
Approximating persistent homology for a cloud of n points in a subquadratic time
The VietorisRips filtration for an npoint metric space is a sequence o...
read it

VietorisRips and Cech Complexes of Metric Gluings
We study VietorisRips and Cech complexes of metric wedge sums and metri...
read it

On universal covers for carpenter's rule folding
We present improved universal covers for carpenter's rule folding in the...
read it

Universal consistency of the kNN rule in metric spaces and Nagata dimension
The k nearest neighbour learning rule (under the uniform distance tie br...
read it

Plurality in Spatial Voting Games with constant β
Consider a set of voters V, represented by a multiset in a metric space ...
read it

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded HopDiameter Spanners
Spanners for metric spaces have been extensively studied, both in genera...
read it

A nonlinear aggregation type classifier
We introduce a nonlinear aggregation type classifier for functional data...
read it
ElderRuleStaircodes for Augmented Metric Spaces
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∈ X has a density function value f_X(x) associated to it. An augmented metric space (X, d_X, f_X) naturally gives rise to a 2parameter filtration 𝒦. However, the resulting 2parameter persistent homology H_∙(𝒦) could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elderrule for the zeroth homology of 1parameter filtration, we propose a barcodelike summary, called the elderrulestaircode, as a way to encode H_0(𝒦). Specifically, if n = X, the elderrulestaircode consists of n number of staircaselike 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 elderrulestaircode. 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 elderrulestaircode is given. Finally, we develop and implement an efficient algorithm to compute the elderrulestaircode in O(n^2log n) time, which can be improved to O(n^2α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
READ FULL TEXT
Comments
There are no comments yet.