Invitation to Topological Data Analysis

Timetable

Module aim (aims)

Topology, a branch of mathematics that studies the qualitative properties of geometric objects, plays an important role in modern data analysis. Thanks to the dynamic development of computational topology methods,some topological invariants have been harnessed to solve data analysis problems. In this course, we will introduce basic topological concepts such as topological spaces, symplectic complexes and persistent homologies. In parallel, we will discuss ways of processing and storing geometric objects, and learn about topological methods of data visualization. During the exercise sessions, you will get acquainted with available programming libraries in Python. During the exercises, students will solve both theoretical and practical problems, including programming.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

Experience in Python programming will be useful for the practical part.

Minimal exposure to abstract algebra notions such as abelian groups, homomorphisms.

Syllabus

Week 1. Overview of topology: Mathematical and computational foundations. Metrics and similarity measures. Topology. metrics suitable for high dimensionality data, curse of dimensionality.

 

Week 2. Overview of data analysis methods, differences between qualitative, descriptive and statistical methods, clustering methods. High dimensionality data, dimensionality reduction and variable selection methods.

 

Week 3. Shape approximation: Cellular complexes for representing complex spaces. Symplicial complexes, singular complexes, point cloud based complexes, cube complexes, regular CW complexes. Introducing efficient data structures to store these complexes.

 

Week 4. Complexes from data: How to get complexes from trees, graphs, etc. (both abstract and inserted into Euclidean spaces). Relationship to network theory.

 

Topological invariants: unsolvability of homeomorphism type, homotopy groups, computational limitations.

 

Week 5. Chains and cycles as a generalization of paths and cycles in graphs, (persistent) homology and cohomology (especially with Z₂ coefficients), complex edge matrix reduction algorithm. Persistence diagrams, distances between them, further requirements for statistics, the need for vectorization.

 

Week 6. Motivation and limitations for multi-parameter persistent homology theory, Euler characteristic curves and profiles. consistency tests in statistics.

 

Week 7. The relationship of DMTs to filtering and persistent homology. Iterated Morse complexes as a way to calculate homology (persistent) with coefficients in bodies using Morse theory.

 

Week 8. Reeb graphs, covering complexes and mapper algorithms from the perspective of relation graphs (A.ee, f(A)), where A is a subset sampled from a high-dimensional set. Standard mapper and Ball mapper, ClusterGraph.

 

Week 9. Computational homotopy groups - relationship to group representation. How to get a simpler representation using discrete Morse theory (DMT).

 

Week 10. Geometric estimators of Riemannian metrics on manifolds, estimators of dimensionality, curvature and range from point clouds.

 

Week 11. Dynamical systems, topology, Ważewski's theorem, Conley's index.

 

Week 12. Applications: Research on brain function, classification of neuron shapes, classification of materials, lung structure in chronic obstructive pulmonary disease (COPD), applications in economics and political science, forecasting markets.

 

Week 13 – 15. Work on projects.

Reading list

• Herbert Edelsbrunner and John Harer, Computational Topology, an introduction, AMS 2011.
• Paweł Dłotko, Applied and computational topology Tutorial, https://arxiv.org/abs/1807.08607
• Mischaikow, Kaczynski, Mrozek, Computational Topology, Springer 2004.
• Gudhi library: gudhi.inria.fr
• Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255-308