Strong collapse and persistent homology

Abstract

In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

Classification of stochastic processes with topological data analysis

“…However, the fact that computations have exponential complexity in both space and time restricts their potential uses. 34,35 Even after the recent computational improvements in various stages in calculating persistent homology of point clouds, [36][37][38] there is still an urgent need for developing methods to reduce the memory…”

“…However, the fact that computations have exponential complexity in both space and time restricts their potential uses 34,35 . Even after the recent computational improvements in various stages in calculating persistent homology of point clouds, 36‐38 there is still an urgent need for developing methods to reduce the memory footprint and run‐time complexity. One such avenue is parallelization where an effective use of parallel HPC may result in substantial reductions in space and time complexity per computation core 16,35,39 …”

Predicting Affinity Through Homology (PATH): Interpretable Binding Affinity Prediction with Persistent Homology

“…In particular, the k -skeleton of a Rips complex (a subcomplex with simplex dimension up to k ) has 𝒪 ( n k +1 ) simplices, where n is the number of points in the input point cloud [18]. The time complexity to compute persistent homology is then , where ω ≈ 2.4 is the matrix multiplication exponent [10, 50, 72]. For a protein-ligand complex, let m be the number of ligand atoms and n be the number of protein atoms.…”