A universal null-distribution for topological data analysis

Area of Science:

• Topological Data Analysis
• Computational Topology
• Data Science

Background:

• Understanding persistence diagram distributions is a key challenge in topological data analysis.
• Current methods struggle to characterize the statistical properties of these diagrams.
• This limits the quantitative assessment of topological features in complex datasets.

Purpose of the Study:

• To investigate the statistical distribution of persistence diagrams from random point-clouds.
• To identify a universal probability law governing these distributions.
• To develop a novel hypothesis testing framework for feature significance.

Main Methods:

• Extensive experimentation on simulated and real-world point-cloud data.
• Normalization of persistence diagrams to identify underlying statistical patterns.
• Statistical analysis to determine a candidate universal probability distribution.
• Development of a hypothesis testing framework based on the discovered law.

Main Results:

• A surprising universal probability law governs normalized persistence diagrams from random point-clouds.
• The discovered law holds across diverse data geometries, topologies, and distributions.
• An explicit, well-known distribution is proposed as a candidate for this universal law.
• A new hypothesis testing framework for computing significance values of topological features was developed.

Conclusions:

• The statistical distribution of persistence diagrams is not arbitrary but follows a universal law.
• This discovery provides a quantitative method for assessing the significance of topological structures.
• The proposed framework advances topological data analysis by offering robust statistical significance testing.