Beyond the Lower-Star Filtration: Alternative Filtration Methods for Persistent Homology
Year: 2026
Type: article
Abstract: In the lower-star filtration (LSF), simplices are assigned filtration values according to the maximum-vertex rule, with each simplex entering the filtration when its highest-valued vertex appears. A limitation of this construction is that higher-dimensional simplices may enter simultaneously with their boundary faces, causing some topological features to be filled immediately upon formation and therefore not contributing to the persistence diagram. To address this limitation, we propose three alternative filtration methods: the ε-lower-star filtration (ELSF), the Lp filtration (LpF), and the softmax filtration (SMF). These methods generalize LSF by assigning simplex filtration values using aggregation functions that depend on all vertex values rather than solely on their maximum. This allows simplices to enter the filtration more progressively, so that topological features such as loops and voids may persist across a wider range of filtration values rather than being filled immediately. Consequently, the resulting persistence diagrams may contain additional and more persistent topological features that are not observed under LSF, potentially providing greater discriminative power. Experimental results on simulated and real-world datasets demonstrate that the proposed filtration methods are computationally efficient and can provide greater sensitivity to structural differences in the data than LSF.
DOI: 10.3390/ijt3040025
Authors: Innocent Abaa, Umar Islambekov
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,207
OpenAlex ID: W7213446735