The Multi-Cover Persistence of Euclidean Balls

Area of Science:

• Computational Topology
• Geometric Data Analysis
• Persistent Homology

Background:

• The k-fold cover is a topological construction used in data analysis.
• Persistent homology is a powerful tool for analyzing the shape of data.
• Standard methods for persistent homology work well for scale-based filtrations.

Purpose of the Study:

• To compute persistence diagrams for two types of filtrations of k-fold covers: scale and depth.
• To develop novel geometric and topological methods for the depth filtration.
• To establish a connection between Delaunay mosaics and the persistence module of multi-covers.

Main Methods:

• Definition of k-fold cover for a point set X and radius r.
• Consideration of two filtrations: scale (fixed k, increasing r) and depth (fixed r, decreasing k).
• Introduction of a rhomboid tiling and order-k Delaunay mosaics for the depth filtration.
• Construction of a zigzag module of Delaunay mosaics.

Main Results:

• Computation of persistence diagrams for both scale and depth filtrations.
• Demonstration that standard methods suffice for scale filtration.
• Development of new geometric and topological concepts for depth filtration.
• Isomorphism between the zigzag module of Delaunay mosaics and the persistence module of multi-covers.

Conclusions:

• The study provides a comprehensive analysis of k-fold covers using persistent homology.
• Novel methods are introduced for analyzing depth-based filtrations, expanding the applicability of persistent homology.
• The connection to Delaunay mosaics offers new insights into the structure of multi-covers.