Geometric Localization of Homology Cycles
This work addresses a fundamental computational geometry problem for researchers in topological data analysis, offering practical algorithms for homology cycle localization, though it is incremental in improving efficiency and stability.
The authors tackled the NP-hard homology localization problem by introducing a geometric optimization of cycles that is computable in polynomial time and stable in an approximate sense, achieving reasonable runtimes and high-quality cycles in experiments on moderate-sized datasets.
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.