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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

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

Paweł Dłotko¹, Davide Gurnari¹

¹Dioscuri Centre in Topological Data Analysis, Mathematical Institute, Polish Academy of Sciences, Warsaw, 00-656, Poland.

|November 15, 2023

Summary

This summary is machine-generated.

Euler characteristic curves and profiles offer a robust alternative to persistent homology for data analysis. These methods provide stable, efficient, and scalable summaries for complex datasets, overcoming limitations of traditional topological data analysis tools.

Keywords:

Euler characteristic distributed computations persistent homology topological data analysis

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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.