--- _id: '1072' abstract: - lang: eng text: Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field. acknowledgement: This research has been supported by the EU project Toposys(FP7-ICT-318493-STREP), by ESF under the ACAT Research Network Programme, by the Russian Government under mega project 11.G34.31.0053, and by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”. article_processing_charge: No article_type: original author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 citation: ama: Bauer U, Edelsbrunner H. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 2017;369(5):3741-3762. doi:10.1090/tran/6991 apa: Bauer, U., & Edelsbrunner, H. (2017). The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/6991 chicago: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” Transactions of the American Mathematical Society. American Mathematical Society, 2017. https://doi.org/10.1090/tran/6991. ieee: U. Bauer and H. Edelsbrunner, “The Morse theory of Čech and delaunay complexes,” Transactions of the American Mathematical Society, vol. 369, no. 5. American Mathematical Society, pp. 3741–3762, 2017. ista: Bauer U, Edelsbrunner H. 2017. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 369(5), 3741–3762. mla: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” Transactions of the American Mathematical Society, vol. 369, no. 5, American Mathematical Society, 2017, pp. 3741–62, doi:10.1090/tran/6991. short: U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762. date_created: 2018-12-11T11:49:59Z date_published: 2017-05-01T00:00:00Z date_updated: 2023-09-20T12:05:56Z day: '01' department: - _id: HeEd doi: 10.1090/tran/6991 ec_funded: 1 external_id: arxiv: - '1312.1231' isi: - '000398030400024' intvolume: ' 369' isi: 1 issue: '5' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1312.1231 month: '05' oa: 1 oa_version: Preprint page: 3741 - 3762 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Transactions of the American Mathematical Society publication_status: published publisher: American Mathematical Society publist_id: '6311' quality_controlled: '1' scopus_import: '1' status: public title: The Morse theory of Čech and delaunay complexes type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 369 year: '2017' ... --- _id: '1424' abstract: - lang: eng text: We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data. acknowledgement: This work was partially supported by the Austrian Science FUnd, project no. KLI 00012. alternative_title: - Advances in Neural Information Processing Systems author: - first_name: Roland full_name: Kwitt, Roland last_name: Kwitt - first_name: Stefan full_name: Huber, Stefan id: 4700A070-F248-11E8-B48F-1D18A9856A87 last_name: Huber orcid: 0000-0002-8871-5814 - first_name: Marc full_name: Niethammer, Marc last_name: Niethammer - first_name: Weili full_name: Lin, Weili last_name: Lin - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 citation: ama: 'Kwitt R, Huber S, Niethammer M, Lin W, Bauer U. Statistical topological data analysis-A kernel perspective. In: Vol 28. Neural Information Processing Systems; 2015:3070-3078.' apa: 'Kwitt, R., Huber, S., Niethammer, M., Lin, W., & Bauer, U. (2015). Statistical topological data analysis-A kernel perspective (Vol. 28, pp. 3070–3078). Presented at the NIPS: Neural Information Processing Systems, Montreal, Canada: Neural Information Processing Systems.' chicago: Kwitt, Roland, Stefan Huber, Marc Niethammer, Weili Lin, and Ulrich Bauer. “Statistical Topological Data Analysis-A Kernel Perspective,” 28:3070–78. Neural Information Processing Systems, 2015. ieee: 'R. Kwitt, S. Huber, M. Niethammer, W. Lin, and U. Bauer, “Statistical topological data analysis-A kernel perspective,” presented at the NIPS: Neural Information Processing Systems, Montreal, Canada, 2015, vol. 28, pp. 3070–3078.' ista: 'Kwitt R, Huber S, Niethammer M, Lin W, Bauer U. 2015. Statistical topological data analysis-A kernel perspective. NIPS: Neural Information Processing Systems, Advances in Neural Information Processing Systems, vol. 28, 3070–3078.' mla: Kwitt, Roland, et al. Statistical Topological Data Analysis-A Kernel Perspective. Vol. 28, Neural Information Processing Systems, 2015, pp. 3070–78. short: R. Kwitt, S. Huber, M. Niethammer, W. Lin, U. Bauer, in:, Neural Information Processing Systems, 2015, pp. 3070–3078. conference: end_date: 2015-12-12 location: Montreal, Canada name: 'NIPS: Neural Information Processing Systems' start_date: 2015-12-07 date_created: 2018-12-11T11:51:56Z date_published: 2015-12-01T00:00:00Z date_updated: 2021-01-12T06:50:38Z day: '01' department: - _id: HeEd intvolume: ' 28' language: - iso: eng main_file_link: - open_access: '1' url: https://papers.nips.cc/paper/5887-statistical-topological-data-analysis-a-kernel-perspective month: '12' oa: 1 oa_version: Submitted Version page: 3070 - 3078 publication_status: published publisher: Neural Information Processing Systems publist_id: '5782' quality_controlled: '1' status: public title: Statistical topological data analysis-A kernel perspective type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 28 year: '2015' ... --- _id: '1483' abstract: - lang: eng text: Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes. author: - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Stefan full_name: Huber, Stefan id: 4700A070-F248-11E8-B48F-1D18A9856A87 last_name: Huber orcid: 0000-0002-8871-5814 - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Roland full_name: Kwitt, Roland last_name: Kwitt citation: ama: 'Reininghaus J, Huber S, Bauer U, Kwitt R. A stable multi-scale kernel for topological machine learning. In: IEEE; 2015:4741-4748. doi:10.1109/CVPR.2015.7299106' apa: 'Reininghaus, J., Huber, S., Bauer, U., & Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning (pp. 4741–4748). Presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA: IEEE. https://doi.org/10.1109/CVPR.2015.7299106' chicago: Reininghaus, Jan, Stefan Huber, Ulrich Bauer, and Roland Kwitt. “A Stable Multi-Scale Kernel for Topological Machine Learning,” 4741–48. IEEE, 2015. https://doi.org/10.1109/CVPR.2015.7299106. ieee: 'J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, “A stable multi-scale kernel for topological machine learning,” presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA, 2015, pp. 4741–4748.' ista: 'Reininghaus J, Huber S, Bauer U, Kwitt R. 2015. A stable multi-scale kernel for topological machine learning. CVPR: Computer Vision and Pattern Recognition, 4741–4748.' mla: Reininghaus, Jan, et al. A Stable Multi-Scale Kernel for Topological Machine Learning. IEEE, 2015, pp. 4741–48, doi:10.1109/CVPR.2015.7299106. short: J. Reininghaus, S. Huber, U. Bauer, R. Kwitt, in:, IEEE, 2015, pp. 4741–4748. conference: end_date: 2015-06-12 location: Boston, MA, USA name: 'CVPR: Computer Vision and Pattern Recognition' start_date: 2015-06-07 date_created: 2018-12-11T11:52:17Z date_published: 2015-10-14T00:00:00Z date_updated: 2021-01-12T06:51:03Z day: '14' department: - _id: HeEd doi: 10.1109/CVPR.2015.7299106 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1412.6821 month: '10' oa: 1 oa_version: Preprint page: 4741 - 4748 publication_identifier: eisbn: - '978-1-4673-6964-0 ' publication_status: published publisher: IEEE publist_id: '5709' scopus_import: 1 status: public title: A stable multi-scale kernel for topological machine learning type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2015' ... --- _id: '1805' abstract: - lang: eng text: 'We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.' author: - first_name: Dominique full_name: Attali, Dominique last_name: Attali - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Olivier full_name: Devillers, Olivier last_name: Devillers - first_name: Marc full_name: Glisse, Marc last_name: Glisse - first_name: André full_name: Lieutier, André last_name: Lieutier citation: ama: 'Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. Homological reconstruction and simplification in R3. Computational Geometry: Theory and Applications. 2015;48(8):606-621. doi:10.1016/j.comgeo.2014.08.010' apa: 'Attali, D., Bauer, U., Devillers, O., Glisse, M., & Lieutier, A. (2015). Homological reconstruction and simplification in R3. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2014.08.010' chicago: 'Attali, Dominique, Ulrich Bauer, Olivier Devillers, Marc Glisse, and André Lieutier. “Homological Reconstruction and Simplification in R3.” Computational Geometry: Theory and Applications. Elsevier, 2015. https://doi.org/10.1016/j.comgeo.2014.08.010.' ieee: 'D. Attali, U. Bauer, O. Devillers, M. Glisse, and A. Lieutier, “Homological reconstruction and simplification in R3,” Computational Geometry: Theory and Applications, vol. 48, no. 8. Elsevier, pp. 606–621, 2015.' ista: 'Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. 2015. Homological reconstruction and simplification in R3. Computational Geometry: Theory and Applications. 48(8), 606–621.' mla: 'Attali, Dominique, et al. “Homological Reconstruction and Simplification in R3.” Computational Geometry: Theory and Applications, vol. 48, no. 8, Elsevier, 2015, pp. 606–21, doi:10.1016/j.comgeo.2014.08.010.' short: 'D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621.' date_created: 2018-12-11T11:54:06Z date_published: 2015-06-03T00:00:00Z date_updated: 2023-02-23T10:59:19Z day: '03' department: - _id: HeEd doi: 10.1016/j.comgeo.2014.08.010 ec_funded: 1 intvolume: ' 48' issue: '8' language: - iso: eng month: '06' oa_version: None page: 606 - 621 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: 'Computational Geometry: Theory and Applications' publication_status: published publisher: Elsevier publist_id: '5305' quality_controlled: '1' related_material: record: - id: '2812' relation: earlier_version status: public scopus_import: 1 status: public title: Homological reconstruction and simplification in R3 type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 48 year: '2015' ... --- _id: '2043' abstract: - lang: eng text: Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node. author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Michael full_name: Kerber, Michael last_name: Kerber orcid: 0000-0002-8030-9299 - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus citation: ama: 'Bauer U, Kerber M, Reininghaus J. Distributed computation of persistent homology. In: McGeoch C, Meyer U, eds. Proceedings of the Workshop on Algorithm Engineering and Experiments. Society of Industrial and Applied Mathematics; 2014:31-38. doi:10.1137/1.9781611973198.4' apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Distributed computation of persistent homology. In C. McGeoch & U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments (pp. 31–38). Portland, USA: Society of Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973198.4' chicago: Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Distributed Computation of Persistent Homology.” In Proceedings of the Workshop on Algorithm Engineering and Experiments, edited by Catherine McGeoch and Ulrich Meyer, 31–38. Society of Industrial and Applied Mathematics, 2014. https://doi.org/10.1137/1.9781611973198.4. ieee: U. Bauer, M. Kerber, and J. Reininghaus, “Distributed computation of persistent homology,” in Proceedings of the Workshop on Algorithm Engineering and Experiments, Portland, USA, 2014, pp. 31–38. ista: 'Bauer U, Kerber M, Reininghaus J. 2014. Distributed computation of persistent homology. Proceedings of the Workshop on Algorithm Engineering and Experiments. ALENEX: Algorithm Engineering and Experiments, 31–38.' mla: Bauer, Ulrich, et al. “Distributed Computation of Persistent Homology.” Proceedings of the Workshop on Algorithm Engineering and Experiments, edited by Catherine McGeoch and Ulrich Meyer, Society of Industrial and Applied Mathematics, 2014, pp. 31–38, doi:10.1137/1.9781611973198.4. short: U. Bauer, M. Kerber, J. Reininghaus, in:, C. McGeoch, U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments, Society of Industrial and Applied Mathematics, 2014, pp. 31–38. conference: end_date: 2014-01-05 location: Portland, USA name: 'ALENEX: Algorithm Engineering and Experiments' start_date: 2014-01-05 date_created: 2018-12-11T11:55:23Z date_published: 2014-01-01T00:00:00Z date_updated: 2021-01-12T06:54:56Z day: '01' department: - _id: HeEd doi: 10.1137/1.9781611973198.4 ec_funded: 1 editor: - first_name: Catherine full_name: ' McGeoch, Catherine' last_name: ' McGeoch' - first_name: Ulrich full_name: Meyer, Ulrich last_name: Meyer language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1310.0710 month: '01' oa: 1 oa_version: Submitted Version page: 31 - 38 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Workshop on Algorithm Engineering and Experiments publication_status: published publisher: Society of Industrial and Applied Mathematics publist_id: '5008' quality_controlled: '1' scopus_import: 1 status: public title: Distributed computation of persistent homology type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2044' abstract: - lang: eng text: We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation. author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Michael full_name: Kerber, Michael last_name: Kerber orcid: 0000-0002-8030-9299 - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus citation: ama: 'Bauer U, Kerber M, Reininghaus J. Clear and Compress: Computing Persistent Homology in Chunks. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Springer; 2014:103-117. doi:10.1007/978-3-319-04099-8_7' apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Clear and Compress: Computing Persistent Homology in Chunks. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III (pp. 103–117). Springer. https://doi.org/10.1007/978-3-319-04099-8_7' chicago: 'Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Clear and Compress: Computing Persistent Homology in Chunks.” In Topological Methods in Data Analysis and Visualization III, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 103–17. Mathematics and Visualization. Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_7.' ieee: 'U. Bauer, M. Kerber, and J. Reininghaus, “Clear and Compress: Computing Persistent Homology in Chunks,” in Topological Methods in Data Analysis and Visualization III, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Springer, 2014, pp. 103–117.' ista: 'Bauer U, Kerber M, Reininghaus J. 2014.Clear and Compress: Computing Persistent Homology in Chunks. In: Topological Methods in Data Analysis and Visualization III. , 103–117.' mla: 'Bauer, Ulrich, et al. “Clear and Compress: Computing Persistent Homology in Chunks.” Topological Methods in Data Analysis and Visualization III, edited by Peer-Timo Bremer et al., Springer, 2014, pp. 103–17, doi:10.1007/978-3-319-04099-8_7.' short: U. Bauer, M. Kerber, J. Reininghaus, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III, Springer, 2014, pp. 103–117. date_created: 2018-12-11T11:55:23Z date_published: 2014-03-19T00:00:00Z date_updated: 2021-01-12T06:54:56Z day: '19' department: - _id: HeEd doi: 10.1007/978-3-319-04099-8_7 ec_funded: 1 editor: - first_name: Peer-Timo full_name: Bremer, Peer-Timo last_name: Bremer - first_name: Ingrid full_name: Hotz, Ingrid last_name: Hotz - first_name: Valerio full_name: Pascucci, Valerio last_name: Pascucci - first_name: Ronald full_name: Peikert, Ronald last_name: Peikert language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1303.0477 month: '03' oa: 1 oa_version: Submitted Version page: 103 - 117 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Topological Methods in Data Analysis and Visualization III publication_status: published publisher: Springer publist_id: '5007' quality_controlled: '1' scopus_import: 1 series_title: Mathematics and Visualization status: public title: 'Clear and Compress: Computing Persistent Homology in Chunks' type: book_chapter user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2153' abstract: - lang: eng text: 'We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).' author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Michael full_name: Lesnick, Michael last_name: Lesnick citation: ama: 'Bauer U, Lesnick M. Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:355-364. doi:10.1145/2582112.2582168' apa: 'Bauer, U., & Lesnick, M. (2014). Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the Annual Symposium on Computational Geometry (pp. 355–364). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582168' chicago: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” In Proceedings of the Annual Symposium on Computational Geometry, 355–64. ACM, 2014. https://doi.org/10.1145/2582112.2582168. ieee: U. Bauer and M. Lesnick, “Induced matchings of barcodes and the algebraic stability of persistence,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 355–364. ista: 'Bauer U, Lesnick M. 2014. Induced matchings of barcodes and the algebraic stability of persistence. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 355–364.' mla: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–64, doi:10.1145/2582112.2582168. short: U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–364. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:01Z date_published: 2014-06-01T00:00:00Z date_updated: 2021-01-12T06:55:38Z day: '01' department: - _id: HeEd doi: 10.1145/2582112.2582168 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1311.3681 month: '06' oa: 1 oa_version: Submitted Version page: 355 - 364 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4853' quality_controlled: '1' scopus_import: 1 status: public title: Induced matchings of barcodes and the algebraic stability of persistence type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2156' abstract: - lang: eng text: We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s). acknowledgement: National Science Foundation under grants CCF-1319406, CCF-1116258. author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Xiaoyin full_name: Ge, Xiaoyin last_name: Ge - first_name: Yusu full_name: Wang, Yusu last_name: Wang citation: ama: 'Bauer U, Ge X, Wang Y. Measuring distance between Reeb graphs. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:464-473. doi:10.1145/2582112.2582169' apa: 'Bauer, U., Ge, X., & Wang, Y. (2014). Measuring distance between Reeb graphs. In Proceedings of the Annual Symposium on Computational Geometry (pp. 464–473). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582169' chicago: Bauer, Ulrich, Xiaoyin Ge, and Yusu Wang. “Measuring Distance between Reeb Graphs.” In Proceedings of the Annual Symposium on Computational Geometry, 464–73. ACM, 2014. https://doi.org/10.1145/2582112.2582169. ieee: U. Bauer, X. Ge, and Y. Wang, “Measuring distance between Reeb graphs,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 464–473. ista: 'Bauer U, Ge X, Wang Y. 2014. Measuring distance between Reeb graphs. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 464–473.' mla: Bauer, Ulrich, et al. “Measuring Distance between Reeb Graphs.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–73, doi:10.1145/2582112.2582169. short: U. Bauer, X. Ge, Y. Wang, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–473. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:02Z date_published: 2014-06-01T00:00:00Z date_updated: 2021-01-12T06:55:39Z day: '01' department: - _id: HeEd doi: 10.1145/2582112.2582169 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1307.2839 month: '06' oa: 1 oa_version: Submitted Version page: 464 - 473 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4850' quality_controlled: '1' scopus_import: 1 status: public title: Measuring distance between Reeb graphs type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2155' abstract: - lang: eng text: Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s). acknowledgement: This research is partially supported by ESF under the ACAT Research Network Programme, and by the Russian Government under mega project 11.G34.31.0053 author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 citation: ama: 'Bauer U, Edelsbrunner H. The morse theory of Čech and Delaunay filtrations. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:484-490. doi:10.1145/2582112.2582167' apa: 'Bauer, U., & Edelsbrunner, H. (2014). The morse theory of Čech and Delaunay filtrations. In Proceedings of the Annual Symposium on Computational Geometry (pp. 484–490). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582167' chicago: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” In Proceedings of the Annual Symposium on Computational Geometry, 484–90. ACM, 2014. https://doi.org/10.1145/2582112.2582167. ieee: U. Bauer and H. Edelsbrunner, “The morse theory of Čech and Delaunay filtrations,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 484–490. ista: 'Bauer U, Edelsbrunner H. 2014. The morse theory of Čech and Delaunay filtrations. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 484–490.' mla: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–90, doi:10.1145/2582112.2582167. short: U. Bauer, H. Edelsbrunner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–490. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:01Z date_published: 2014-06-01T00:00:00Z date_updated: 2021-01-12T06:55:38Z day: '01' department: - _id: HeEd doi: 10.1145/2582112.2582167 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1312.1231 month: '06' oa: 1 oa_version: Submitted Version page: 484 - 490 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4851' quality_controlled: '1' scopus_import: 1 status: public title: The morse theory of Čech and Delaunay filtrations type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '10894' abstract: - lang: eng text: PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations. article_processing_charge: No author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Michael full_name: Kerber, Michael last_name: Kerber - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Hubert full_name: Wagner, Hubert last_name: Wagner citation: ama: 'Bauer U, Kerber M, Reininghaus J, Wagner H. PHAT – Persistent Homology Algorithms Toolbox. In: ICMS 2014: International Congress on Mathematical Software. Vol 8592. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg; 2014:137-143. doi:10.1007/978-3-662-44199-2_24' apa: 'Bauer, U., Kerber, M., Reininghaus, J., & Wagner, H. (2014). PHAT – Persistent Homology Algorithms Toolbox. In ICMS 2014: International Congress on Mathematical Software (Vol. 8592, pp. 137–143). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_24' chicago: 'Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “PHAT – Persistent Homology Algorithms Toolbox.” In ICMS 2014: International Congress on Mathematical Software, 8592:137–43. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. https://doi.org/10.1007/978-3-662-44199-2_24.' ieee: 'U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “PHAT – Persistent Homology Algorithms Toolbox,” in ICMS 2014: International Congress on Mathematical Software, Seoul, South Korea, 2014, vol. 8592, pp. 137–143.' ista: 'Bauer U, Kerber M, Reininghaus J, Wagner H. 2014. PHAT – Persistent Homology Algorithms Toolbox. ICMS 2014: International Congress on Mathematical Software. ICMS: International Congress on Mathematical SoftwareLNCS vol. 8592, 137–143.' mla: 'Bauer, Ulrich, et al. “PHAT – Persistent Homology Algorithms Toolbox.” ICMS 2014: International Congress on Mathematical Software, vol. 8592, Springer Berlin Heidelberg, 2014, pp. 137–43, doi:10.1007/978-3-662-44199-2_24.' short: 'U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, in:, ICMS 2014: International Congress on Mathematical Software, Springer Berlin Heidelberg, Berlin, Heidelberg, 2014, pp. 137–143.' conference: end_date: 2014-08-09 location: Seoul, South Korea name: 'ICMS: International Congress on Mathematical Software' start_date: 2014-08-05 date_created: 2022-03-21T07:12:16Z date_published: 2014-09-01T00:00:00Z date_updated: 2023-09-20T09:42:40Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-662-44199-2_24 intvolume: ' 8592' language: - iso: eng month: '09' oa_version: None page: 137-143 place: Berlin, Heidelberg publication: 'ICMS 2014: International Congress on Mathematical Software' publication_identifier: eisbn: - '9783662441992' eissn: - 1611-3349 isbn: - '9783662441985' issn: - 0302-9743 publication_status: published publisher: Springer Berlin Heidelberg quality_controlled: '1' related_material: record: - id: '1433' relation: later_version status: public scopus_import: '1' series_title: LNCS status: public title: PHAT – Persistent Homology Algorithms Toolbox type: conference user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 8592 year: '2014' ... --- _id: '2812' abstract: - lang: eng text: 'We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H* (X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.' acknowledgement: Some of the authors were partially supported by the GIGA ANR grant (contract ANR-09-BLAN-0331-01) and the European project CG-Learning (contract 255827). author: - first_name: Dominique full_name: Attali, Dominique last_name: Attali - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Olivier full_name: Devillers, Olivier last_name: Devillers - first_name: Marc full_name: Glisse, Marc last_name: Glisse - first_name: André full_name: Lieutier, André last_name: Lieutier citation: ama: 'Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. Homological reconstruction and simplification in R3. In: Proceedings of the 29th Annual Symposium on Computational Geometry. ACM; 2013:117-125. doi:10.1145/2462356.2462373' apa: 'Attali, D., Bauer, U., Devillers, O., Glisse, M., & Lieutier, A. (2013). Homological reconstruction and simplification in R3. In Proceedings of the 29th annual symposium on Computational Geometry (pp. 117–125). Rio de Janeiro, Brazil: ACM. https://doi.org/10.1145/2462356.2462373' chicago: Attali, Dominique, Ulrich Bauer, Olivier Devillers, Marc Glisse, and André Lieutier. “Homological Reconstruction and Simplification in R3.” In Proceedings of the 29th Annual Symposium on Computational Geometry, 117–25. ACM, 2013. https://doi.org/10.1145/2462356.2462373. ieee: D. Attali, U. Bauer, O. Devillers, M. Glisse, and A. Lieutier, “Homological reconstruction and simplification in R3,” in Proceedings of the 29th annual symposium on Computational Geometry, Rio de Janeiro, Brazil, 2013, pp. 117–125. ista: 'Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. 2013. Homological reconstruction and simplification in R3. Proceedings of the 29th annual symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 117–125.' mla: Attali, Dominique, et al. “Homological Reconstruction and Simplification in R3.” Proceedings of the 29th Annual Symposium on Computational Geometry, ACM, 2013, pp. 117–25, doi:10.1145/2462356.2462373. short: D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, in:, Proceedings of the 29th Annual Symposium on Computational Geometry, ACM, 2013, pp. 117–125. conference: end_date: 2013-06-20 location: Rio de Janeiro, Brazil name: 'SoCG: Symposium on Computational Geometry' start_date: 2013-06-17 date_created: 2018-12-11T11:59:44Z date_published: 2013-06-01T00:00:00Z date_updated: 2023-02-23T10:15:15Z day: '01' department: - _id: HeEd doi: 10.1145/2462356.2462373 language: - iso: eng main_file_link: - open_access: '1' url: http://hal.archives-ouvertes.fr/hal-00833791/ month: '06' oa: 1 oa_version: Submitted Version page: 117 - 125 publication: Proceedings of the 29th annual symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4072' quality_controlled: '1' related_material: record: - id: '1805' relation: later_version status: public scopus_import: 1 status: public title: Homological reconstruction and simplification in R3 type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2013' ...