Persistence Terrace for Topological Inference of Point Cloud Data

Moon, Chul; Giansiracusa, Noah; Lazar, Nicole A.

doi:10.1080/10618600.2017.1422432

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…Particularly, Beksi and Papnikolopoulos have shown the utility of clustering 3D point clouds [2,4] and designing signatures for points [3]. Moon et al design the persistence terrace to help provide a summary plot to guide the inference process [20]. These methods all focus on low-dimensional point sets (typically two-and three-dimensional data).…”

Section: Related Workmentioning

confidence: 99%

Implementing Persistence-Based Clustering of Point Clouds in the Topology ToolKit

Cotsakis¹,

Shaw²,

Tierny³

et al. 2021

Mathematics and Visualization

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We show how the scalar field topology features of the Topology ToolKit (TTK) can be leveraged in a pipeline for persistence-based clustering of point clouds.While TTK provides numerous features for computing topological structures of scalar fields on unstructured meshes, prior to this work, it allowed for only basic point cloud input. In this work, we implemented two new modules in TTK: one for sampling scalar fields based on either distance or density of the point cloud and a second for computing persistence-based clusters. Both modules provide heuristics for automatically specifying key thresholds so as to simplify user interaction. This document outlines the implementation details of the two modules and provides experimental results that demonstrate their modularity and utility.

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Section: Related Workmentioning

confidence: 99%

Implementing Persistence-Based Clustering of Point Clouds in the Topology ToolKit

Cotsakis¹,

Shaw²,

Tierny³

et al. 2021

Mathematics and Visualization

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“…For example one could detect significant i-dimensional holes in the distribution by considering the H i module in a similar setup. For related work see Persistence Terraces [MGL17].…”

Section: Modal Estimationmentioning

confidence: 99%

Multiparameter Persistence Landscapes

Vipond¹

2018

Preprint

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An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this paper we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance, and that the collection of multiparameter landscapes faithfully represents the rank invariant. Finally we provide example calculations and statistical tests to demonstrate a range of potential applications and how one can interpret the landscapes associated to a multiparameter module.

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“…The information that persistent homology yields on the change of topological features of a given point cloud can be presented in various different ways such as barcodes, 9 persistence diagrams, 10 landscapes, 11 images, 12 terraces, 13 entropy, 14 and curves 15 . However, the common difficulty one encounters in TDA calculations is that persistent homology, in all of the representations we enumerated above, is computationally expensive.…”

Section: Introductionmentioning

confidence: 99%

Classification of stochastic processes with topological data analysis

Güzel

Kaygun

2023

Concurrency and Computation

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SummaryIn this study, we demonstrate that engineered topological features can distinguish time series sampled from different stochastic processes with different noise characteristics, in both balanced and unbalanced sampling schemes. We compare our classification results against the results of the same classification on features coming from descriptive statistics and the wavelet transform. We conclude that machine learning models built on engineered topological features alone perform consistently better than those built on the standard statistical and wavelet features for time series classification tasks. We also apply dimension reduction techniques to our engineered features and compare the result of our classification models before and after dimensionality reduction. Finally, we also show that in our calculations of the engineered topological features, employing parallel computing methods does yield significant improvements in run time and memory footprint.

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Persistence Terrace for Topological Inference of Point Cloud Data

Cited by 7 publications

References 24 publications

Implementing Persistence-Based Clustering of Point Clouds in the Topology ToolKit

Implementing Persistence-Based Clustering of Point Clouds in the Topology ToolKit

Multiparameter Persistence Landscapes

Classification of stochastic processes with topological data analysis

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