--- _id: '9630' abstract: - lang: eng text: Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context. acknowledgement: This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF). article_processing_charge: Yes article_type: original author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Ziga full_name: Virk, Ziga id: 2E36B656-F248-11E8-B48F-1D18A9856A87 last_name: Virk - first_name: Hubert full_name: Wagner, Hubert id: 379CA8B8-F248-11E8-B48F-1D18A9856A87 last_name: Wagner citation: ama: Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. Journal of Computational Geometry. 2020;11(2):162-182. doi:10.20382/jocg.v11i2a7 apa: Edelsbrunner, H., Virk, Z., & Wagner, H. (2020). Topological data analysis in information space. Journal of Computational Geometry. Carleton University. https://doi.org/10.20382/jocg.v11i2a7 chicago: Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” Journal of Computational Geometry. Carleton University, 2020. https://doi.org/10.20382/jocg.v11i2a7. ieee: H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” Journal of Computational Geometry, vol. 11, no. 2. Carleton University, pp. 162–182, 2020. ista: Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182. mla: Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” Journal of Computational Geometry, vol. 11, no. 2, Carleton University, 2020, pp. 162–82, doi:10.20382/jocg.v11i2a7. short: H. Edelsbrunner, Z. Virk, H. Wagner, Journal of Computational Geometry 11 (2020) 162–182. date_created: 2021-07-04T22:01:26Z date_published: 2020-12-14T00:00:00Z date_updated: 2021-08-11T12:26:34Z day: '14' ddc: - '510' - '000' department: - _id: HeEd doi: 10.20382/jocg.v11i2a7 file: - access_level: open_access checksum: f02d0b2b3838e7891a6c417fc34ffdcd content_type: application/pdf creator: asandaue date_created: 2021-08-11T11:55:11Z date_updated: 2021-08-11T11:55:11Z file_id: '9882' file_name: 2020_JournalOfComputationalGeometry_Edelsbrunner.pdf file_size: 1449234 relation: main_file success: 1 file_date_updated: 2021-08-11T11:55:11Z has_accepted_license: '1' intvolume: ' 11' issue: '2' language: - iso: eng license: https://creativecommons.org/licenses/by/3.0/ month: '12' oa: 1 oa_version: Published Version page: 162-182 project: - _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316 grant_number: I4887 name: Discretization in Geometry and Dynamics publication: Journal of Computational Geometry publication_identifier: eissn: - 1920180X publication_status: published publisher: Carleton University quality_controlled: '1' scopus_import: '1' status: public title: Topological data analysis in information space tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/3.0/legalcode name: Creative Commons Attribution 3.0 Unported (CC BY 3.0) short: CC BY (3.0) type: journal_article user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf volume: 11 year: '2020' ... --- _id: '521' abstract: - lang: eng text: Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f:X→Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f:X→Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y)≤asdim(X)+n−1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension. author: - first_name: Kyle full_name: Austin, Kyle last_name: Austin - first_name: Ziga full_name: Virk, Ziga id: 2E36B656-F248-11E8-B48F-1D18A9856A87 last_name: Virk citation: ama: Austin K, Virk Z. Higson compactification and dimension raising. Topology and its Applications. 2017;215:45-57. doi:10.1016/j.topol.2016.10.005 apa: Austin, K., & Virk, Z. (2017). Higson compactification and dimension raising. Topology and Its Applications. Elsevier. https://doi.org/10.1016/j.topol.2016.10.005 chicago: Austin, Kyle, and Ziga Virk. “Higson Compactification and Dimension Raising.” Topology and Its Applications. Elsevier, 2017. https://doi.org/10.1016/j.topol.2016.10.005. ieee: K. Austin and Z. Virk, “Higson compactification and dimension raising,” Topology and its Applications, vol. 215. Elsevier, pp. 45–57, 2017. ista: Austin K, Virk Z. 2017. Higson compactification and dimension raising. Topology and its Applications. 215, 45–57. mla: Austin, Kyle, and Ziga Virk. “Higson Compactification and Dimension Raising.” Topology and Its Applications, vol. 215, Elsevier, 2017, pp. 45–57, doi:10.1016/j.topol.2016.10.005. short: K. Austin, Z. Virk, Topology and Its Applications 215 (2017) 45–57. date_created: 2018-12-11T11:46:56Z date_published: 2017-01-01T00:00:00Z date_updated: 2021-01-12T08:01:21Z day: '01' department: - _id: HeEd doi: 10.1016/j.topol.2016.10.005 intvolume: ' 215' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1608.03954v1 month: '01' oa: 1 oa_version: Submitted Version page: 45 - 57 publication: Topology and its Applications publication_identifier: issn: - '01668641' publication_status: published publisher: Elsevier publist_id: '7299' quality_controlled: '1' status: public title: Higson compactification and dimension raising type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 215 year: '2017' ... --- _id: '737' abstract: - lang: eng text: We generalize Brazas’ topology on the fundamental group to the whole universal path space X˜ i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action. article_processing_charge: No author: - first_name: Ziga full_name: Virk, Ziga id: 2E36B656-F248-11E8-B48F-1D18A9856A87 last_name: Virk - first_name: Andreas full_name: Zastrow, Andreas last_name: Zastrow citation: ama: Virk Z, Zastrow A. A new topology on the universal path space. Topology and its Applications. 2017;231:186-196. doi:10.1016/j.topol.2017.09.015 apa: Virk, Z., & Zastrow, A. (2017). A new topology on the universal path space. Topology and Its Applications. Elsevier. https://doi.org/10.1016/j.topol.2017.09.015 chicago: Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” Topology and Its Applications. Elsevier, 2017. https://doi.org/10.1016/j.topol.2017.09.015. ieee: Z. Virk and A. Zastrow, “A new topology on the universal path space,” Topology and its Applications, vol. 231. Elsevier, pp. 186–196, 2017. ista: Virk Z, Zastrow A. 2017. A new topology on the universal path space. Topology and its Applications. 231, 186–196. mla: Virk, Ziga, and Andreas Zastrow. “A New Topology on the Universal Path Space.” Topology and Its Applications, vol. 231, Elsevier, 2017, pp. 186–96, doi:10.1016/j.topol.2017.09.015. short: Z. Virk, A. Zastrow, Topology and Its Applications 231 (2017) 186–196. date_created: 2018-12-11T11:48:14Z date_published: 2017-11-01T00:00:00Z date_updated: 2023-09-27T12:53:01Z day: '01' department: - _id: HeEd doi: 10.1016/j.topol.2017.09.015 external_id: isi: - '000413889100012' intvolume: ' 231' isi: 1 language: - iso: eng month: '11' oa_version: None page: 186 - 196 publication: Topology and its Applications publication_identifier: issn: - '01668641' publication_status: published publisher: Elsevier publist_id: '6930' quality_controlled: '1' scopus_import: '1' status: public title: A new topology on the universal path space type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 231 year: '2017' ...