and real datasets demonstrate the higher accuracy and precision of the proposed method.
  arXiv  Detail & Related papers  (2021-03-06T08:22:24Z)
• Manifold Learning via Manifold Deflation [105.7418091051558]
  dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. Many popular methods can fail dramatically, even on simple two-dimensional Manifolds. This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates. Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
  arXiv  Detail & Related papers  (2020-07-07T10:04:28Z)
• Semiparametric Nonlinear Bipartite Graph Representation Learning with Provable Guarantees [106.91654068632882]
  We consider the bipartite graph and formalize its representation learning problem as a statistical estimation problem of parameters in a semiparametric exponential family distribution. We show that the proposed objective is strongly convex in a neighborhood around the ground truth, so that a gradient descent-based method achieves linear convergence rate. Our estimator is robust to any model misspecification within the exponential family, which is validated in extensive experiments.
  arXiv  Detail & Related papers  (2020-03-02T16:40:36Z)
• Improved guarantees and a multiple-descent curve for Column Subset Selection and the Nystr\"om method [76.73096213472897]
  We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees. Our approach leads to significantly better bounds for datasets with known rates of singular value decay. We show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.
  arXiv  Detail & Related papers  (2020-02-21T00:43:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.