Elder-Rule-Staircodes for Augmented Metric Spaces

Abstract

An augmented metric space is a metric space (X, d(x)) equipped with a function f(x) : X -> R This type of data arises commonly in practice, e.g., a point cloud X in R-D where each point x is an element of X has a density function value fx(x) associated to it. An augmented metric space (X, dx, fx) naturally gives rise to a 2-parameter filtration K. However, the resulting 2-parameter persistent homology H.(K) 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 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode H-0(K). Specifically, if n = vertical bar X vertical bar the elder-rule-staircode consists of n number of staircase-like blocks in the plane. We show that if H-0(K) is interval decomposable, then the barcode of H-0(K) is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of H-0(K) can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n(2) log n) time, which can be improved to O (n(2) alpha(n)) if X is from a fixed dimensional Euclidean space R-D, where alpha(n) is the inverse Ackermann function.