Field choice problem in persistent homology

Abstract: Mathematically, persistent homology is formalized on the homology vector spaces of a filtration and linear maps between the vector spaces induced by the inclusion maps. By encoding the scale information in the filtration, we can capture the geometric features of data. The structure theorem of persistent homology ensures the existence and uniqueness of the interval decomposition. A persistence diagram is given by the interval decomposition. The diagram has the complete information about the algebraic structure of persistent homology.

When we fix the field of the homology vector spaces, the uniqueness of the decomposition is ensured. However, the uniqueness is broken when the field is changed. One easy example is a filtration including a Klein bottle. A more interesting example is given by a Möbius ring.

From the above examples, the following questions naturally arise.

What condition does ensure the independence of the choice of the field?

Is there an efficient algorithm to check the above condition?

How often does a persistence diagram change as the field changes?

The aim of our research is to answer the above questions. The result is published on Arxiv[1]. This is joint work with M. Yoshiwaki.

[1] Ippei Obayashi and Michio Yoshiwaki. Field choice problem in persistent homology. arXiv:1911.11350, 2019.

computational geometryalgebraic topologycombinatoricsgeometric topologyprobability

Audience: advanced learners

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Asia Pacific Seminar on Applied Topology and Geometry

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