Graph Neural Networks on Symmetric Positive Definite Manifold

Yu Wang, Xiaofeng Cao, Yi Chang

Primary Area: learning on graphs and other geometries & topologies

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Keywords: Non-Euclidean Geometry, Graph Neural Network, Symmetric Positive Definite Manifold, Log-Cholesky Metric

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Abstract:

Geometric deep learning equips graph neural networks (GNNs) with some symmetry aesthetics from its underlying principles, which draw the structural properties of graphs.
However, modeling in Euclidean or hyperbolic geometry, or even their combinations, usually hypothesizes that the graph nodes satisfy the preferred geometric properties, which ignores the actual graph structures.
This prompted us to consider a more solid expression to relieve the above significant hypothesis for the geometric graph embeddings.
In this study, we generalize the fundamental components of GNNs on the Symmetric Positive Definite (SPD) manifold, which could be approximately observed by the integration of Euclidean and non-Euclidean geometric structures.
This motivates us to reconstruct the GNNs with manifold-preserving linear transformation, neighborhood aggregation, non-linear activation, and multinomial logistic regression, in which the Log-Cholesky metric derives the closed-form Fréchet mean representation for neighborhood aggregation and computational tractability for learning geometric embeddings.
Experiments demonstrate that the SPDGNN can learn superior representations for grid and hierarchical node structures, leading to significant performance improvements in subsequent classifications compared to the Euclidean and Hyperbolic analogs.

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Submission Number: 7342