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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.04274 |
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| _version_ | 1866911666402230272 |
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| author | Levada, Alexandre L. M. |
| author_facet | Levada, Alexandre L. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning Levada, Alexandre L. M. Machine Learning Artificial Intelligence 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. |
| title | A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.04274 |