A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning

• URL: http://arxiv.org/abs/2605.04274v1
• Date: Tue, 05 May 2026 20:19:09 GMT
• Title: A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning
• Abstract summary: We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
• Score: 52.452902154360565
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning that departs from traditional density-based approaches by explicitly modeling the intrinsic curvature of the data manifold. The method relies on a discrete approximation of the shape operator, estimated from local k-nearest neighbor patches, to compute pointwise mean curvature without requiring explicit manifold parametrization. The key insight of MCBP is to use mean curvature as a principled descriptor of boundary structure: high-curvature regions naturally correspond to transitions between clusters, geometric irregularities, and low-density interfaces. This yields a unified geometric interpretation of boundary, outlier, and transition points. We further introduce an adaptive percentile-based thresholding scheme that enables multiscale boundary extraction without relying on ad hoc density parameters. Beyond detection, we propose a curvature-driven data decomposition that separates samples into smooth (low-curvature) and boundary (high-curvature) subsets, effectively acting as a non-linear geometric filtering mechanism. This representation enhances cluster separability and improves the robustness of downstream unsupervised algorithms. Extensive experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance, particularly in complex and high-dimensional scenarios. These results position MCBP as a concrete contribution to Geometric Machine Learning, highlighting the potential of curvature-aware analysis as a unifying paradigm bridging differential geometry and data-driven modeling.

Related papers

• CuBAS: Information Geometric Curvature-Based Adaptive Sampling for Supervised Classification [52.452902154360565]
  CuBAS is an information-geometric framework for adaptive data selection in supervised classification.<n>A labeled dataset can be viewed as a statistical manifold, on which local curvature, estimated via the ratio of second to first-order observed Fisher information, faithfully encodes the geometric complexity of the data distribution.
  arXiv  Detail & Related papers  (2026-07-03T09:36:02Z)
• Generalization in Nonlinear Least Squares via Learned Feature Geometry [15.966075413898336]
  We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability.<n>We deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model.
  arXiv  Detail & Related papers  (2026-06-07T19:33:36Z)
• Fitting Unknown Number of Hyperplanes with Manifold Optimization [57.48093263119306]
  Fitting unknown number of linearplanes to data is a fundamental yet challenging problem in machine learning.<n>Existing approaches often struggle with optimum optima or lack geometric consistency.
  arXiv  Detail & Related papers  (2026-05-27T14:02:20Z)
• Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning [52.452902154360565]
  GTSA-PCA is a geometric extension of Principal Component Analysis.<n>It integrates curvature awareness and geodesic consistency within a unified spectral framework.<n>Our results position GTSA-PCA as a principled bridge between statistical and geometric approaches to dimensionality reduction.
  arXiv  Detail & Related papers  (2026-04-20T20:36:36Z)
• Understanding and Improving UMAP with Geometric and Topological Priors: The JORC-UMAP Algorithm [1.7484982792736636]
  dimensionality reduction techniques, particularly UMAP, are widely used for visualizing high-dimensional data.<n>We introduce Ollivier-Ricci curvature as a geometric prior, reinforcing edges at geometric bottlenecks and reducing redundant links.<n>Experiments on synthetic and real-world datasets show that JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods.
  arXiv  Detail & Related papers  (2026-01-23T08:42:56Z)
• GeoGNN: Quantifying and Mitigating Semantic Drift in Text-Attributed Graphs [59.61242815508687]
  Graph neural networks (GNNs) on text--attributed graphs (TAGs) encode node texts using pretrained language models (PLMs) and propagate these embeddings through linear neighborhood aggregation.<n>This work introduces a local PCA-based metric that measures the degree of semantic drift and provides the first quantitative framework to analyze how different aggregation mechanisms affect manifold structure.
  arXiv  Detail & Related papers  (2025-11-12T06:48:43Z)
• Variational Geometric Information Bottleneck: Learning the Shape of Understanding [0.0]
  Variational Geometric Information Bottleneck (V-GIB) is a variational estimator that unifies mutual-information compression and curvature regularization.<n>V-GIB provides a principled and measurable route to representations that are geometrically coherent, data-efficient, and aligned with human-understandable structure.
  arXiv  Detail & Related papers  (2025-11-04T11:33:54Z)
• Curvature Learning for Generalization of Hyperbolic Neural Networks [51.888534247573894]
  Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures.<n>Inappropriate curvatures may cause HNNs to converge to suboptimal parameters, degrading overall performance.<n>We propose a sharpness-aware curvature learning method to smooth the loss landscape, thereby improving the generalization of HNNs.
  arXiv  Detail & Related papers  (2025-08-24T07:14:30Z)
• Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models [63.331590876872944]
  We propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models.<n>These metrics define spatially varying distances, enabling the computation of geodesics.<n>We show that EBM-derived metrics consistently outperform established baselines.
  arXiv  Detail & Related papers  (2025-05-23T12:18:08Z)
• What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold [0.053713376045563095]
  We introduce a score-based Riemannian metric to characterize the intrinsic geometry of a data manifold.<n>Our approach creates a geometry where geodesics naturally follow the manifold's contours.<n>We show that our score-based geodesics capture meaningful perpendicular transformations that respect the underlying data distribution.
  arXiv  Detail & Related papers  (2025-05-16T11:19:57Z)
• Weighted Riesz Particles [0.0]
  We consider the target distribution as a mapping where the infinite-dimensional space of the parameters consists of a number of deterministic submanifolds. We study the properties of the point, called Riesz, and embed it into sequential MCMC. We find that there will be higher acceptance rates with fewer evaluations.
  arXiv  Detail & Related papers  (2023-12-01T14:36:46Z)

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.