3 Search Results for "Scoccola, Luis"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.56
FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles

Authors: Luis Scoccola and Jose A. Perea

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

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.
Cite as

Luis Scoccola and Jose A. Perea. FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 56:1-56:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@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/entities/document/10.4230/LIPIcs.SoCG.2023.56},
  URN =		{urn:nbn:de:0030-drops-179068},
  doi =		{10.4230/LIPIcs.SoCG.2023.56},
  annote =	{Keywords: topological inference, dimensionality reduction, vector bundle, cocycle}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.57
Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction

Authors: Luis Scoccola, Hitesh Gakhar, Johnathan Bush, Nikolas Schonsheck, Tatum Rask, Ling Zhou, and Jose A. Perea

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a 1-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra-Lenstra-Lovász algorithm from computational number theory.
Cite as

Luis Scoccola, Hitesh Gakhar, Johnathan Bush, Nikolas Schonsheck, Tatum Rask, Ling Zhou, and Jose A. Perea. Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 57:1-57:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{scoccola_et_al:LIPIcs.SoCG.2023.57,
  author =	{Scoccola, Luis and Gakhar, Hitesh and Bush, Johnathan and Schonsheck, Nikolas and Rask, Tatum and Zhou, Ling and Perea, Jose A.},
  title =	{{Toroidal Coordinates: Decorrelating Circular Coordinates with Lattice Reduction}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{57:1--57:20},
  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/entities/document/10.4230/LIPIcs.SoCG.2023.57},
  URN =		{urn:nbn:de:0030-drops-179073},
  doi =		{10.4230/LIPIcs.SoCG.2023.57},
  annote =	{Keywords: dimensionality reduction, lattice reduction, Dirichlet energy, harmonic, cocycle}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.58
The Degree-Rips Complexes of an Annulus with Outliers

Authors: Alexander Rolle

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.
Cite as

Alexander Rolle. The Degree-Rips Complexes of an Annulus with Outliers. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 58:1-58:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{rolle:LIPIcs.SoCG.2022.58,
  author =	{Rolle, Alexander},
  title =	{{The Degree-Rips Complexes of an Annulus with Outliers}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.58},
  URN =		{urn:nbn:de:0030-drops-160664},
  doi =		{10.4230/LIPIcs.SoCG.2022.58},
  annote =	{Keywords: multi-parameter persistent homology, stability, homology inference}
}
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