The Christoffel-Darboux kernel for topological data analysis

TL;DR

This paper introduces a robust, computationally efficient persistence module based on Christoffel-Darboux kernels for topological data analysis, improving outlier resistance and scalability in studying point cloud topology.

Contribution

It presents a novel persistence module leveraging Christoffel-Darboux kernels, offering robustness to outliers and linear-time computation in data size, expanding geometric inference methods.

Findings

The new module is robust to statistical outliers.

Persistent homology can be computed in linear time relative to data points.

The approach is stable under Wasserstein distance and scalable in ambient dimension.

Abstract

Persistent homology has been widely used to study the topology of point clouds in $\mathbb{R}^n$. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this paper we introduce a novel persistence module for a point cloud using the theory of Christoffel-Darboux kernels. This module is robust to (statistical) outliers in the data, and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in $\mathbb{R}^n$, for $n=1, 2, 3$. Our work expands upon recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference (Lasserre, Pauwels and Putinar, 2022). There, these kernels are used to construct a polynomial whose level sets capture the geometry of a point cloud in a precise sense. We show that the persistent homology associated to the sublevel set filtration of this polynomial is stable with respect to the Wasserstein distance. Moreover, we show that the persistent homology of this filtration can be computed in singly exponential time in the ambient dimension $n$, using a recent algorithm of Basu & Karisani (2022).