Efficient computation of multi-parameter persistent homology

Multi-parameter persistent (multi-persistent) homology is an extension of persistent homology, which is a multiscale approach to homological shape analysis, to the case where several scalar functions are associated with the data (multifield data). The objective of our research on multi-persistent homology is to devise algorithms for efficiently computing it on real-world data sets. This is a challenging problem, since very few results exist in the literature on multi-persistent homology, both from a computational and a theoretical point of view. We have proposed a pre-processing approach which computes a Morse-like discrete vector field compatible with the multifield. Such algorithm is well suited to be used with both simplicial complexes and regular grids, it scales well when the size of the input complex increases and is well suited for a parallel implementation. Moreover, we have shown that the use of such pre-processing provides an improvement of at least one order of magnitude in the computation of multi-persistent homology.

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