Basic metric geometry of the bottleneck distance (Q6566404)

The paper presents a thorough analysis of the basic metric geometry of the bottleneck distance, a critical measure of similarity in persistent homology. The main focus is on investigating the topological and geometric properties of spaces of persistence diagrams equipped with the bottleneck distance.\N\NKey findings include:\N\N1. Characterization of metrizability (Theorem A), completeness (Theorem B), and separability (Theorem D) of persistence diagram spaces with the bottleneck distance.\N\N2. A criterion for when these spaces form a geodesic pseudometric space, given that the underlying metric space is proper (Theorem E).\N\NThe significance of these results lies in their contribution to a deeper understanding of persistence diagram spaces' properties, which is fundamental to persistent homology applications. Specifically:\N\N1. The characterization of metrizability, completeness, and separability establishes a solid mathematical foundation for using persistence diagrams in various applications.\N\N2. The criterion for geodesic pseudometric spaces enhances the theoretical underpinnings of persistent homology, potentially enabling more robust and reliable analyses.\N\NThese insights into the metric geometry of the bottleneck distance are crucial for effectively applying persistent homology techniques across diverse fields such as data analysis, shape recognition, and topological data analysis.\N\NIn conclusion, this paper strengthens the theoretical framework of persistent homology, thereby enhancing its applicability and impact across various domains. By providing a comprehensive analysis of the bottleneck distance's properties, it lays the groundwork for more sophisticated applications and further theoretical developments in the field.