Topological Methods in Data Analysis - Journal Club (Winter 2019/20)
Topological Data Analysis (TDA) is a recent development in mathematics and data analysis that actually offers real world applications. The basic idea is using a homology theory - called persistent homology - to unveil and identify structures in data via a notion of its topology. However, interpreting these structures is by no means an easy task, and depends on the specific details of the underlying system.
In this Journal Club we will take a detailed look at foundational articles, specific applications and recent developments in the field, providing a link between the mathematical theory and scientific studies.
In case of questions, do not hesitate to contact us, the organizers of this Journal Club, via mail at structures-hiwi@mathi.uni-heidelberg.de.
Coordinates and Organization
Time: Wednesdays, from 9.15 am to 10.45 am.
Location: Mathematikon, Seminar Room 9.
Proceeding: Each week, a presenter discusses on the basis of the named paper(s) the given topic, taking into account further references whenever necessary. The presenter may set a focus following his or her individual interests.
Organizers: Michael Bleher, Daniel Spitz. Mail: structures-hiwi@mathi.uni-heidelberg.de.
Please consider registering on Müsli to join our mailing list.
Schedule
| Date | Topic | Article | Speaker | Slides |
|---|---|---|---|---|
| 16.10. | Organizational Meeting | Michael Bleher & Daniel Spitz | Slides | |
| 23.10. | The Mapper Algorithm and its Application | M. Nicolau, A. J. Levine, and G. Carlsson (2011). Topology Based Data Analysis Identifies a Subgroup of Breast Cancers with a Unique Mutational Profile and Excellent Survival; online available via doi [1] | Ruth Lang Fuentes | Slides |
| 30.10. | Stability Theorems I | D. Cohen-Steiner, H. Edelsbrunner, and J. Harer (2007). Stability of Persistence Diagrams; online available via doi [2]
D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko (2010): Lipschitz Functions Have L^p -Stable Persistence; online available via doi [3] |
Daniel Spitz | Essay |
| 6.11. | Stability Theorems II | U. Bauer, M. Lesnick (2016). Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem; online available on the arXiv [4] | Maximilian Schmahl | Notes |
| 13.11. | Persistent Homology applied to Dynamical Quantum Phenomena; including Statistics of Persistence Diagrams | Resources include E. Berry, Y.-C. Chen, J. Cisewski-Kehe, B. T. Fasy (2018): Functional Summaries of Persistence Diagrams; online available on the arXiv [5] | Daniel Spitz | |
| 20.11. | --- | --- | --- | --- |
| 27.11. | Machine Learning & TDA I | C.S. Pun, K. Xia, and S.X. Lee (2018). Persistent-Homology-based Machine Learning and its Applications--A Survey. arXiv [6] | Sebastian Damrich | Slides |
| 4.12. | Differential Calculus on Persistence Barcodes | J. Leygonie, S. Oudot, U. Tillmann (2019). A Framework for Differential Calculus on Persistence Barcodes; online available via arXiv[7] | Arnaud Maret | |
| 11.12. | Graph Embedding in Symmetric Spaces | M. Nickel, D. Kiela (2017). Poincaré Embeddings for Learning Hierarchical Representations. NIPS, 2017. arXiv[8]
B. Wilson and M. Leimeister (2018). Gradient descent in hyperbolic space. arXiv |
Clemens Fruboese | |
| 18.12. | Structure Theorems | M.B. Botnan and W. Crawley-Boevey (2019). Decomposition of persistence modules. arXiv[9] | Michael Bleher | |
| 08.01. | Machine Learning & TDA II (cancelled) | --- | --- | |
| 15.01. | Multipersistence I - Invariants | G. Carlsson and A. Zomorodian (2009). The Theory of Multidimensional Persistence. doi [10] (also see here for an earlier and substantially different version) | Maximilian Neumann | Essay |
| 22.01. | Multipersistence II - Real Multiparameter Persistence | Ezra Miller (2017). Data structures for real multiparameter persistence modules. arXiv | Lukas Waas | |
| 29.01. | Topological Exploration of Neuronal Morphologies (cancelled) | J.-B. Bardin, G. Spreemann and K. Hess (2018). Topological exploration of artificial neuronal network dynamics. arXiv | --- | |
| 05.02. | Künneth Theorem | H. Gakhar and J. A. Perea (2019). Künneth Formulae in Persistent Homology. arXiv | Johannes Feldmann |
Further Content
Interesting directions for later sessions include the discussion of further scientific TDA applications, as well as an introduction to multiparameter persistence, to the relation between machine learning and TDA, and a primer on discrete Morse theory.
In this Journal Club, further interesting literature and content proposals from the audience are welcome.
References
- ↑ Nicolau, Monica, Arnold J. Levine, and Gunnar Carlsson. ‘Topology Based Data Analysis Identifies a Subgroup of Breast Cancers with a Unique Mutational Profile and Excellent Survival’. Proceedings of the National Academy of Sciences of the United States of America 108, no. 17 (26 April 2011): 7265–70. https://doi.org/10.1073/pnas.1102826108
- ↑ Cohen-Steiner, David, Herbert Edelsbrunner, and John Harer. ‘Stability of Persistence Diagrams’. Discrete & Computational Geometry 37, no. 1 (1 January 2007): 103–20. https://doi.org/10.1007/s00454-006-1276-5.
- ↑ Cohen-Steiner, David, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko. ‘Lipschitz Functions Have L p -Stable Persistence’. Foundations of Computational Mathematics 10, no. 2 (April 2010): 127–39. https://doi.org/10.1007/s10208-010-9060-6.
- ↑ Bauer, Ulrich, and Michael Lesnick. ‘Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem’, 31 October 2016. https://arxiv.org/abs/1610.10085v2.
- ↑ Berry, Eric, Yen-Chi Chen, Jessi Cisewski-Kehe, and Brittany Terese Fasy. ‘Functional Summaries of Persistence Diagrams’. 4 April 2018. http://arxiv.org/abs/1804.01618
- ↑ Pun, Chi Seng, Kelin Xia, und Si Xian Lee. „Persistent-Homology-based Machine Learning and its Applications -- A Survey“. arXiv:1811.00252 [math], 1. November 2018. http://arxiv.org/abs/1811.00252.
- ↑ Leygonie, Jacob, Steve Oudot, and Ulrike Tillmann. "A Framework for Differential Calculus on Persistence Barcodes." https://arxiv.org/abs/1910.00960 (2019).
- ↑ Nickel, Maximillian, und Douwe Kiela. „Poincaré Embeddings for Learning Hierarchical Representations“. In Advances in Neural Information Processing Systems 30, 6338–6347. http://papers.nips.cc/paper/7213-poincare-embeddings-for-learning-hierarchical-representations
- ↑ Botnan, Magnus Bakke, und William Crawley-Boevey. „Decomposition of persistence modules“. arXiv:1811.08946 [math], 4. Oktober 2019. http://arxiv.org/abs/1811.08946
- ↑ Carlsson, G. & Zomorodian, A. Discrete Comput Geom (2009) 42: 71. https://doi.org/10.1007/s00454-009-9176-0