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. | Persistent Homology in Hockey
Stability Theorems I |
-
D. Goldfarb (2014): An Application of Topological Data Analysis to Hockey Analytics; online available on the arXiv[2] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer (2007): Stability of Persistence Diagrams; online available via doi [3] D. Cohen-Steiner, H. Edelsbrunner, J. Harer (2010): Lipschitz Functions Have L^p -Stable Persistence; online available via doi [4] |
Lukas Hahn
Daniel Spitz |
|
| 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 [5] | Maximilian Schmahl | |
| 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 [6] | Daniel Spitz | |
| 20.11. | Machine Learning & TDA I | Sebastian Damrich | ||
| 27.11. | ||||
| 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 | Clemens Fruboese | ||
| 18.12. | Structure Theorems | Michael Bleher
Aljosa Marjanovic |
||
| 08.01. | Machine Learning & TDA II | Sebastian Dobrzynski | ||
| 15.01. | Multipersistence I | Maximilian Neumann | ||
| ... | Multipersistence II | ... | Lukas Waas |
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
- ↑ Goldfarb, Daniel. "An application of topological data analysis to hockey analytics." https://arxiv.org/abs/1409.7635 (2014).
- ↑ 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
- ↑ Leygonie, Jacob, Steve Oudot, and Ulrike Tillmann. "A Framework for Differential Calculus on Persistence Barcodes." https://arxiv.org/abs/1910.00960 (2019).