Filter Learning for Subgraphs: Algebras and Performance Risk Bounds

2026-07-23

AI Summary

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.

Abstract

Graph signal processing tasks that leverage spectral information typically assume access to the complete graph topology, which is often unavailable in practice. We propose a systematic framework for subgraph filter learning (SFL), where subgraph-supported operators approximate ambient graph filters under partial observations. We formulate SFL as a statistical learning problem in which optimal subgraph operators are inherently data-dependent. To address the difficulty of directly estimating such operators, we develop a subgraph filter algebra based on distance-aware Laplacian constructions, defining a structured and controllable class of filters for effective approximation. We further establish performance risk bounds under the least squares loss, quantifying how well the learned operator approximates the restricted ambient mapping. Experiments real-world datasets show that, for SFL tasks, the proposed algebraic models consistently outperform polynomial filters, distribution-agnostic operators, and direct numerical filter learning baselines that attempt to recover the underlying structure from data.

Problem

• subgraph filter learning
• graph signal processing
• partial observations
• spectral information
• graph topology

Innovation

• subgraph filter learning
• graph signal processing
• filter algebra
• risk bounds
• distance-aware Laplacian