2 research outputs found
The Christoffel-Darboux kernel for topological data analysis
Persistent homology has been widely used to study the topology of point
clouds in . 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 , for . 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 , using a recent algorithm
of Basu & Karisani (2022).Comment: 22 pages, 11 figures, 1 tabl