• Corpus ID: 141460541

# Coordinatizing Data With Lens Spaces and Persistent Cohomology

@article{Polanco2019CoordinatizingDW,
title={Coordinatizing Data With Lens Spaces and Persistent Cohomology},
author={Luis Polanco and Jose A. Perea},
journal={ArXiv},
year={2019},
volume={abs/1905.00350}
}
• Published 1 May 2019
• Mathematics
• ArXiv
We introduce here a framework to construct coordinates in \emph{finite} Lens spaces for data with nontrivial 1-dimensional $\mathbb{Z}_q$ persistent cohomology, $q\geq 3$. Said coordinates are defined on an open neighborhood of the data, yet constructed with only a small subset of landmarks. We also introduce a dimensionality reduction scheme in $S^{2n-1}/\mathbb{Z}_q$ (Lens-PCA: $\mathsf{LPCA}$), and demonstrate the efficacy of the pipeline $PH^1(\;\cdot\; ; \mathbb{Z}_q)$ class $\Rightarrow… 6 Citations ## Figures and Tables from this paper • Computer Science Foundations of Data Science • 2021 This paper proposes a method to adapt the circular coordinate framework to take into account the roughness of circular coordinates in change-point and high-dimensional applications and uses a generalized penalty function instead of an L_{2}$\end{document} penalty in the traditional circular coordinate algorithm.
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## References

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• Jose A. Perea
• Mathematics, Computer Science
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It is shown that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks, so that the coordinates are sparse.
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A framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations, and shows how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes.
It is shown that the language of principal bundles leads to coordinates defined on an open neighborhood of the data, but computed using only a smaller subset of landmarks, so that the coordinates are sparse.
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An approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set and efficiently computes a globally optimal solution, and is guaranteed to converge asymptotically to the true structure.
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A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ.
The definition and computation of homology in the standard setting of simplicial complexes and topological spaces are discussed, then it is shown how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object.
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A brief survey on the evolution of persistent homology is presented, starting from the subject's computational inception more than 20 years ago, to the more modern categorical and representation-theoretic point of view.
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