2021
DOI: 10.1142/s1793525321500291 View full text |Buy / Rent full text
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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 filtra… Show more

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“…We foresee that the subject of homological skeletonization is not yet concluded. Other approaches to finding canonical generators of homology are possible (for example in 56,85 ), and we plan to investigate them further in subsequent works.…”
Section: Discussionmentioning
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“…We foresee that the subject of homological skeletonization is not yet concluded. Other approaches to finding canonical generators of homology are possible (for example in 56,85 ), and we plan to investigate them further in subsequent works.…”
Section: Discussionmentioning
“…We will focus primarily on sequential approaches to persistent homology computation. Other, non-sequential approaches include the chunk algorithm [3], spectral sequence procedures [46,22], Morse-theoretic batch reduction [32,33,58,6,29,34,48,59,21], distributed algorithms [4,53,44], GPU acceleration [63,38], streaming [41], and homotopy collapse [9,20,8]. There are closely related techniques in matrix factorization and zigzag persistence [50,11,10].…”
Section: Related Literaturementioning
“…Proof. By Equation (8), the matrix M ρκ is a generalized permutation matrix. Since M = R −1 DC one argues, by induction, that each non-pivot row of D (working from the bottom up) can be deleted without reducing the rank of D. A similar argument holds for columns, working left to right.…”
Section: Block Submatricesmentioning
“…Also, the efficiency of state-of-the-art software for flag complexes, such as Ripser [Bau19] and Eirene [HHSS19] is to a large extent due to implicit compression. Another recent line of research uses the concept of strong collapses to arrive at a small complex quickly, in general [BPP18], and for flag complexes [BP20].…”
Section: Related Workmentioning