Spectral Graph Coarsening Using Inner Product Preservation and the Grassmann Manifold

Ido Cohen, Ronen Talmon

Keywords: Graph coarsening, Graph signal processing, Grassmann manifold, Node classification

TL;DR: A novel graph coarsening method that preserves inner products between node features and demonstrates superior performance on graph coarsening and node classification benchmarks.

Abstract:

In this work, we propose a new functorial graph coarsening approach that preserves inner products between node features.
Existing graph coarsening methods often overlook the mutual relationships between node features, focusing primarily on the graph structure.
By treating node features as functions on the graph and preserving their inner products, our method ensures that the coarsened graph retains both structural and feature relationships, facilitating substantial benefits for downstream tasks.
To this end, we present the Inner Product Error (IPE) that quantifies how well inner products between node features are preserved. By leveraging the underlying geometry of the problem on the Grassmann manifold, we formulate an optimization objective that minimizes the IPE, even for unseen smooth functions. We show that minimizing the IPE also promotes improvements in other standard coarsening metrics. We demonstrate the effectiveness of our method through visual examples that highlight its clustering ability. Additionally, empirical results on benchmarks for graph coarsening and node classification show superior performance compared to state-of-the-art methods.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 7891