AA5 – Variational Problems in Data-Driven Applications


AA5-3 (was EF1-18)

Manifold-Valued Graph Neural Networks

Project Heads

Christoph von Tycowicz, Gabriele Steidl

Project Members

Martin Hanik

Project Duration

01.01.2022 − 31.12.2023

Located at

FU Berlin


Geometry-aware, data-analytic approaches improve understanding and assessment of pathophysiological processes. In this project, we derive a new theoretical framework for deep neural networks that can cope with geometric data and apply it for classification of musculoskeletal illness from both shape and movement patterns. As part of the process, we develop the necessary software in Python and make it accessible through the Morphomatics library. The source code is freely available on GitHub, and there are tutorials that can be run live on Binder.


In an application of graph neural networks to manifolds, we studied how well the latter can predict cognitive measures from brain connectomes. The latter are graphs with edge weights determined by correlations in brain activity, and they can be thought of as symmetric positive-definite matrices, which constitute a cone-like manifold. In our work, we could show that (a) graph neural networks outperform state-of-the-art in the prediction of two common cognitive scores and (b) these results can sometimes even be improved by only using a learned representative subset of samples for training.


To generalize layers for manifold-valued features, we are investigating diffusion processes as they allow for information transfer in embedded graphs. In the figures below, you can see a graph that is embedded in the two-dimensional sphere and the flow of (other) vertices of a graph under a diffusion-like process.

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