Geometric Data Analysis

Recommended Prerequisites

Linear Algebra, Algorithms.

Teaching Methods

The lectures will be of expository nature, while promoting student participation and where the student is expected to develop on herhis own applications of the methods learned.

Learning Outcomes

In this curricular unit we introduce topological data analysis. Data can exist in high dimensional spaces, but concentrated around lowdimensional geometric structures that are important to uncover. These topological structures are challenging to explore using classical machine learning methods. These emergent methods, lying in the intersection of algebraic topology and computer science, have shown capable of identifying some of this geometric information. In this course some of these methods will be studied, namely clustering, persistent homology, reconstruction and data visualization. We aim to use these methods on real data. We aim to develop the following skills: knowledge of algorithms and mathematical results; research; learning autonomy; creative and critical thinking; implementation of computational methods.

Work Placement(s)

No

Syllabus

1. Introductory concepts (point clouds, graphs, connectedness, topological space / manifold, homeomorphism, homotopy)
2. Simplicial complexes (combinatorial structures in point clouds)
3. Clustering and dimension reduction
4. Homology and persistent homology (topological and statistical aspects)
5. Morse functions and Reeb graphs
6. Structural inference and reconstruction from data
7. Topological Data Analysis for visualization
8. Practical applications of the methods learned

Head Lecturer(s)

António Manuel Freitas Gomes Cunha Salgueiro

Assessment Methods

Assessment
Frequency: 25.0%
Project: 75.0%

Bibliography

Notas da disciplina.
Carlsson, G., Vejdemo-Johansson, M. (2022). Topological Data Analysis with Applications.
Dey, T., Wang, W. (2022). Computational Topology for Data Analysis.
Edelsbrunner, H. (2010). Computational topology : an introduction.
Hatcher, A. (2003). Algebraic Topology.
Schenk, H. (2022). Algebraic Foundations for Applied Topology and Data Analysis.