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Session I.4 - Computational Geometry and Topology - Semi-plenary talk

Monday, June 12, 16:30 ~ 17:30

Curse of dimensionality in persistence diagrams — How to characterize topology in single-cell resolution —

Yasu Hiraoka

Kyoto, Japan   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

It is well known that persistence diagrams stably behave under small perturbations to the input data. This is the consequence of stability theorems, firstly proved by Cohen-Steiner, Edelsbrunner, and Harer (2007), and then extended by several researchers. On the other hand, if the input data is realized in a high-dimensional space with a small noise, the curse of dimensionality (CoD) causes serious adverse effects on data analysis, especially leading to inconsistency of distances.

In this talk, I will show several examples of CoD appearing in persistence diagrams and mappers (e.g., from single-cell RNA sequencing data in biology). Those examples demonstrate that the classical stability theorems are not sufficient to guarantee stable behaviors of persistence diagrams for high-dimensional data. Then I will show several mathematical results about the existence and the (partial) resolution of CoD in persistence diagrams. This is a joint work with Enhao Liu, Yusuke Imoto and Shu Kanazawa.

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