Learning Graph from Smooth Signals under Partial Observation: A Robustness Analysis

2025-09-18

AI Summary

This work addresses the problem of learning the underlying graph structure from smooth graph signals when only a subset of node signals is observable—i.e., under partial observability with hidden nodes. To overcome the limitations of naive “ignore-hidden-nodes” approaches, we first extend the Restricted Isometry Property (RIP) to the Dirichlet energy functional and theoretically establish its implicit robustness under signal smoothness assumptions: accurate recovery of the true topology of the observable subgraph is guaranteed even from subsampled signals. Our method integrates graph signal smoothness modeling, Dirichlet energy minimization, and the GL-SigRep framework, tailored for low-pass filtered signals. Both theoretical analysis and experiments on synthetic and real-world datasets validate this robustness, revealing an intrinsic adaptability of smoothness-based graph learning methods to partial observations.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Social networks and social learning

Abstract

Learning the graph underlying a networked system from nodal signals is crucial to downstream tasks in graph signal processing and machine learning. The presence of hidden nodes whose signals are not observable might corrupt the estimated graph. While existing works proposed various robustifications of vanilla graph learning objectives by explicitly accounting for the presence of these hidden nodes, a robustness analysis of "naive", hidden-node agnostic approaches is still underexplored. This work demonstrates that vanilla graph topology learning methods are implicitly robust to partial observations of low-pass filtered graph signals. We achieve this theoretical result through extending the restricted isometry property (RIP) to the Dirichlet energy function used in graph learning objectives. We show that smoothness-based graph learning formulation (e.g., the GL-SigRep method) on partial observations can recover the ground truth graph topology corresponding to the observed nodes. Synthetic and real data experiments corroborate our findings.

Problem

• Analyzing robustness of naive graph learning under hidden nodes
• Extending RIP to Dirichlet energy for theoretical guarantees
• Recovering true graph topology from partial smooth signals

Innovation

• Robust graph learning from partial observations
• Extending RIP to Dirichlet energy function
• Recovering topology via smoothness-based formulations