Elder-Rule-Staircodes for Augmented Metric Spaces
An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables.
In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
Persistent homology
Multiparameter persistence
Barcodes
Elder rule
Hierarchical clustering
Graded Betti numbers
Mathematics of computing~Topology
Theory of computation~Computational geometry
26:1-26:17
Regular Paper
This work is supported by NSF grants DMS-1723003, CCF-1740761, DMS-1547357, and IIS-1815697.
A full version of this paper is available at https://arxiv.org/abs/2003.04523.
The authors thank to the anonymous reviewers who made a number of helpful comments to improve the paper. Also, CC and WK thank Cheng Xin for helpful discussions.
Chen
Cai
Chen Cai
Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Woojin
Kim
Woojin Kim
Department of Mathematics, The Ohio State University, Columbus, OH, USA
Facundo
Mémoli
Facundo Mémoli
Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Yusu
Wang
Yusu Wang
Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
10.4230/LIPIcs.SoCG.2020.26
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Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang
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