Region-Aware Wasserstein Distances of Persistence Diagrams and Merge Trees

Area of Science:

• Computational Topology
• Data Analysis
• Scientific Visualization

Background:

• Topological Data Analysis (TDA) relies on summarizing data shape using features like persistence diagrams and merge trees.
• Classical Wasserstein distance provides a metric for comparing these topological features but can be limited in discriminative power.
• Analyzing large-scale ensemble data requires efficient and robust methods for comparing and tracking topological structures.

Purpose of the Study:

• To generalize the Wasserstein distance for 0th persistence diagrams and merge trees by incorporating domain region properties.
• To develop a more discriminative metric than the classical Wasserstein distance for topological features.
• To propose efficient computational strategies and demonstrate applications in time-varying ensemble analysis and dimensionality reduction.

Main Methods:

• Redefinition of topological feature comparison as a distance between extrema-aligned region values.
• Introduction of an input parameter to adjust the influence of region properties on the distance.
• Implementation of strategies for computational efficiency, including subsetting regions and compressing properties.
• Development of temporal persistence curves for tracking feature evolution and distance matrices for dimensionality reduction.

Main Results:

• The proposed generalized Wasserstein distance offers improved discriminative power compared to the classical approach.
• Efficient methods for computation time and memory storage were demonstrated.
• Experiments on ensemble data showed average running times in the order of minutes.
• Successful application in tracking temporal evolution of topological features and in dimensionality reduction for ensemble visualization.

Conclusions:

• The generalized Wasserstein distance provides a powerful and efficient tool for analyzing topological features in domain regions.
• The method facilitates understanding of feature dynamics in time-varying data and aids in ensemble data exploration.
• The C++ implementation allows for reproducibility and practical application of the proposed techniques.