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Invitation to Topological Data Analysis

Informacje ogólne

Kod przedmiotu: 1000-1M21TDA Kod Erasmus / ISCED: (brak danych) / (brak danych)
Nazwa przedmiotu: Invitation to Topological Data Analysis
Jednostka: Wydział Matematyki, Informatyki i Mechaniki
Grupy: Przedmioty monograficzne dla matematyki 2 stopnia
Punkty ECTS i inne: 6.00
zobacz reguły punktacji
Język prowadzenia: angielski
Rodzaj przedmiotu:

monograficzne

Skrócony opis:

Modern topology has found its way through computations into applications. In this lecture we will discuss basic tools of topological data analysis: homology and persistent homology theory, discrete

Morse theory, Reeb graphs and mapper algorithms and more. In particular, a number of applications

of those techniques will be highlighted. We will show how those methods can be integrated with

statistics and machine learning. By doing so will lay down solid theoretical foundations and learn

to use the techniques in practice.

Prior knowledge of algebraic topology is desired. Ability to program in Python or R is required.

Pełny opis:

In this lecture we will discuss the following topics:

Topological foundations

1. Overview of topology. Mathematical and computational preliminaries.

2. Approximating shapes: Complexes as examples of simple shapes that provide combinatorial

representations of complex spaces. Examples: simplicial, singular, point cloud-based, cubical,

regular CW complexes, nerve complexes etc. Introduce efficient data structures to store those

complexes.

3. Complexes from data: How to obtain complexes from trees, graphs etc (both abstract and

embedded). Relation to network theory, clustering coefficients and similar concepts.

Homotopy equivalence: homotopy equivalent to a sublevel set of single or multi varied functions.

Persistent homology

4. Chains and cycles as generalization of paths and cycles in graphs.

5. Z_2 (persistent) homology and cohomology, reduction algorithm.

6. Z-(co)homology and Smith Normal Form algorithm. Why persistence cannot be defined for non-field

coefficients.

7. Introduction to Discrete Morse Theory (DMT). Link of DMT and filtrations / persistent homology.

Iterated Morse complexes as a way to compute field (persistent) homology.

8. Computational homotopy groups - relation to group representation. How to get a simpler

representation using DMT.

9. Geometrical estimators of Rieman metric on a manifold, its reach and curvature from point clouds

Mapper-type algorithms

10. Reeb graphs, cover complexes and mapper type algorithms. Present it as a way to plot a relation

(x,f(x)) for x being a subsample of a high dimensional set.

11. Standard mapper

12. Ball mapper

Topology and Dynamics

13. Dynamical systems, topology, Wazewski principle, Conley index.

Applications

14. Brain function

15. Classification of neuron shapes, brain plasticity and response.

16. Classification of materials

17. Lung structure in COPD

18. Applications to economics and political sciences. Market prediction.

Literatura:

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

Efekty uczenia się:

After this lecture you will know the basic techniques of topological data analysis, you will be

able to use them in practice (via python) and apply to real-world problems. You will be able to

read basic papers in topological data analysis.

Metody i kryteria oceniania:

Programming or theoretical project on labs (50%).

Written and oral exam (50%).

Lab project + oral / written exam.

Zajęcia w cyklu "Semestr letni 2021/22" (jeszcze nie rozpoczęty)

Okres: 2022-02-21 - 2022-06-15
Wybrany podział planu:


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Typ zajęć: Laboratorium, 30 godzin więcej informacji
Wykład monograficzny, 30 godzin więcej informacji
Koordynatorzy: Paweł Dłotko
Prowadzący grup: Paweł Dłotko
Lista studentów: (nie masz dostępu)
Zaliczenie: Egzamin
Opisy przedmiotów w USOS i USOSweb są chronione prawem autorskim.
Właścicielem praw autorskich jest Uniwersytet Warszawski.