• Corpus ID: 8863126

Interactive Visualization of 2-D Persistence Modules

@article{Lesnick2015InteractiveVO,
  title={Interactive Visualization of 2-D Persistence Modules},
  author={Michael Lesnick and Matthew L. Wright},
  journal={ArXiv},
  year={2015},
  volume={abs/1512.00180}
}
The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes of 1-D affine slices of a 2-D persistence module $M$. It also computes and visualizes the dimension of… 
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117 Citations
Highly Influential Citations
11
Background Citations
41
Methods Citations
45
Results Citations
2

117 Citations

Citation Type

Citation Type

A flat persistence diagram for improved visualization of persistent homology
TLDR
A modification to the conventional persistence diagram is proposed - the flat persistence diagram - that more efficiently displays information relevant to persistent homology and simultaneously corrects for visual bias present in the former.
Efficient Approximation of Multiparameter Persistence Modules
TLDR
This article presents the first approximation scheme, which is based on bered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence, for computing and decomposing general multi-parameters persistence modules.
Morse-based Fibering of the Persistence Rank Invariant
TLDR
It is shown how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field.
Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules
  • T. Dey, Cheng Xin
  • Computer Science
    Journal of Applied and Computational Topology
  • 2022
TLDR
A generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in $O(n^{2\omega})$ time where $\omega<2.373$ is the exponent for matrix multiplication.
Pseudo-Multidimensional Persistence and Its Applications
TLDR
This new approach is able to differentiate between topologically equivalent geometric objects and offers insight into the study of the Kuramoto–Sivashinsky partial differential equation and Lissajous knots.
A User’s Guide to Topological Data Analysis
TLDR
This article introduces two of the most commonly used topological signatures, the persistence diagram and the mapper graph, which represent loops and holes in the space by considering connectivity of the data points for a continuum of values rather than a single fixed value.
Persistence and Sheaves : from Theory to Applications
TLDR
This manuscript establishes explicit links between the theory of derived sheaves equipped with the convolu-tion distance (after Kashiwara-Schapira) and persistence theory, and proves that the observable category of persistence mo-dules is equivalent to the well-known category of -sheaves.
Efficient Approximation of the Matching Distance for 2-parameter persistence
TLDR
This work designs efficient algorithms for approximating the matching distance of two bi-filtered complexes to any desired precision $\epsilon>0$ based on a quad-tree refinement strategy introduced by Biasotti et al., but their approach is recast entirely in geometric terms.
Asymptotic Improvements on the Exact Matching Distance for 2-parameter Persistence
TLDR
This work improves significantly on the previous fastest algorithm to compute dM by describing an algorithm with expected running time O(n log n), where T (n) is an upper bound for the complexity of deciding if dM ≤ λ.
A T ] 3 0 N ov 2 02 0 Morse-based Fibering of the Persistence Rank Invariant Asilata
Although there is no doubt that multi-parameter persistent homology is a useful tool to analyse multi-variate data, efficient ways to compute these modules are still lacking in the available…
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