Project of the Applied Category Theory Adjoint School 2020
Mentor: Nina Otter
TAs: Barbara Giunti and Lukas Waas
Project description
Techniques and ideas from topology are being applied to the study of data with increasing frequency and success. One of the most well-known methods of this type is persistent homology, in which one associates a multiparameter family of spaces to a data set and studies how the holes evolve across the parameter space. The resulting algebraic object is called a persistence module. For one-parameter families of spaces, isomorphism classes of persistence modules can be completely classified by discrete invariants, while in the multiparameter case the situation is more delicate.
Distances between persistence modules are crucial both from a theoretical point of view — for instance, to study the stability of invariants associated to persistence modules —, as well as for applications. The standard distances used for one-parameter persistence modules are L^p distances called Wasserstein distances. For p = ∞ this distance is also called bottleneck distance, and it has arguably been the most used and studied distance for persistence modules. While the original definition is combinatorial, it has equivalent categorical and algebraic formulations called interleaving distance [2, 3].
In this project we will study different ways to give diagrammatic and algebraic formulations of Wasserstein distances for multiparameter persistence modules. Our motivation will be to investigate the stability of the invariants for multiparameter persistence modules introduced in [4], and our point of departure will be [1], in which the authors study how to generalise Wasserstein distances for p < ∞.
Readings:
- P. Bubenik, J. A. Scott, and D. Stanley, Wasserstein distance for generalized persistence modules and abelian categories (2018). Preprint, available on arXiv:1809.09654.
- P. Bubenik, J. A. Scott, Categorification of Persistent Homology, Discrete & Computational Geometry, 51(3):600–627 (2014), https://doi.org/10.1007/s00454-014-9573-x.
- F. Chazal, D. Cohen-Steiner, M. Glisse, L.J. Guibas, and S. Oudot, Proximity of persistence modules and their diagrams, Proceedings of SCG ’09, 237–246 (2009), https://doi.org/10.1145/1542362.1542407.
- H. Harrington, N. Otter, H. Schenck, U. Tillmann, Stratifying multiparameter persistent homology, SIAM Journal on Applied Algebraic Geometry, 3(3):439–471 (2019), https://doi.org/10.1137/18M1224350.
The selected readings for the presentations will be [1] and [4].