Wasserstein Stability of Persistence Landscapes and Barcodes

TL;DR

This work addresses the dependence of $p$-Wasserstein distances on the choice of metric for interval modules in persistent topology by introducing a rank-based metric $d_{ m rank}$ that leverages the rank invariant. It proves stability results connecting filtered CW complexes to barcodes and barcodes to persistence landscapes under $W_1^{\rm rank}$, and derives a sharp $L^1$-stability bound for persistence landscapes with respect to the rank-based Wasserstein distance. A key contribution is the demonstration that landscapes provide a robust, parameter-free 1-Lipschitz vectorization of barcodes, while graded persistence diagrams offer an additional stability framework via Möbius-inverted graded ranks. Overall, the paper strengthens the theoretical foundations of persistence summaries and provides practical stability guarantees for vectorizations used in data analysis and machine learning contexts.

Abstract

Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the $p-$Wasserstein distances. However, the $p-$Wasserstein distances depend on a choice of a metric on the set of interval modules, and there is no canonical choice. One convention is to use the length of the symmetric difference between 2 intervals, which equals to the 1-norm of the difference between their Hilbert functions. We propose a new metric for interval modules based on the rank invariant instead of the dimension invariant. Our metric is topologically equivalent to the metrics induced by the $p-$norms on $\R^2$. We establish stability results from filtered CW complexes to barcodes, as well as from barcodes to persistence landscapes. In particular, we show that vectorization via persistence landscapes is 1-Lipschitz with a sharp bound, with respect to the 1-Wasserstein distance on barcodes and the 1-norm on persistence landscapes.

Figures (8)

• Figure 1: Let $I \oplus J$ denote the direct sum of two interval modules for two overlapping intervals $I$ and $J$. We plot the function $\operatorname{rank} \: I\oplus J: \mathbb{R}_\leq^2 \to \mathbb{Z}$. By restricting the domain to the diagonal, the rank function recovers the Hilbert function, i.e. $\dim I \oplus J (t) = \operatorname{rank} I \oplus J (t,t)$.
• Figure 2: We plot the boundaries of the balls of radius $r$ centered at $x$ for each of $d_{\operatorname{rank}}$ (red), $d_{\operatorname{dim}}$ (blue), and $d_{\infty}$ (green). The radius increases from left to right.
• Figure 3: Each ball is divided into 4 regions corresponding to the 4 cases. As the radius $r$ changes, each of the 4 curves can change convexity.
• Figure 4: The metric ball's unconventional shape is due to $d_{\operatorname{rank}}$ uses not the dimension invariants but the rank invariants, dependent on the persistence of the center. The boundaries are always given by 2 hyperbolas and 2 lines of constant persistence. In the edge case $r = d_{\operatorname{rank}}(x,0)$, each of the hyperbolas is degenerate, intersecting at a point on the diagonal.
• Figure 5: When $r < \min(d_{\operatorname{dim}}(x, \Delta),d_{\operatorname{rank}}(x, \Delta))$, the balls are of this shape. We show any ball is nested inside a ball defined by the other metric.

Theorems & Definitions (59)

• Definition 1.1
• Theorem 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.6
• Corollary 2.7
• Definition 3.1
• Lemma 3.2
• proof