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Filtered covers in topological data analysis

Mortern Brun (Bergen)
Mandag, 15 maj, 2017, at 15:00-16:00, in Aud. D2 (1531-119)
I will begin the talk by sharing my understanding of the objectives of topological data analysis with emphasis on divisive covers, a method we have developed to construct filtered covers.

Given a point cloud P in a metric space, topological data analysis proceeds in two steps. First a filtered space is constructed from P, and secondly this filtered space is analyzed. Mostly by persistent homology but also by other means, like visual representations the graph given by its one-skeleton.

The most common way of constructing a filtered space associated to P is by constructing coverings of P whose associated nerves form a filtered space. A popular example of this is the filtered Cech complex obtained from covers by balls of growing size.

The defining properties of our filtered covers imply that their nerves are filtered simplicial complexes which are log-interleaved with the filtered Cech complex. This is similar to the Vietoris Rips complex being log-interleaved with the Cech complex.

I will present the divisive cover algorithm providing filtered covers. The advantage of divisive covers is that their nerves are quite small, and hence it is practical to compute their persistent homology, and to visualize their one-skeleton. Moreover, they are constructed top-down, so that if we disregard fine scale structure of the metric space, then we obtain a very small filtered simplicial complex.
Kontaktperson: Marcel Bökstedt