Stability of multidimensional persistent homology with respect to domain perturbations

Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets. Moreover we prove that our construction maintains information about the original set. Other well known methods to compare sets are considered, such as the symmetric difference distance between classical sets and the sup-distance between fuzzy sets. Also in these cases we present results stating that the multidimensional matching distance between rank invariants of persistent homology groups is upperly bounded by these distances. An experiment showing the potential of our approach concludes the paper.

💡 Research Summary

The paper addresses a fundamental limitation of current persistent homology theory: while stability results are well‑established for perturbations of the filtering function, they do not directly cover situations where the underlying domain itself is altered. Such domain perturbations are ubiquitous in computer vision and graphics, where data acquisition is often incomplete, noisy, or imprecise. The authors propose a systematic framework that extends multidimensional persistent homology stability to these domain changes by encoding sets through distance functions and by measuring set differences with several well‑known metrics.

Core Construction.
Given a compact subset (A\subset\mathbb{R}^d), the authors define the distance function (\rho_A(x)=\operatorname{dist}(x,A)). This scalar field is 1‑Lipschitz and captures the geometry of (A) in a way that is robust to small changes in the set. To obtain a multidimensional filtration, (\rho_A) is combined with any auxiliary real‑valued functions ((g_1,\dots,g_{k})) that may already be used in the application. The resulting vector‑valued map ((\rho_A,g_1,\dots,g_k):\mathbb{R}^d\to\mathbb{R}^{k+1}) induces a nested family of sublevel sets, and the associated homology groups give rise to the rank invariant (R_A).

Stability Theorems.
The main theoretical contribution is a set of upper‑bound inequalities linking the multidimensional matching distance (D_{\text{match}}(R_A,R_B)) between rank invariants to classical set distances:

1. Hausdorff distance. For any two compact sets (A,B), \