On the Hausdorff stability of barcodes over posets

TL;DR

The paper addresses stability of barcodes for multiparameter persistence by establishing a Lipschitz bound between Hausdorff and interleaving distances for interval-decomposable poset modules under structural assumptions. The authors introduce an $\mathbb{R}$-flow and focus on intersection-closed families of intervals to derive the bound $d_{\mathrm{H}}(M,N) \le 2 \cdot d_{\mathrm{I}}(M,N)$, together with geometric criteria for interleavings of interval modules. They also demonstrate that bottleneck and Hausdorff distances can be unstable in general, while showing that the stability result holds in the intersection-closed setting, with tightness shown by a convex-interval construction. These results provide a robust, geometry-driven path toward stability for generalized barcode invariants and connect to conjectures about stability under additional interval-structure assumptions, with potential applications to convex-interval and upperset decompositions in multiparameter persistence.

Abstract

The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their barcodes. Significant effort has been devoted to extending this result to modules defined over more general posets. As these modules do not generally admit nice decompositions, one must restrict attention to the class of interval-decomposable modules in order to define an appropriate notion of bottleneck distance. Even with this assumption, it is known that bottleneck distance may not be equivalent to interleaving distance, but that it is Lipschitz stable under certain, fairly restrictive, assumptions. In this paper, we consider the more basic question of stability of the Hausdorff distance with respect to interleaving distance for interval-decomposable modules. Our main theorem is a Lipschitz stability result, which holds in a fairly general setting of interval-decomposable modules over arbitrary posets, where intervals are assumed to be taken from any family satisfying certain closure conditions. Along the way, we develop some new tools and results for interval-decomposable modules over arbitrary posets, in the form of geometrically-flavored characterizations of the existence of morphisms and interleavings between interval modules.

Figures (6)

• Figure 1: Examples in $\mathbb{R}^2$: (a) intervals, shown in blue; (b) non-intervals, shown in red.
• Figure 2: An example of $g\circ f=0$, where $0\neq f: C(I)\to C(J)$ and $0 \ne g: C(J)\to C(K)$. Here, each $f_p$ is the identity map on $I\cap J$ and zero elsewhere, and each $g_p$ is the identity map on $J\cap K$ and zero elsewhere, but their composition $g_p\circ f_p$ is always zero.
• Figure 3: (a) Geometrically convex intervals in $\mathbb{R}^2$. (b) An interval in $\mathbb{R}^2$ that is not geometrically convex. (c) Geometrically convex subsets of $\mathbb{R}^2$ that are not intervals.
• Figure 4: An open and geometrically convex $I$, two points $p,q\in I$, subdivision points $p_i$ on the line $\overline{pq}$, and a pruned chain of open $\ell_\infty$-balls $B_\infty(p_i,\mathbf{\varepsilon})\subset I$ with overlapping intersections along the segment.
• Figure 5: Intervals illustrating Hausdorff instability; see text for details. (a) Original intervals, generating persistence modules $C(I) \oplus C(J)$ and $C(K)$. (b) Interval $K$ shifted by $\pm \varepsilon$. The unshifted $K$ is shown with reduced opacity and the intersection $K(\varepsilon) \cap K(-\varepsilon)$ is shown with increased opacity. (c) Intervals $I$ and $J$ each shifted by $\pm \mathbf{\varepsilon}$. The unshifted $I$ and $J$ are shown with reduced opacity and the intersection of the shifted versions is shown with increased opacity.

Theorems & Definitions (53)

• Theorem
• Definition 2.1: Interval in a Poset
• Remark 2.2
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 2.6: $\mathcal{P}$-Module
• Definition 2.7: Characteristic Module