HomologyBasis: Fast Computation of Persistent Homology

Abstract

Simplicial complexes are used in topological data analysis (TDA) to extract topological features of the data. The HomologyBasis algorithm is proposed as an efficient method for the computation of the topological features of a finite filtered simplicial complex. We build up the implementation and intuition of this algorithm from its theoretical foundation ensuring this schema produces the desired simplicial homlogy groups as claimed. HomlogyBasis implemented and compared with the GUHDI algorithm to determine the HomologyBasis' efficiency at computing persistence pairs for finite filtered simplicial complexes. We find the HomologyBasis algorithm performs much better than GUHDI on large low-dimensional simplicial complexes but needs further refinement before it can more efficiently work with high-dimensional complexes.