[{"department":[{"_id":"HeEd"}],"date_updated":"2023-09-20T12:05:56Z","status":"public","article_type":"original","type":"journal_article","_id":"1072","volume":369,"issue":"5","ec_funded":1,"language":[{"iso":"eng"}],"publication_status":"published","month":"05","intvolume":" 369","scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1312.1231","open_access":"1"}],"oa_version":"Preprint","abstract":[{"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.","lang":"eng"}],"title":"The Morse theory of Čech and delaunay complexes","publist_id":"6311","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"external_id":{"isi":["000398030400024"],"arxiv":["1312.1231"]},"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"ista":"Bauer U, Edelsbrunner H. 2017. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 369(5), 3741–3762.","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.","short":"U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762.","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.","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","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."},"project":[{"call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493"}],"date_published":"2017-05-01T00:00:00Z","doi":"10.1090/tran/6991","date_created":"2018-12-11T11:49:59Z","page":"3741 - 3762","day":"01","publication":"Transactions of the American Mathematical Society","isi":1,"year":"2017","quality_controlled":"1","publisher":"American Mathematical Society","oa":1,"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”."},{"author":[{"first_name":"Roland","last_name":"Kwitt","full_name":"Kwitt, Roland"},{"first_name":"Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87","full_name":"Huber, Stefan","orcid":"0000-0002-8871-5814","last_name":"Huber"},{"first_name":"Marc","last_name":"Niethammer","full_name":"Niethammer, Marc"},{"last_name":"Lin","full_name":"Lin, Weili","first_name":"Weili"},{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer"}],"publist_id":"5782","title":"Statistical topological data analysis-A kernel perspective","department":[{"_id":"HeEd"}],"citation":{"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.","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.","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.","short":"R. Kwitt, S. Huber, M. Niethammer, W. Lin, U. Bauer, in:, Neural Information Processing Systems, 2015, pp. 3070–3078.","mla":"Kwitt, Roland, et al. Statistical Topological Data Analysis-A Kernel Perspective. Vol. 28, Neural Information Processing Systems, 2015, pp. 3070–78.","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.","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."},"date_updated":"2021-01-12T06:50:38Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","conference":{"name":"NIPS: Neural Information Processing Systems","start_date":"2015-12-07","location":"Montreal, Canada","end_date":"2015-12-12"},"type":"conference","status":"public","_id":"1424","page":"3070 - 3078","date_created":"2018-12-11T11:51:56Z","date_published":"2015-12-01T00:00:00Z","volume":28,"year":"2015","publication_status":"published","language":[{"iso":"eng"}],"day":"01","oa":1,"main_file_link":[{"url":"https://papers.nips.cc/paper/5887-statistical-topological-data-analysis-a-kernel-perspective","open_access":"1"}],"quality_controlled":"1","publisher":"Neural Information Processing Systems","alternative_title":["Advances in Neural Information Processing Systems"],"intvolume":" 28","month":"12","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.","oa_version":"Submitted Version"},{"type":"conference","conference":{"name":"CVPR: Computer Vision and Pattern Recognition","start_date":"2015-06-07","end_date":"2015-06-12","location":"Boston, MA, USA"},"status":"public","_id":"1483","author":[{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","full_name":"Reininghaus, Jan","last_name":"Reininghaus"},{"full_name":"Huber, Stefan","orcid":"0000-0002-8871-5814","last_name":"Huber","first_name":"Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bauer","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Kwitt","full_name":"Kwitt, Roland","first_name":"Roland"}],"publist_id":"5709","department":[{"_id":"HeEd"}],"title":"A stable multi-scale kernel for topological machine learning","citation":{"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.","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.","short":"J. Reininghaus, S. Huber, U. Bauer, R. Kwitt, in:, IEEE, 2015, pp. 4741–4748.","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.","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","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","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."},"date_updated":"2021-01-12T06:51:03Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":1,"publisher":"IEEE","main_file_link":[{"url":"http://arxiv.org/abs/1412.6821","open_access":"1"}],"oa":1,"month":"10","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."}],"oa_version":"Preprint","page":"4741 - 4748","doi":"10.1109/CVPR.2015.7299106","date_published":"2015-10-14T00:00:00Z","date_created":"2018-12-11T11:52:17Z","publication_identifier":{"eisbn":["978-1-4673-6964-0 "]},"year":"2015","publication_status":"published","day":"14","language":[{"iso":"eng"}]},{"citation":{"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.","short":"D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621.","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","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.","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.","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."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publist_id":"5305","author":[{"last_name":"Attali","full_name":"Attali, Dominique","first_name":"Dominique"},{"full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","last_name":"Bauer","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Devillers","full_name":"Devillers, Olivier","first_name":"Olivier"},{"first_name":"Marc","last_name":"Glisse","full_name":"Glisse, Marc"},{"last_name":"Lieutier","full_name":"Lieutier, André","first_name":"André"}],"title":"Homological reconstruction and simplification in R3","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493","name":"Topological Complex Systems"}],"year":"2015","day":"03","publication":"Computational Geometry: Theory and Applications","page":"606 - 621","doi":"10.1016/j.comgeo.2014.08.010","date_published":"2015-06-03T00:00:00Z","date_created":"2018-12-11T11:54:06Z","quality_controlled":"1","publisher":"Elsevier","date_updated":"2023-02-23T10:59:19Z","department":[{"_id":"HeEd"}],"_id":"1805","type":"journal_article","status":"public","publication_status":"published","language":[{"iso":"eng"}],"volume":48,"issue":"8","related_material":{"record":[{"relation":"earlier_version","id":"2812","status":"public"}]},"ec_funded":1,"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."}],"oa_version":"None","scopus_import":1,"month":"06","intvolume":" 48"},{"publication_status":"published","language":[{"iso":"eng"}],"ec_funded":1,"abstract":[{"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.","lang":"eng"}],"oa_version":"Submitted Version","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1310.0710"}],"scopus_import":1,"month":"01","date_updated":"2021-01-12T06:54:56Z","department":[{"_id":"HeEd"}],"_id":"2043","conference":{"end_date":"2014-01-05","location":"Portland, USA","start_date":"2014-01-05","name":"ALENEX: Algorithm Engineering and Experiments"},"type":"conference","status":"public","year":"2014","publication":"Proceedings of the Workshop on Algorithm Engineering and Experiments","day":"01","page":"31 - 38","date_created":"2018-12-11T11:55:23Z","doi":"10.1137/1.9781611973198.4","date_published":"2014-01-01T00:00:00Z","oa":1,"quality_controlled":"1","publisher":"Society of Industrial and Applied Mathematics","citation":{"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","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","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.","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.","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.","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.","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."},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","publist_id":"5008","author":[{"full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","last_name":"Bauer","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich"},{"first_name":"Michael","last_name":"Kerber","orcid":"0000-0002-8030-9299","full_name":"Kerber, Michael"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","full_name":"Reininghaus, Jan","last_name":"Reininghaus"}],"title":"Distributed computation of persistent homology","editor":[{"last_name":" McGeoch","full_name":" McGeoch, Catherine","first_name":"Catherine"},{"last_name":"Meyer","full_name":"Meyer, Ulrich","first_name":"Ulrich"}],"project":[{"name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}]},{"date_updated":"2021-01-12T06:54:56Z","department":[{"_id":"HeEd"}],"series_title":"Mathematics and Visualization","_id":"2044","status":"public","type":"book_chapter","language":[{"iso":"eng"}],"publication_status":"published","ec_funded":1,"oa_version":"Submitted Version","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."}],"month":"03","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1303.0477"}],"scopus_import":1,"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","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","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.","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.","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.","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.","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."},"editor":[{"full_name":"Bremer, Peer-Timo","last_name":"Bremer","first_name":"Peer-Timo"},{"last_name":"Hotz","full_name":"Hotz, Ingrid","first_name":"Ingrid"},{"first_name":"Valerio","full_name":"Pascucci, Valerio","last_name":"Pascucci"},{"first_name":"Ronald","last_name":"Peikert","full_name":"Peikert, Ronald"}],"title":"Clear and Compress: Computing Persistent Homology in Chunks","author":[{"first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich"},{"last_name":"Kerber","orcid":"0000-0002-8030-9299","full_name":"Kerber, Michael","first_name":"Michael"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"5007","project":[{"name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"publication":"Topological Methods in Data Analysis and Visualization III","day":"19","year":"2014","date_created":"2018-12-11T11:55:23Z","date_published":"2014-03-19T00:00:00Z","doi":"10.1007/978-3-319-04099-8_7","page":"103 - 117","oa":1,"quality_controlled":"1","publisher":"Springer"},{"scopus_import":1,"publisher":"ACM","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1311.3681"}],"oa":1,"month":"06","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)."}],"oa_version":"Submitted Version","page":"355 - 364","doi":"10.1145/2582112.2582168","date_published":"2014-06-01T00:00:00Z","date_created":"2018-12-11T11:56:01Z","ec_funded":1,"year":"2014","publication_status":"published","day":"01","publication":"Proceedings of the Annual Symposium on Computational Geometry","language":[{"iso":"eng"}],"type":"conference","conference":{"start_date":"2014-06-08","location":"Kyoto, Japan","end_date":"2014-06-11","name":"SoCG: Symposium on Computational Geometry"},"status":"public","project":[{"name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"_id":"2153","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","last_name":"Bauer"},{"first_name":"Michael","full_name":"Lesnick, Michael","last_name":"Lesnick"}],"publist_id":"4853","title":"Induced matchings of barcodes and the algebraic stability of persistence","department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:38Z","citation":{"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.","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.","short":"U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–364.","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","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","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. 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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)."}],"oa_version":"Submitted Version","scopus_import":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1307.2839"}],"month":"06","citation":{"ama":"Bauer U, Ge X, Wang Y. Measuring distance between Reeb graphs. In: Proceedings of the Annual Symposium on Computational Geometry. 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SoCG: Symposium on Computational Geometry, 464–473.","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."},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer"},{"last_name":"Ge","full_name":"Ge, Xiaoyin","first_name":"Xiaoyin"},{"first_name":"Yusu","full_name":"Wang, Yusu","last_name":"Wang"}],"publist_id":"4850","title":"Measuring distance between Reeb graphs","project":[{"grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"year":"2014","day":"01","publication":"Proceedings of the Annual Symposium on Computational Geometry","page":"464 - 473","date_published":"2014-06-01T00:00:00Z","doi":"10.1145/2582112.2582169","date_created":"2018-12-11T11:56:02Z","acknowledgement":"National Science Foundation under grants CCF-1319406, CCF-1116258.","publisher":"ACM","quality_controlled":"1","oa":1},{"ec_funded":1,"language":[{"iso":"eng"}],"publication_status":"published","month":"06","scopus_import":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1312.1231"}],"oa_version":"Submitted Version","abstract":[{"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).","lang":"eng"}],"department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:38Z","status":"public","type":"conference","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2014-06-11","location":"Kyoto, Japan","start_date":"2014-06-08"},"_id":"2155","doi":"10.1145/2582112.2582167","date_published":"2014-06-01T00:00:00Z","date_created":"2018-12-11T11:56:01Z","page":"484 - 490","day":"01","publication":"Proceedings of the Annual Symposium on Computational Geometry","year":"2014","quality_controlled":"1","publisher":"ACM","oa":1,"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","title":"The morse theory of Čech and Delaunay filtrations","publist_id":"4851","author":[{"first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","last_name":"Bauer"},{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"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.","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.","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.","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","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"},"project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493","name":"Topological Complex Systems"}]},{"citation":{"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.","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.","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","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","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.","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."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","author":[{"last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Kerber, Michael","last_name":"Kerber","first_name":"Michael"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan"},{"first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert"}],"article_processing_charge":"No","title":"PHAT – Persistent Homology Algorithms Toolbox","quality_controlled":"1","publisher":"Springer Berlin Heidelberg","year":"2014","day":"01","publication":"ICMS 2014: International Congress on Mathematical Software","page":"137-143","doi":"10.1007/978-3-662-44199-2_24","date_published":"2014-09-01T00:00:00Z","date_created":"2022-03-21T07:12:16Z","_id":"10894","series_title":"LNCS","type":"conference","conference":{"name":"ICMS: International Congress on Mathematical Software","location":"Seoul, South Korea","end_date":"2014-08-09","start_date":"2014-08-05"},"status":"public","date_updated":"2023-09-20T09:42:40Z","department":[{"_id":"HeEd"}],"abstract":[{"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.","lang":"eng"}],"oa_version":"None","scopus_import":"1","month":"09","place":"Berlin, Heidelberg","intvolume":" 8592","publication_identifier":{"issn":["0302-9743"],"isbn":["9783662441985"],"eissn":["1611-3349"],"eisbn":["9783662441992"]},"publication_status":"published","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"later_version","status":"public","id":"1433"}]},"volume":8592},{"publist_id":"4072","author":[{"first_name":"Dominique","last_name":"Attali","full_name":"Attali, Dominique"},{"last_name":"Bauer","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich"},{"first_name":"Olivier","last_name":"Devillers","full_name":"Devillers, Olivier"},{"last_name":"Glisse","full_name":"Glisse, Marc","first_name":"Marc"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"}],"title":"Homological reconstruction and simplification in R3","department":[{"_id":"HeEd"}],"citation":{"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.","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.","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","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.","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."},"date_updated":"2023-02-23T10:15:15Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"conference","conference":{"location":"Rio de Janeiro, Brazil","end_date":"2013-06-20","start_date":"2013-06-17","name":"SoCG: Symposium on Computational Geometry"},"status":"public","_id":"2812","page":"117 - 125","doi":"10.1145/2462356.2462373","related_material":{"record":[{"relation":"later_version","status":"public","id":"1805"}]},"date_published":"2013-06-01T00:00:00Z","date_created":"2018-12-11T11:59:44Z","publication_status":"published","year":"2013","day":"01","publication":"Proceedings of the 29th annual symposium on Computational Geometry","language":[{"iso":"eng"}],"quality_controlled":"1","publisher":"ACM","scopus_import":1,"main_file_link":[{"open_access":"1","url":"http://hal.archives-ouvertes.fr/hal-00833791/"}],"oa":1,"month":"06","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."}],"oa_version":"Submitted Version","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)."}]