Title: The Mathematics of UMAP-like Algorithms

Duration: until 30.06.2025

Research Area: Mathematical Foundations and Statistical Learning

We develop the mathematics from Riemannian and the metric geometry and category theory underlying machine learning algorithms for dimension reduction and clustering of metric data.

Aims

We aim to better understand and improve the geometric machine learning algorithms.

Problem

The decomposing and merging of metric spaces requires sophisticated tools from the category theory. We want to develop these tools and identify those places in the algorithm where choices or modifications are possible, and to find optimal such choices to improve the performance on empirical data sets.

Practical example

In a later stage of the project, we would like to apply our improved algorithms to neuroscience data, such as spike trains, hippocampal grid and place cells.

Outlook

We are improving one of the most widely used machine learning algorithms, UMAP. Potentially, this can be applied to all data sets investigated with UMAP.

Publications

• P. Joharinad, J. Jost, Mathematical principles of topological and geometric data analysis, monograph, series: Mathematics of Data, Springer, 2023.

Team

Lead

• Prof. Dr. Jürgen Jost

Team Members

• Parvaneh Joharinad
• Fatemeh Fahimi
• Lukas Barth
• Janis Keck

Partners

• MPI for Mathematics in the Sciences