Robust Barycenters of Persistence Diagrams

Area of Science:

• Computational Topology
• Geometric Data Analysis
• Machine Learning

Background:

• Persistence diagrams are key in topological data analysis.
• Classical Wasserstein barycenter computation is limited to q=2 Wasserstein distance.
• Robustness to outliers is crucial for real-world data.

Purpose of the Study:

• To generalize Wasserstein barycenter computation for persistence diagrams beyond q=2.
• To develop a method robust to outliers in persistence diagram analysis.
• To demonstrate the utility of robust barycenters in clustering and dictionary encoding.

Main Methods:

• Adaptation of a fixed-point method for Wasserstein barycenter computation.
• Generalization of transportation costs for q > 1, specifically q in (1,2).
• Application of the robust barycenter method to persistence diagram clustering and dictionary encoding.

Main Results:

• A generalized approach for computing robust Wasserstein barycenters of persistence diagrams is presented.
• The method successfully computes barycenters for generic transportation costs (q > 1).
• Demonstrated enhanced robustness to outliers in clustering and dictionary encoding applications.

Conclusions:

• The proposed fixed-point method provides a robust and generalized framework for Wasserstein barycenters.
• This generalization extends the applicability of barycenters to outlier-prone datasets.
• The method offers significant advantages for topological data analysis tasks like clustering and encoding.