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

TL;DR

This paper introduces a geometric boundary detection method based on mean curvature, improving unsupervised learning in high-dimensional, non-linear data by leveraging intrinsic data manifold properties.

Contribution

It presents a novel curvature-based boundary detection framework that does not rely on density estimates, enhancing clustering robustness in complex data scenarios.

Findings

Consistently improves clustering performance on synthetic and real datasets.

Effectively separates boundary and smooth data points using curvature-driven decomposition.

Provides a geometric interpretation of data boundaries through mean curvature analysis.

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.