Geometry and Representation Learning

29. June 2021
Analysis of the structure of graphs for real world datasets, BIO-DISEASOME (left), CSPHD (center) and FACEBOOK (right). Our method distinguishes denser connected parts of the graphs (blue/green in BIO-DISEASOME and FACEBOOK) from lesser connected, tree-structure-like parts (yellow in CSPHD).

In recent years, interest in hyperbolic geometry has grown in the fields of data science and machine learning for representation learning via graph embeddings and for the construction of so-called hyperbolic neural networks.

Learning graph representations via low-dimensional embeddings is an important problem in machine learning since many situations (e.g., in linguistics, evolutionary biology, computer networks, etc.) involve data that have a hierarchical structure, such as a graph structure. Hyperbolic spaces are well-known to geometers as typically being better ambient spaces for embeddings of graphs than are Euclidean spaces because the latter do not have “as much room” for the exponential growth of many graphs and trees.

In 2020, a collaboration between the GRG and the NLP groups started to explore the more-intricate non-Euclidean geometries that arise from symmetric spaces. A first paper, in which we propose a systematic framework and metrics for learning graph embeddings in symmetric spaces, was published at the International Conference on Machine Learning (ICML 2021).

Contributors:

Michael Strube (NLP)
Anna Wienhard (GRG)

Member:

Federico López (NLP)

Publications:

“Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach” (accepted to ICML 2021 Conference) https://arxiv.org/abs/2106.04941
“Hermitian Symmetric Spaces for Graph Embeddings”, presentation in the NeurIPS 2020 workshop on Differential Geometry meets Deep Learning, arXiv:2105.05275
“Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices” (accepted for a spotlight presentation at NeurIPS2021) https://arxiv.org/abs/2110.13475   


Collaborators:

Maria Beatrice Pozzetti (Heidelberg University)
Steve Trettel (Stanford University)


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