Open AccessJournal Article
Statistical topological data analysis using persistence landscapes
TLDR
In this article, the persistence landscape is defined as a topological summary for data that is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries.Abstract:
We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.read more
Citations
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Journal ArticleDOI
A roadmap for the computation of persistent homology
Nina Otter,Nina Otter,Mason A. Porter,Mason A. Porter,Ulrike Tillmann,Ulrike Tillmann,Peter Grindrod,Heather A. Harrington +7 more
TL;DR: A friendly introduction to PH is given, the pipeline for the computation of PH is navigated with an eye towards applications, and a range of synthetic and real-world data sets are used to evaluate currently available open-source implementations for the computations of PH.
Posted Content
An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists
Frédéric Chazal,Bertrand Michel +1 more
TL;DR: This paper is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of TDA for non experts.
Proceedings ArticleDOI
A stable multi-scale kernel for topological machine learning
TL;DR: In this paper, a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data, is proposed for 3D shape classification/retrieval and texture recognition.
Journal Article
Persistence images: a stable vector representation of persistent homology
Henry Adams,Tegan Emerson,Michael Kirby,Rachel Neville,Chris Peterson,Patrick D. Shipman,Sofya Chepushtanova,Eric Hanson,Francis C. Motta,Lori Ziegelmeier +9 more
TL;DR: In this article, a persistence diagram (PD) is converted to a finite-dimensional vector representation which is called a persistence image (PI) and proved the stability of this transformation with respect to small perturbations in the inputs.
Journal ArticleDOI
Visualizing High-Dimensional Data: Advances in the Past Decade
TL;DR: This work provides guidance for data practitioners to navigate through a modular view of the recent advances in high-dimensional data visualization, inspiring the creation of new visualizations along the enriched visualization pipeline, and identifying future opportunities for visualization research.
References
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Journal ArticleDOI
Topology and data
TL;DR: This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources.
Journal ArticleDOI
Topological Persistence and Simplification
Edelsbrunner,Letscher,Zomorodian +2 more
TL;DR: Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Journal ArticleDOI
Computing Persistent Homology
TL;DR: In this article, it was shown that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology over a polynomial ring of a particular graded module.
Journal ArticleDOI
Barcodes: The persistent topology of data
TL;DR: In this paper, a survey of the use of algebraic topology for feature detection and shape recognition in high-dimensional data is presented. But the main focus of the survey is on the application of topology to the classification of natural images.