Pseudo metrics between persistence modules can be defined starting from Noise Systems [1]. Such metrics are used to compare the modules directly or to extract stable vectorisations. While the stability property directly follows from the axioms of Noise Systems, finding algorithms or closed formulas to compute the distances or associated vectorizations is often a difficult problem, especially in the multi-parameter setting. In this seminar I will show how extra properties of Noise Systems can be used to define algorithms. In particular I will describe how to compute stable vectorisations with respect to Wasserstein distances [2]. Lastly I will discuss ongoing work (with D. Lundin and R. Corbet) for the computation of a geometric distance (the Volume Noise distance) and associated invariants on interval modules.
[1] M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam, S. Oberg. Multidimensional Persistence and Noise, (2016) Foundations of Computational Mathematics, Vol 17, Issue 6, pages 1367-1406. doi:10.1007/s10208-016-9323-y.
[2] J. Agerberg, A. Guidolin, I. Ren and M. Scolamiero. Algebraic Wasserstein distances and stable homological invariants of data. (2023) arXiv: 2301.06484.
Further Information
Martina Scolamiero is an Assistant Professor in Mathametics with specialization in Geometry and Mathematical Statistics in Artificial Intelligence.
Her research is in Applied and Computational Topology, mainly working on defining topological invariants which are suitable for data analysis, understanding their statistical properties and their applicability in Machine Learning. Martina is also interested in applications of topological methods to Neuroscience and Psychiatry.