Topo-Geometric Analysis of Variability in Point Clouds Using Persistence Landscapes.

Area of Science:

• Computational Topology
• Data Science
• Geometric Data Analysis

Background:

• Topological data analysis (TDA) uses tools like persistence homology to find low-dimensional structures in noisy point clouds.
• Persistence homology represents features using persistence diagrams, which can be converted to persistence landscapes for statistical analysis.
• Variability in point clouds confounds topological and geometric information in both diagrams and landscapes, hindering reliable conclusions.

Purpose of the Study:

• To develop a framework for decomposing variability in persistence diagrams into topological signal and topological noise.
• To enable more reliable statistical analysis of point cloud data by distinguishing true topological features from noise.
• To improve the interpretation of results from persistence homology in TDA.

Main Methods:

• Developed a framework to decompose persistence diagram variability into signal and noise.
• Utilized persistence landscapes and an elastic Riemannian metric for alignment.
• Isolated topological signal through aligned landscapes (amplitude) and identified topological noise via reparameterizations (phase).

Main Results:

• The proposed framework successfully decouples topological signal from topological noise in persistence diagrams.
• Aligned landscapes capture the topological signal, while reparameterizations reveal geometric, scaling, and sampling variabilities as topological noise.
• Demonstrated the framework's effectiveness on simulated data and provided novel insights in two real-world data studies.

Conclusions:

• Distinguishing topological signal from noise is crucial for drawing reliable conclusions from persistence homology.
• The developed framework offers a robust method for variability decomposition in persistence diagrams.
• This approach enhances the application of TDA in analyzing complex, noisy datasets across various scientific domains.