Persistent Homology via Ellipsoids

TL;DR

This work introduces Rips-type ellipsoid complexes that replace Euclidean balls with tangent-aligned ellipsoids, estimated via local PCA to better approximate manifold-structured data. It proves stability of the resulting ellipsoid filtrations under small $\delta$-perturbations, establishing interleaving between filtrations and a bound on bottleneck distance for the corresponding barcodes. The authors derive inclusions between Rips and ellipsoid complexes, provide an efficient algorithm for computing ellipsoid barcodes via ellipsoid intersections, and demonstrate through diverse experiments that ellipsoid barcodes yield stronger topological signals and improved classification performance, especially on manifolds and bottleneck-rich datasets. The work also releases code and discusses future directions, including potential integration with Alpha complexes and alternatives to PCA for tangent estimation, broadening practical applicability in topological data analysis.

Abstract

Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from, with Čech, Rips, alpha, and witness complexes being popular choices. In this manuscript, we build a novel type of geometrically informed simplicial complex, called a Rips-type 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, as used in the construction of Rips and Alpha complexes. We use Principal Component Analysis to estimate tangent spaces directly from samples and present an algorithm for computing Rips-type ellipsoid barcodes, i.e., topological descriptors based on Rips-type ellipsoid complexes. Additionally, we show that the ellipsoid barcodes depend continuously on the input data so that small perturbations of a k-generic point cloud lead to proportionally small changes in the resulting ellipsoid barcodes. This provides a theoretical guarantee analogous, if somewhat weaker, to the classical stability results for Rips and Čech filtrations. We also conduct extensive experiments and compare Rips-type ellipsoid barcodes with standard Rips barcodes. Our findings indicate that Rips-type ellipsoid complexes are particularly effective for estimating the homology of manifolds and spaces with bottlenecks from samples. In particular, the persistence intervals corresponding to ground-truth topological features are longer compared to those obtained using the Rips complex of the data. Furthermore, Rips-type ellipsoid barcodes lead to better classification results in sparsely sampled point clouds. Finally, we demonstrate that Rips-type ellipsoid barcodes outperform Rips barcodes in classification tasks.

Figures (14)

• Figure 1: Four stages of a Rips complex construction for a point cloud, showing simplices up to dimension $2$.
• Figure 2: Example of a persistence barcode and a persistence diagram. The underlying point cloud dataset is shown in Figure \ref{['fig:Rips complex example']}.
• Figure 3: Four stages of a Rips-type ellipsoid complex for a point cloud, showing simplices up to dimension 2.
• Figure 4: A graphical representation of the nesting property between balls (dashed circles) and ellipsoids (red) that implies the nesting relation between Rips and Rips-type ellipsoid complexes.
• Figure 5: Schematic depiction showing that adding multiple nearby points to a single point can lead to a swapping of the axes.

Theorems & Definitions (57)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6: $\delta$-interleaving
• Remark 1.7
• Definition 1.8
• Theorem 1.9
• Remark 1.10