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Short course on Topological Data Analysis (TDA)

 

Christian Hirsch, Aarhus University

will give a presentation on

Statistical foundations of topological data analysis

Dates :

  • Wednesday December 13, 2023, 10h45-12h45 and 14h-16h ISBA, Room c115
  • Thursday December 14, 2023, 10h45-12h45 and 14h-16h ISBA, Room c115

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Extended abstract:

Topological data analysis (TDA) is an emerging field at the interface between algebraic topology and data science. The philosophy behind TDA is to leverage invariants from algebraic topology to gain insights into data sets. While initially, TDA was developed and promoted by mathematicians, it is now applied in a variety of disciplines such as biology, chemistry, and materials science. The key tool in TDA is the persistence diagram, which captures the appearance and disappearance of topological features at multiple scales. The goal of this course is to explain how the tools provided by TDA can be combined with rigorous statistical methods to enable a scientific analysis of complex geometric data.

Lecture 1: The persistence diagram. We discuss the topological foundations of the persistence diagram and discuss its computability. We also present practical use cases with the GUDHI package.

Lecture 2: Distance-to-measure. While the persistence diagram can unearth refined geometric structures, it can be sensitive to the presence of outliers. We discuss the distance-to-measure as one way to make the persistence diagram more robust.

Lecture 3: Point processes. To carry out formal hypothesis testing, it is essential to work with well-defined null hypotheses. We introduce point processes as a flexible model for point clouds.

Lecture 4: Persistence-based summary statistics. We combine the methods from the first three lectures to define statistical tests based on the persistence diagram. These include simulation-based tests that are feasible for moderately large data as well as asymptotic approaches that become exact for large windows.

Registration (free for university members and students): please fill in the following form