License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.56
URN: urn:nbn:de:0030-drops-179068
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17906/
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Scoccola, Luis ; Perea, Jose A.

FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles

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LIPIcs-SoCG-2023-56.pdf (3 MB)

Abstract

Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view and, as an application, we describe a dimensionality reduction algorithm based on topological inference for vector bundles. The algorithm takes as input a dataset together with an initial representation in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations obtained through local linear dimensionality reduction. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.

BibTeX - Entry

@InProceedings{scoccola_et_al:LIPIcs.SoCG.2023.56,
  author =	{Scoccola, Luis and Perea, Jose A.},
  title =	{{FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{56:1--56:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17906},
  URN =		{urn:nbn:de:0030-drops-179068},
  doi =		{10.4230/LIPIcs.SoCG.2023.56},
  annote =	{Keywords: topological inference, dimensionality reduction, vector bundle, cocycle}
}

Keywords: topological inference, dimensionality reduction, vector bundle, cocycle
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023
Supplementary Material: Proof-of-concept implementation [Luis Scoccola and Jose A. Perea, 2022]

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