DTM-Based Filtrations

Anai, Hirokazu; Chazal, Frédéric; Glisse, Marc; Ike, Yuichi; Inakoshi, Hiroya; Tinarrage, Raphaël; Umeda, Yuhei

doi:10.4230/lipics.socg.2019.58

Cited by 4 publications

(2 citation statements)

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“…To overcome this difficulty, approximations of the k-distance have been proposed recently that led to certified (although rather poor) approximations of PH (Guibas et al [20]; Buchet et al [7]). Recently, improved approximations have been reported using this approach [3].…”

Section: Introductionmentioning

confidence: 99%

Dimensionality Reduction for Persistent Homology with Gaussian Kernels

Boissonnat,

Dutta

2023

Preprint

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Computing persistent homology using Gaussian kernels is useful in the domains of topological data analysis and machine learning as shown by Phillips, Wang and Zheng [SoCG 2015]. However, contrary to the case of computing persistent homology using the Euclidean distance or even the k-distance, it is not known how to compute the persistent homology of high dimensional data using Gaussian kernels. In this paper, we consider a power distance version of the Gaussian kernel distance (GKPD) given by Phillips, Wang and Zheng, and show that the persistent homology of the Čech filtration of P computed using the GKPD is approximately preserved. For datasets in R D , under a relative error bound of ε ∈ (0, 1], we obtain a dimensionality of (i) O(ε −2 log 2 n) for n-point datasets and (ii) O(Dε −2 log(Dr/ε)) for datasets having diameter r (up to a scaling factor).We use two main ingredients. The first one is a new decomposition of the squared radii of Čech simplices using the kernel power distance, in terms of the pairwise GKPDs between the vertices, which we state and prove. The second one is the Random Fourier Features (RFF) map of Rahimi and Recht [NeurIPS 2007], as used by Chen and Phillips [ALT 2017]. * Supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.† Supported by the Polish NCN-SONATA Grant no. 2019/35/D/ST6/04525 (Probabilistic tools for highdimensional geometric inference, topological data analysis and large-scale networks).

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Section: Introductionmentioning

confidence: 99%

Dimensionality Reduction for Persistent Homology with Gaussian Kernels

Boissonnat,

Dutta

2023

Preprint

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“…Within the framework of 1-parameter persistent homology, there have been many proposals for alternative constructions which address these issues. These approaches include the removal of low density outliers [9], filtering by a density function [7,15,16], distance to measure constructions [1,8,14,29], kernel density functions [42], and subsampling [4]. A detailed overview of these approaches can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

Computing the Multicover Bifiltration

Corbet¹,

Kerber²,

Lesnick³

et al. 2021

Preprint

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Given a finite set A ⊂ R d , let Cov r,k denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy [44]. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang [26], and can be efficiently computed using a variant of an algorithm given by these authors in [27]. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.

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Detection of small holes by the scale-invariant robust density-aware distance (RDAD) filtration

Siu,

Samorodnitsky,

et al. 2024

J Appl. and Comput. Topology

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DTM-Based Filtrations

Cited by 4 publications

References 0 publications

Dimensionality Reduction for Persistent Homology with Gaussian Kernels

Dimensionality Reduction for Persistent Homology with Gaussian Kernels

Computing the Multicover Bifiltration

Detection of small holes by the scale-invariant robust density-aware distance (RDAD) filtration

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