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Spherical coordinates from persistent cohomology

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Nikolas C. Schonsheck
Nikolas C. Schonsheck
Nikolas C. Schonsheck
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Stefan Schonsheck at University of California, Davis
Stefan Schonsheck
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Abstract and Figures

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} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} from any significant feature. From this cocycle, we define an associated map α:VR(X)→S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha : VR(X) \rightarrow S^2$$\end{document} and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of 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}, preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach.
Energy Minimization
Energy Minimization
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Nonlinear dimensionality reduction of a noisy sphere embedded in R50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{50}$$\end{document} using several popular techniques and our methodology
Nonlinear dimensionality reduction of a noisy sphere embedded in R50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{50}$$\end{document} using several popular techniques and our methodology
… 
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}
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}
… 
Left: Our choice of canonical directions for positive and negative orientations for a triangle with nonzero winding (w=±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w = \pm 1)$$\end{document}, vertices at [1, 0, 0] and barycenter [-1,0,0]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,0,0]$$\end{document}. The outward normal is chosen to be positive and the inward negative. Right: Equilateral Triangle mapped to circle and regular, triangular pyramid mapped to the unit sphere using harmonic energy minimizing maps
Left: Our choice of canonical directions for positive and negative orientations for a triangle with nonzero winding (w=±1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w = \pm 1)$$\end{document}, vertices at [1, 0, 0] and barycenter [-1,0,0]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,0,0]$$\end{document}. The outward normal is chosen to be positive and the inward negative. Right: Equilateral Triangle mapped to circle and regular, triangular pyramid mapped to the unit sphere using harmonic energy minimizing maps
… 
First Row: Evenly sampled circle, Second Row: Noisy, sparsely sampled circle, Third Row: Trefoil knot in 3D (visualized in 2D) with spring energy, Fourth row: Noisy Ellipse in 50D (visualized in 2D without noise) with harmonic energy, Last row: Same as above but with spring energy
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First Row: Evenly sampled circle, Second Row: Noisy, sparsely sampled circle, Third Row: Trefoil knot in 3D (visualized in 2D) with spring energy, Fourth row: Noisy Ellipse in 50D (visualized in 2D without noise) with harmonic energy, Last row: Same as above but with spring energy
… 
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Journal of Applied and Computational Topology (2024) 8:149–173
https://doi.org/10.1007/s41468-023-00141-w
Spherical coordinates from persistent cohomology
Nikolas C. Schonsheck1,2 ·Stefan C. Schonsheck2
Received: 6 October 2022 / Revised: 14 August 2023 / Accepted: 31 August 2023 /
Published online: 21 October 2023
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023
Abstract
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 Xand extract a cocycle αfrom any significant feature. From this cocycle,
we define an associated map α:VR(X)→S2and use this map as an infeasible
initialization for a variational model, which we show has a unique solution (up to
rigid motion). We then employ an alternating gradient descent/Möbius transformation
update method to solve the problem and generate a more suitable, i.e., smoother,
representative of the homotopy class of α, preserving the relevant topological feature.
Finally, we conduct numerical experiments on both synthetic and real-world data sets
to show the efficacy of our proposed approach.
Keywords Persistent cohomology ·Nonlinear dimensionality reduction ·Variational
optimization
1 Introduction
Representing high dimensional data in a lower dimension through nonlinear dimen-
sionality reduction (NLDR) algorithms is a key step in understanding complex data.
Different NLDR methods attempt to preserve different key properties of the data,
Nikolas C. Schonsheck and Stefan C. Schonsheck have contributed equally to this work.
BNikolas C. Schonsheck
nischon@udel.edu
Stefan C. Schonsheck
scschonsheck@ucdavis.edu
1Department of Mathematical Sciences, University of Delaware, 15 Orchard Road, Newark
19716, Delaware, USA
2TETRAPODS Institute of Data Science, University of California Davis, 1 Shields Avenue, Davis
95616, California, USA
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... 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 ). ...
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