Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

TL;DR

This paper introduces a novel graph embedding method using symmetric spaces and Finsler-Riemannian metrics, improving structural adaptation and performance in graph reconstruction, recommendation, and classification tasks.

Contribution

It systematically applies symmetric spaces to graph embeddings and develops a Finsler-Riemannian approach, enhancing structural representation and task performance.

Findings

Outperforms baselines in graph reconstruction tasks.

Effective in recommender systems and node classification.

Provides tools for analyzing embedding structural properties.

Abstract

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.