Grassmanian Interpolation of Low-Pass Graph Filters: Theory and Applications

2025-10-27

AI Summary

Computing low-pass graph filters over parametric graph families is computationally expensive due to repeated eigenvalue decompositions. Method: This paper introduces a novel low-frequency subspace interpolation method grounded in Riemannian geometry on the Grassmann manifold. It pioneers the use of normal coordinate interpolation for low-frequency subspace estimation, provides a theoretical error bound, incorporates a similarity correction mechanism to accommodate dynamic topology evolution, and constructs a dot-product-based graph-family-enhanced message-passing framework that enables node-feature-driven adaptive graph structure updates. Results: Experiments demonstrate substantial reduction in eigendecomposition overhead and improved message-passing performance on node classification tasks, effectively supporting dynamic graph modeling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Low-pass graph filters are fundamental for signal processing on graphs and other non-Euclidean domains. However, the computation of such filters for parametric graph families can be prohibitively expensive as computation of the corresponding low-frequency subspaces, requires the repeated solution of an eigenvalue problem. We suggest a novel algorithm of low-pass graph filter interpolation based on Riemannian interpolation in normal coordinates on the Grassmann manifold. We derive an error bound estimate for the subspace interpolation and suggest two possible applications for induced parametric graph families. First, we argue that the temporal evolution of the node features may be translated to the evolving graph topology via a similarity correction to adjust the homophily degree of the network. Second, we suggest a dot product graph family induced by a given static graph which allows to infer improved message passing scheme for node classification facilitated by the filter interpolation.

Problem

• Interpolating low-pass graph filters efficiently on Grassmann manifold
• Reducing eigenvalue problem computations for parametric graph families
• Developing applications for evolving graph topology and message passing

Innovation

• Riemannian interpolation on Grassmann manifold
• Error bound estimation for subspace interpolation
• Applications in evolving graph topology