Beyond the Lower-Star Filtration: Alternative Filtration Methods for Persistent Homology

Innocent Abaa, Umar Islambekov

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 some higher-dimensional simplices may enter simultaneously with their boundary faces, causing certain topological features to be filled immediately upon formation and thus not appear in the resulting persistence diagram. To address this limitation, we propose three alternative filtration methods: the \(\varepsilon\)-lower-star filtration (ELSF), the \(L_p\) filtration (LpF), and the softmax filtration (SMF). {ELSF introduces a dimension-dependent delay to the maximum-vertex rule,} allowing higher-dimensional simplices to enter the filtration after their boundary faces, whereas {LpF and SMF use aggregation functions that depend on all vertex values rather} than solely on their maximum. By modifying how filtration values are assigned, these methods allow simplices to enter the filtration more progressively, so that topological features such as loops and voids may persist over longer filtration intervals rather than being filled immediately.…