Euler characteristic curves and profiles: a stable shape invariant for big data problems.

Area of Science:

• Topological Data Analysis
• Computational Topology
• Data Science

Background:

• Persistent homology is a standard tool for data shape summarization but faces computational and scalability challenges.
• Limitations include difficulties in distributed computation, generalization to multifiltrations, and prohibitive costs for large datasets.

Purpose of the Study:

• To introduce and analyze Euler characteristic curves for 1-parameter filtrations and Euler characteristic profiles for multiparameter filtrations.
• To demonstrate the advantages of Euler characteristic-based methods over persistent homology for data analysis.
• To highlight the stability and practical applicability of these novel topological invariants.

Main Methods:

• Development of efficient algorithms for computing Euler characteristic curves and profiles.
• Demonstration of distributed computation strategies for these methods.
• Generalization of Euler characteristic approaches to multiparameter filtrations (multifiltrations).

Main Results:

• Euler characteristic curves and profiles overcome key limitations of persistent homology, including computational expense and distribution challenges.
• These methods are shown to be generalizable to multifiltrations.
• The stability of Euler curves and profiles is proven, confirming their robustness for data analysis.

Conclusions:

• Euler characteristic-based methods provide a powerful and scalable alternative to persistent homology for topological data analysis.
• Their efficiency, generalizability, and stability make them suitable for analyzing large and complex datasets.
• The study validates the practical applicability of Euler curves and profiles through various use cases.