Optimizing Polynomial Graph Filters: A Novel Adaptive Krylov Subspace Approach

TL;DR

The paper tackles the limited adaptability of polynomial graph filters across graphs with varying heterophily by unifying existing filters within the Krylov subspace framework and introducing an adaptive Krylov design. It develops a spectrum-controllable propagation matrix $\\mathbf{P}_\\tau$ from the graph heat equation to reshape the spectrum and proposes AdaptKry, a polynomial filter built from adaptive Krylov bases, with an extension to multiple bases that shares weights to avoid extra training cost. Theoretical results show that while AdaptKry retains the expressive power of traditional filters, spectrum shaping improves approximation of the optimal spectral filter, and convergence properties are governed by the Krylov basis and the second-largest eigenvalue of the propagation matrix. Empirically, AdaptKry achieves state-of-the-art node classification accuracy across six real-world datasets with diverse homophily levels, and ablations confirm the benefits of the adaptive parameter $\\tau$ and multi-basis extensions, highlighting practical impact for spectrum-aware graph learning.

Abstract

Graph Neural Networks (GNNs), known as spectral graph filters, find a wide range of applications in web networks. To bypass eigendecomposition, polynomial graph filters are proposed to approximate graph filters by leveraging various polynomial bases for filter training. However, no existing studies have explored the diverse polynomial graph filters from a unified perspective for optimization. In this paper, we first unify polynomial graph filters, as well as the optimal filters of identical degrees into the Krylov subspace of the same order, thus providing equivalent expressive power theoretically. Next, we investigate the asymptotic convergence property of polynomials from the unified Krylov subspace perspective, revealing their limited adaptability in graphs with varying heterophily degrees. Inspired by those facts, we design a novel adaptive Krylov subspace approach to optimize polynomial bases with provable controllability over the graph spectrum so as to adapt various heterophily graphs. Subsequently, we propose AdaptKry, an optimized polynomial graph filter utilizing bases from the adaptive Krylov subspaces. Meanwhile, in light of the diverse spectral properties of complex graphs, we extend AdaptKry by leveraging multiple adaptive Krylov bases without incurring extra training costs. As a consequence, extended AdaptKry is able to capture the intricate characteristics of graphs and provide insights into their inherent complexity. We conduct extensive experiments across a series of real-world datasets. The experimental results demonstrate the superior filtering capability of AdaptKry, as well as the optimized efficacy of the adaptive Krylov basis.

Figures (6)

• Figure 1: Illustrations of polynomial approximations to four common graph filters
• Figure 2: Accuracy of AdaptKry with varying $\tau$ values.
• Figure 3: Bases reshaping by $\tau$.
• Figure 4: Running times of orthgonal / non-orthogonal polynomial bases.
• Figure 5: Graph filters learnt by AdaptKry on $6$ graphs with various homophily ratios

Theorems & Definitions (8)

• Definition 1: Homophily Ratio $h$
• Proposition 1
• Definition 2: Grade of $\mathbf{x}\xspace$ with respect to $\mathbf{P}\xspace$ liesen2013krylov
• Proposition 2
• Theorem 1: Mixing time of Markov chain chung1997spectrallevin2017markov
• Theorem 2
• Theorem 3
• Theorem 4