Computing the Multicover Bifiltration

Area of Science:

• Computational Topology
• Geometric Analysis
• Data Science

Background:

• The multicover bifiltration models spatial data by considering points within a certain distance (r) to a minimum number of data points (k).
• Existing Čech-based models for this bifiltration are computationally intensive and large.
• Efficient computation of topological features for evolving spatial datasets is a significant challenge.

Purpose of the Study:

• To develop a computationally efficient and topologically equivalent alternative to the multicover bifiltration.
• To introduce novel combinatorial constructions (polyhedral and simplicial) for the multicover bifiltration.
• To facilitate the computation of homology for the multicover bifiltration.

Main Methods:

• Introduction of a polyhedral bifiltration based on the rhomboid tiling, utilizing a modified algorithm for efficient computation.
• Development of a related simplicial bifiltration to aid in understanding and validating the polyhedral construction.
• Implementation and experimental evaluation of the constructions in dimensions 2 and 3.

Main Results:

• The proposed polyhedral and simplicial bifiltrations are topologically equivalent to the multicover bifiltration.
• These combinatorial constructions are significantly smaller and more computationally efficient than previous Čech-based models.
• Experimental results in dimensions 2 and 3 demonstrate the practical applicability and efficiency of the new methods.

Conclusions:

• The novel combinatorial bifiltrations offer a more efficient approach to computing the homology of the multicover bifiltration.
• These methods provide a valuable tool for analyzing complex spatial data and topological structures.
• The research advances computational topology by offering practical and scalable solutions for data analysis.