Continuous persistence landscapes

TL;DR

This paper introduces continuous persistence landscapes, a measure-theoretic extension of traditional persistence landscapes, providing a stable, invertible, and more faithful vector summary for large data sets in topological data analysis.

Contribution

It generalizes persistence landscapes to a measure-based framework, ensuring stability and invertibility, and better capturing the structure of persistence diagrams in large data regimes.

Findings

Continuous persistence landscapes are bijective and L^1-stable.

The method applies to diagrams with high point multiplicities.

It allows reconstruction of the original measure from the landscape.

Abstract

As the size of data increase, persistence diagrams often exhibit structured asymptotic behavior, converging weakly to a Radon measure. However, conventional vector summaries such as persistence landscapes are not well-behaved in this setting, particularly for diagrams with high point multiplicities. We introduce continuous persistence landscapes, a new vectorization defined on a special class of Borel measures, which we call q-tame measures. It includes both the persistence diagrams and their weak limits. Our construction generalizes persistence landscapes to a measure-theoretic setting, preserving the intrinsic structure of persistence measures. We show that this vector summary is bijective and L^1-stable under mild assumptions, and that the original measure can be uniquely reconstructed. This approach gives a more faithful description of the shape of data in the limit and provides a stable, invertible way to analyze topological features in large systems.