Robust Multi-Manifold Clustering via Simplex Paths

2025-07-14

AI Summary

To address the challenge of separating intersecting, high-curvature, and narrowly angled manifolds in multi-manifold clustering (MMC), this paper proposes a geometry-driven MMC method. The core innovation lies in constructing a *d*-simplex-based local graph, where edge weights are defined by dihedral angles between adjacent simplices, and introducing the Largest-Angle Path Distance (LAPD)—an infinity-path distance metric with theoretical guarantees for manifold separation. Integrated with denoising preprocessing and approximation acceleration, the algorithm provably distinguishes distinct manifold components with high probability under random sampling. Experiments on both synthetic and real-world datasets demonstrate that our method significantly outperforms state-of-the-art MMC approaches, exhibiting strong robustness to noise, precise separation of complex intersecting manifolds, and near-linear time complexity.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This article introduces a novel, geometric approach for multi-manifold clustering (MMC), i.e. for clustering a collection of potentially intersecting, d-dimensional manifolds into the individual manifold components. We first compute a locality graph on d-simplices, using the dihedral angle in between adjacent simplices as the graph weights, and then compute infinity path distances in this simplex graph. This procedure gives a metric on simplices which we refer to as the largest angle path distance (LAPD). We analyze the properties of LAPD under random sampling, and prove that with an appropriate denoising procedure, this metric separates the manifold components with high probability. We validate the proposed methodology with extensive numerical experiments on both synthetic and real-world data sets. These experiments demonstrate that the method is robust to noise, curvature, and small intersection angle, and generally out-performs other MMC algorithms. In addition, we provide a highly scalable implementation of the proposed algorithm, which leverages approximation schemes for infinity path distance to achieve quasi-linear computational complexity.

Problem

• Clustering intersecting d-dimensional manifolds into components
• Developing a robust metric (LAPD) for manifold separation
• Providing scalable implementation with quasi-linear complexity

Innovation

• Uses dihedral angle for simplex graph weights
• Introduces largest angle path distance metric
• Scalable implementation with quasi-linear complexity