Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

Heterogeneous manifolds for curvature-aware graph embedding

“…In general, homogeneous manifolds other than Euclidean space have already been investigated in deep architectures Huang et al (2018), for attention mechanisms Gulcehre et al (2018), general optimization frameworks Bonnabel (2013); Becigneul and Ganea (2018) and variational autoencoders Skopek et al (2019), to mention few. Our work continues the line of research of Wilson et al (2014); Nickel and Kiela (2017); ; Cruceru et al (2021); López et al (2021a) and generalizes these methods by proposing a new class of ambient spaces and embedding algorithms.…”

“…Homogeneity of existing models. In this paragraph we briefly review that the main existing models analysed recently in Nickel and Kiela (2017); ; Cruceru et al (2021); López et al (2021a) are indeed homogeneous and hence their curvature information cannot generally match the graph discrete one. In fact, as observed in López et al (2021a), the homogeneity follows from the stronger requirement of symmetry Petersen (2006)[Chapter 8.1]: symmetric spaces are characterized by the property that for each point p there exists an isometry fixing p whose Jacobian is minus the identity.…”

“…The homogeneity is a consequence of their Lie group structure (see Lin (2019) and Petersen (2006)[Chapter 8.2] respectively). Finally another class of symmetric spaces -Siegel manifolds -have been studied in López et al (2021a). In particular, we refer to the Appendix of López et al (2021a) for a more detailed discussion of symmetric manifolds and why they are generally advantageous in representation learning tasks López et al (2021b).…”

“…The existing graph embedding strategies in symmetric manifolds as per Nickel and Kiela (2017); ; Cruceru et al (2021); López et al (2021a) rely on Riemannian versions of stochastic gradient descent Bonnabel (2013). Accordingly, we need to compute Riemannian gradients of losses defined on manifolds.…”

“…Such findings have sparked interest in exploring different manifold classes such as products of constant curvature spaces , generalizing Wilson et al (2014)), and matrix manifolds Cruceru et al (2021), that could better accommodate the structural properties of graphs. Only very recently these approaches have been shown to be special instances of graph embeddings into symmetric manifolds López et al (2021a). This family of manifolds are usually amenable to optimization techniques, partly because all points 'look the same', a feature known as homogeneity.…”