Spectral Graph Coarsening Using Inner Product Preservation and the Grassmann Manifold

Ido Cohen, Ronen Talmon

Keywords: Graph coarsening, Grassmann manifold, Graph signals, Spectral methods

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

Abstract:

We propose a novel functorial graph coarsening method that preserves inner products between node features, a property often overlooked by existing approaches focusing primarily on structural fidelity.
By treating node features as functions on the graph and preserving their inner products, our method retains both structural and feature relationships, facilitating substantial benefits for downstream tasks.
To formalize this, we introduce the Inner Product Error (IPE), which quantifies how the inner products between node features are preserved.
Leveraging the underlying geometry of the problem on the Grassmann manifold, we formulate an optimization objective that minimizes the IPE, also for unseen smooth functions. We show that minimizing the IPE improves standard coarsening metrics, and illustrate our method’s properties through visual examples that highlight its clustering ability. Empirical results on benchmarks for graph coarsening and node classification show that our approach outperforms existing state-of-the-art methods.

Primary Area: General machine learning (supervised, unsupervised, online, active, etc.)

Submission Number: 16454