  Learning faithful graph representations as sets of vertex embeddings has
become a fundamental intermediary step in a wide range of machine learning
applications. We propose the systematic use of symmetric spaces in
representation learning, a class encompassing many of the previously used
embedding targets. This enables us to introduce a new method, the use of
Finsler metrics integrated in a Riemannian optimization scheme, that better
adapts to dissimilar structures in the graph. We develop a tool to analyze the
embeddings and infer structural properties of the data sets. For
implementation, we choose Siegel spaces, a versatile family of symmetric
spaces. Our approach outperforms competitive baselines for graph reconstruction
tasks on various synthetic and real-world datasets. We further demonstrate its
applicability on two downstream tasks, recommender systems and node
classification.
