Strong Collapse for Persistence

Boissonnat, Jean‐Daniel; Pritam, Siddharth; Pareek, Divyansh

doi:10.4230/lipics.esa.2018.67

Cited by 6 publications

(3 citation statements)

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“…Other approaches follow a more flexible strategy and omit the need to define a filtration, instead focusing on arbitrary simplicial maps between consecutive complexes (Dey et al, 2014). This in turn gives rise to algorithms successively collapsing simplices of the sequence of simplicial complexes (Dey et al, 2019;Botnan and Spreemann, 2015;Boissonnat et al, 2018;Attali et al, 2012).…”

Section: Related Work and Contributionmentioning

confidence: 99%

“…This implies that we can use existing software to compute their persistent homology and profit from some of the accelerations described here, as long as they are working on simplicial filtrations and do not require metric spaces as input. Techniques that provide such accelerations and thus might be used in combination with our technique include (Dey and Slechta, 2020;Boissonnat et al, 2018;D lotko and Wagner, 2014;Attali et al, 2012;Dey et al, 2014;Chen and Kerber, 2011).…”

Section: Related Work and Contributionmentioning

confidence: 99%

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Adaptive Approximation of Persistent Homology

Herick,

Joachim,

Vahrenhold

2023

Preprint

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We study an important preprocessing step for the efficient calculation of persistent homology: coarsening a set of points while controlling the quality of the induced persistence diagram. This coarsening step is instrumental in reducing the overall runtime of state-of-the-art algorithms such as Ripser, GUDHI, or PHAT. For this, we adaptively sparsify the set of points and carefully define a dissimilarity function between the remaining points. This function takes into account local properties given by a separate function defined on the point set as well as the relation of the removed points to the subsample. It is then used to build simplicial filtrations and calculate their persistent homology. We assess the quality of our approach both theoretically by proving topological approximation guarantees and empirically by using the bottleneck distance. Since our subsample is calculated adaptively, we also prove adaptive properties of our result. Our results show that we can significantly reduce the size of the point sample with only very moderate bottleneck distance to the ground truth; in particular, we still are able to capture the salient topological features of the input through our approximated persistence diagrams.

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Section: Related Work and Contributionmentioning

confidence: 99%

Section: Related Work and Contributionmentioning

confidence: 99%

Adaptive Approximation of Persistent Homology

Herick,

Joachim,

Vahrenhold

2023

Preprint

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“…In this section, we first recap the theory of simplicial complexes and collapses. For this, we use the manuscripts by Forman [14] and Boissonnat et al [6]. For a detailed introduction to these topics, we refer to the publications by Whitehead [28], Milnor [23], and Cohens [8].…”

Section: Introductionmentioning

confidence: 99%

Reduced Connectivity for Local Bilinear Jacobi Sets

Klötzl¹,

Krake²,

Zhou³

et al. 2022

Preprint

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Figure 1: Comparison of the original connectivity construction for a local bilinear Jacobi set (left) and our novel reduced connectivity (right). The color coding (blue-to-red) shows the gradient alignment field of two analytic functions, and the extracted/computed Jacobi set is displayed by solid black lines. The center images show zoomed-in details of two regions in the visualizations (marked by white outlines). It can be observed that the reduced connectivity construction results in fewer edges and, therefore, in a clearer visual representation.

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