Continuous persistence landscapes

TL;DR

The paper introduces continuous persistence landscapes as a measure-theoretic vectorization for $q$-tame measures, unifying persistence diagrams and their weak limits. It proves that this mapping is bijective and $L^1$-stable, enabling exact reconstruction of the original measure via Carathéodory’s extension. The framework provides a principled, invertible descriptor that remains faithful in the limit and under sampling, with a rank-based $W_1$ stability bound. It also analyzes the relationship to average persistence landscapes and extends to signed measures through Jordan decomposition, broadening applicability to a wider class of persistence measures.

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.

Figures (4)

• Figure 1: A persistence diagram with $Q_{t,h} = (-\infty, t-h)\times(t+h, \infty)$ shaded.
• Figure 2: We order the quadrants with reverse containment. This choice is consistent with interval modules with the containment order.
• Figure 3: Fix $t\in \mathbb{R}$. We plot an example of the continuous persistence landscape $\lambda(a,t)$ varying $a$ and show it is a left-continuous generalized inverse of the function $\mu(Q_{t,h})$, where $\mu$ is the $q-$tame measure whose continuous landscape is $\lambda$. For discrete landscapes, $\lambda(a,t)$ is a decreasing left-continuous step function, constant on $(n,n+1]$ for $n\in \mathbb{N}$.
• Figure 4: Fix $t\in \mathbb{R}$. We plot $\nu_0(Q_{t,h})$, the right-continuous generalized inverse of $\lambda(a,t)$, which is a decreasing function with respect to $h$. We will show $\nu_0 = \mu$ on $\langle \mathcal{E} \rangle$. For discrete landscapes, $\nu_0(Q_{t,h})$ is a decreasing right-continuous step function.

Theorems & Definitions (17)

• Theorem 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3