Geometric Localization of Homology Cycles

TL;DR

This paper introduces a polynomial-time, stable geometric optimization method for homology cycles that approximates optimal solutions and performs well on practical datasets, addressing computational challenges in homology localization.

Contribution

It proposes a new stable geometric optimization framework for homology cycles that is computationally feasible and adaptable to various homology problems.

Findings

Approximation algorithms perform well on moderate datasets.

Computed cycles are of high quality in experiments.

The method offers a practical alternative to NP-hard exact solutions.

Abstract

Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.