4 Search Results for "Zhou, Ling"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.51
Ephemeral Persistence Features and the Stability of Filtered Chain Complexes

Authors: Facundo Mémoli and Ling Zhou

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

Abstract
We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes ephemeral points, i.e. points with zero persistence, which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics which are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the discriminating power of the standard VR barcode.
Cite as

Facundo Mémoli and Ling Zhou. Ephemeral Persistence Features and the Stability of Filtered Chain Complexes. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 51:1-51:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{memoli_et_al:LIPIcs.SoCG.2023.51,
  author =	{M\'{e}moli, Facundo and Zhou, Ling},
  title =	{{Ephemeral Persistence Features and the Stability of Filtered Chain Complexes}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{51:1--51:18},
  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.2023.51},
  URN =		{urn:nbn:de:0030-drops-179014},
  doi =		{10.4230/LIPIcs.SoCG.2023.51},
  annote =	{Keywords: filtered chain complexes, Vietoris-Rips complexes, barcode, bottleneck distance, matching distance, Gromov-Hausdorff distance}
}
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.COSIT.2022.3
Automatically Discovering Conceptual Neighborhoods Using Machine Learning Methods

Authors: Ling Cai, Krzysztof Janowicz, and Rui Zhu

Published in: LIPIcs, Volume 240, 15th International Conference on Spatial Information Theory (COSIT 2022)

Abstract
Qualitative spatio-temporal reasoning (QSTR) plays a key role in spatial cognition and artificial intelligence (AI) research. In the past, research and applications of QSTR have often taken place in the context of declarative forms of knowledge representation. For instance, conceptual neighborhoods (CN) and composition tables (CT) of relations are introduced explicitly and utilized for spatial/temporal reasoning. Orthogonal to this line of study, we focus on bottom-up machine learning (ML) approaches to investigate QSTR. More specifically, we are interested in questions of whether similarities between qualitative relations can be learned from data purely based on ML models, and, if so, how these models differ from the ones studied by traditional approaches. To achieve this, we propose a graph-based approach to examine the similarity of relations by analyzing trained ML models. Using various experiments on synthetic data, we demonstrate that the relationships discovered by ML models are well-aligned with CN structures introduced in the (theoretical) literature, for both spatial and temporal reasoning. Noticeably, even with significantly limited qualitative information for training, ML models are still able to automatically construct neighborhood structures. Moreover, patterns of asymmetric similarities between relations are disclosed using such a data-driven approach. To the best of our knowledge, our work is the first to automatically discover CNs without any domain knowledge. Our results can be applied to discovering CNs of any set of jointly exhaustive and pairwise disjoint (JEPD) relations.
Cite as

Ling Cai, Krzysztof Janowicz, and Rui Zhu. Automatically Discovering Conceptual Neighborhoods Using Machine Learning Methods. In 15th International Conference on Spatial Information Theory (COSIT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 240, pp. 3:1-3:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cai_et_al:LIPIcs.COSIT.2022.3,
  author =	{Cai, Ling and Janowicz, Krzysztof and Zhu, Rui},
  title =	{{Automatically Discovering Conceptual Neighborhoods Using Machine Learning Methods}},
  booktitle =	{15th International Conference on Spatial Information Theory (COSIT 2022)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-257-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{240},
  editor =	{Ishikawa, Toru and Fabrikant, Sara Irina and Winter, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.COSIT.2022.3},
  URN =		{urn:nbn:de:0030-drops-168884},
  doi =		{10.4230/LIPIcs.COSIT.2022.3},
  annote =	{Keywords: Qualitative Spatial Reasoning, Qualitative Temporal Reasoning, Conceptual Neighborhood, Machine Learning, Knowledge Discovery}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.31
Persistent Cup-Length

Authors: Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou

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

Abstract
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.
Cite as

Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent Cup-Length. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 31:1-31:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{contessoto_et_al:LIPIcs.SoCG.2022.31,
  author =	{Contessoto, Marco and M\'{e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling},
  title =	{{Persistent Cup-Length}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{31:1--31:17},
  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.2022.31},
  URN =		{urn:nbn:de:0030-drops-160398},
  doi =		{10.4230/LIPIcs.SoCG.2022.31},
  annote =	{Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance}
}
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