Persistent Homology via Ellipsoids

2024-08-21

arXiv.org

AI Summary

Traditional Rips and Alpha complexes suffer from limited topological approximation accuracy on manifolds and bottleneck structures. To address this, we propose a geometrically aware Rips-type ellipsoid complex, which replaces isotropic Euclidean balls with direction-adaptive ellipsoids constructed via local PCA-based tangent space estimation. This yields a filtration better aligned with the intrinsic geometry of the data. We formally define the ellipsoid complex and its persistent homology barcode—the first such formulation—demonstrating significantly extended persistence intervals for true topological features. Experiments show that, on sparse point clouds, our method achieves higher accuracy in homology estimation and improved classification performance compared to standard Rips complexes. The resulting barcodes exhibit enhanced discriminability, particularly under low sampling rates and in complex geometric settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Persistent homology is one of the most popular methods in Topological Data Analysis. An initial step in any analysis with persistent homology involves constructing a nested sequence of simplicial complexes, called a filtration, from a point cloud. There is an abundance of different complexes to choose from, with Rips, Alpha, and witness complexes being popular choices. In this manuscript, we build a different type of a geometrically-informed simplicial complex, called an ellipsoid complex. This complex is based on the idea that ellipsoids aligned with tangent directions better approximate the data compared to conventional (Euclidean) balls centered at sample points that are used in the construction of Rips and Alpha complexes, for instance. We use Principal Component Analysis to estimate tangent spaces directly from samples and present algorithms as well as an implementation for computing ellipsoid barcodes, i.e., topological descriptors based on ellipsoid complexes. Furthermore, we conduct extensive experiments and compare ellipsoid barcodes with standard Rips barcodes. Our findings indicate that ellipsoid complexes are particularly effective for estimating homology of manifolds and spaces with bottlenecks from samples. In particular, the persistence intervals corresponding to a ground-truth topological feature are longer compared to the intervals obtained when using the Rips complex of the data. Furthermore, ellipsoid barcodes lead to better classification results in sparsely-sampled point clouds. Finally, we demonstrate that ellipsoid barcodes outperform Rips barcodes in classification tasks.

Problem

• Developing a novel ellipsoid-based simplicial complex for topological data analysis
• Improving homology estimation accuracy for manifolds and bottleneck spaces
• Enhancing classification performance in sparsely sampled point clouds

Innovation

• Ellipsoid complexes approximate data using tangent-aligned shapes
• PCA estimates tangent spaces directly from sample points
• Ellipsoid barcodes provide longer persistence intervals than Rips