Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral GCNs
This work addresses a specific limitation in graph learning for directed graphs, offering an incremental improvement by generalizing existing Laplacian matrices to better preserve graph topology.
The authors tackled the problem of handling directed graphs with antiparallel edges (digons) of different weights in spectral graph convolutional networks by introducing a quaternion-valued Laplacian, resulting in QuaterGCN achieving superior performance compared to state-of-the-art GCNs in scenarios where digon information is crucial.
We introduce QuaterGCN, a spectral Graph Convolutional Network (GCN) with quaternion-valued weights at whose core lies the Quaternionic Laplacian, a quaternion-valued Laplacian matrix by whose proposal we generalize two widely-used Laplacian matrices: the classical Laplacian (defined for undirected graphs) and the complex-valued Sign-Magnetic Laplacian (proposed to handle digraphs with weights of arbitrary sign). In addition to its generality, our Quaternionic Laplacian is the only Laplacian to completely preserve the topology of a digraph, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weights without reducing them to a single (directed or undirected) edge as done with other Laplacians. Experimental results show the superior performance of QuaterGCN compared to other state-of-the-art GCNs, particularly in scenarios where the information the digons carry is crucial to successfully address the task at hand.