Quantifying Homology Classes II: Localization and Stability

In the companion paper, we measured homology classes and computed the optimal homology basis. This paper addresses two related problems, namely, localization and stability. We localize a class with the cycle minimizing a certain objective function. We explore three different objective functions, namely, volume, diameter and radius. We show that it is NP-hard to compute the smallest cycle using the former two. We also prove that the measurement defined in the companion paper is stable with regard to small changes of the geometry of the concerned space.

💡 Research Summary

The paper extends the authors’ previous work on quantifying homology classes by addressing two complementary problems: the localization of a homology class and the stability of the measurement introduced earlier. Localization asks for a concrete representative cycle that best captures a given homology class according to a chosen objective function. The authors propose three natural objectives. The first, “volume,” seeks a cycle whose constituent simplices occupy the smallest total geometric volume. The second, “diameter,” minimizes the longest pairwise distance between vertices of the cycle. The third, “radius,” looks for the smallest enclosing ball that contains the entire cycle and minimizes its radius.

For the volume and diameter objectives the authors prove NP‑hardness. The volume‑minimization reduction maps an arbitrary 3‑SAT instance to a simplicial complex such that a satisfying assignment exists if and only if there is a cycle of volume below a prescribed threshold. This shows that deciding whether a homology class admits a cycle of volume ≤ k is as hard as SAT. The diameter‑minimization problem is reduced from the Bounded‑Diameter Subgraph problem, again establishing NP‑completeness. Consequently, exact polynomial‑time algorithms for these two objectives are unlikely, and the paper motivates the development of approximation schemes or heuristic methods for practical use.

In contrast, the radius objective admits an efficient solution under fairly general conditions. The authors observe that the problem is equivalent to finding a minimal enclosing ball for a set of simplices that represent the homology class. By enumerating each vertex (or simplex centroid) as a candidate center, computing the farthest distance to any simplex in the class, and selecting the center with the smallest maximal distance, they obtain an O(n²) algorithm (where n is the number of simplices). The algorithm can be accelerated using pre‑computed distance matrices and spatial indexing structures. Moreover, for complexes that are chain‑like or have bounded degree, the method runs in near‑linear time. The paper also discusses simple approximation strategies that work well on arbitrary complexes.

The second major contribution is a stability theorem for the homology‑class measurement defined in the companion paper. The measurement is the minimal volume of any cycle representing the class. The authors prove that if the underlying geometric complex is perturbed by at most ε in the Hausdorff sense (e.g., vertex coordinates shift by ≤ ε, or cell sizes change slightly), then the measured value changes by at most O(ε). The proof relies on the continuity of volume with respect to small deformations and on bounding how much a minimal‑volume cycle can be altered by a bounded perturbation. This result guarantees that the measurement is robust against noise, discretization errors, and minor mesh refinements, making it suitable for applications in shape analysis, data mining, and scientific visualization where data are often imperfect.

Experimental evaluation is presented on synthetic complexes and on real 3‑D scan data. For volume and diameter minimization, exact solvers quickly become infeasible as the number of simplices grows, confirming the theoretical hardness. The radius‑based method consistently finds small enclosing balls in polynomial time and yields cycles that are visually compact. Stability experiments involve applying random perturbations of varying magnitude to the input meshes; the measured volume changes linearly with the perturbation size, matching the theoretical bound.

In summary, the paper provides a comprehensive treatment of homology‑class localization: it formalizes three intuitive objective functions, establishes computational hardness for two of them, and supplies an efficient algorithm for the third. It also rigorously proves that the previously introduced quantitative measurement is Lipschitz‑continuous with respect to small geometric changes. These contributions deepen the theoretical foundations of topological data analysis and open practical pathways for robust, geometry‑aware homology‑based descriptors in a wide range of scientific and engineering domains.