Figure - available from: Journal of Applied and Computational Topology
This content is subject to copyright. Terms and conditions apply.
![Given a 2-cocycle α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, we consider the corresponding map on the boundary of a generic 3-simplex σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}](publication/374901539/figure/fig3/AS:11431281225342733@1708660880945/Given-a-2-cocycle-adocumentclass12ptminimal-usepackageamsmath.png)
Given a 2-cocycle α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, we consider the corresponding map on the boundary of a generic 3-simplex σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}
Source publication
![Given a 2-cocycle α\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/374901539/figure/fig3/AS:11431281225342733@1708660880945/Given-a-2-cocycle-adocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set X and extract a cocycle α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepac...
Similar publications
The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorizatio...
The growth of Big Data has resulted in an overwhelming increase in the volume of data available, including the number of features. Feature selection, the process of selecting relevant features and discarding irrelevant ones, has been successfully used to reduce the dimensionality of datasets. However, with numerous feature selection approaches in t...
Citations
... In 2011 as well, the authors of [55] gave the equivalence of persistent homology and persistent cohomology where duality was greatly executed. [65] in 2022 constructed spherical coordiantes from persistent cohomology. Authors of [57] presented an approach to topological motion planning which is fully data driven in nature and which relies solely on the knowledge of samples in the free configuration space. ...
... Lemma 3.6.[55,65,68] The persistence diagram (barcode) is the multi set of ordered pairs [p, q] in the decomposition, or alternatively the multi set of half open intervals [a p , a q+1 ). ...
Persistent homology is an important tool in non-linear data reduction. Its sister theory, persistent cohomology, has attracted less attention in the past years eventhough it has many advantages. Several literatures dealing with theory and computations of persistent homology and cohomology were surveyed. Reasons why cohomology has been neglected over time are identified and, few possible solutions to the identified problems are made available. Furthermore, using Ripserer, the computation of persistent homology and cohomology using 2-sphere both manually and computationally are carried out. In both cases, same result was obtained, particularly in the computation of their barcodes which visibly revealed the point where the two coincides. Conclusively, it is observed that persistent cohomology is not only faster in computation than persistent homology, but also uses less memory in a little time.
Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally $$\kappa $$ κ -dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of $$\kappa $$ κ -dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.
