# On the Expectation of a Persistence Diagram by the Persistence Weighted Kernel

## TL;DR

This paper investigates the statistical properties of the expectation of persistence diagrams using the persistence weighted kernel, establishing theoretical results and demonstrating practical applications through numerical experiments.

## Contribution

It provides a theoretical framework linking probability distributions to the persistence weighted kernel, including laws of large numbers, confidence intervals, and stability results.

## Key findings

- Established strong law of large numbers for the expectation
- Developed a method for numerical confidence intervals
- Provided a counterexample challenging common views in TDA

## Abstract

In topological data analysis, persistent homology characterizes robust topological features in data and it has a summary representation, called a persistence diagram. Statistical research for persistence diagrams have been actively developed, and the persistence weighted kernel shows several advantages over other statistical methods for persistence diagrams. If data is drawn from some probability distribution, the corresponding persistence diagram have randomness. Then, the expectation of the persistence diagram by the persistence weighted kernel is well-defined. In this paper, we study relationships between a probability distribution and the persistence weighted kernel in the viewpoint of (1) the strong law of large numbers and the central limit theorem, (2) a confidence interval to estimate the expectation of the persistence weighted kernel numerically, and (3) the stability theorem to ensure the continuity of the map from a probability distribution to the expectation. In numerical experiments, we demonstrate our method gives an interesting counterexample to a common view in topological data analysis.