Learning Regularization for Graph Inverse Problems

TL;DR

This work addresses the challenge of Graph Inverse Problems (GRIP), where states on graphs must be inferred from indirect and possibly noisy observations via the forward map $F({\bf x},{\cal G})$, with optional meta-data ${\bf f}_{\rm M}$. It proposes a unified GRIP framework that blends data fidelity with learned graph priors using Graph Neural Networks, introducing neural regularizers such as Var-GNN, ISS-GNN, and Prox-GNN. The paper defines four representative GRIPs—Property Completion, Inverse Source Estimation, Inverse Graph Transport, and Edge Property Recovery—and demonstrates across multiple datasets that neural graph regularization outperforms classical approaches, with meta-data providing additional gains. Overall, the framework enables robust reconstruction on graphs with changing topology and rich auxiliary information, broadening the applicability of GNNs to inverse problems.

Abstract

In recent years, Graph Neural Networks (GNNs) have been utilized for various applications ranging from drug discovery to network design and social networks. In many applications, it is impossible to observe some properties of the graph directly; instead, noisy and indirect measurements of these properties are available. These scenarios are coined as Graph Inverse Problems (GRIP). In this work, we introduce a framework leveraging GNNs to solve GRIPs. The framework is based on a combination of likelihood and prior terms, which are used to find a solution that fits the data while adhering to learned prior information. Specifically, we propose to combine recent deep learning techniques that were developed for inverse problems, together with GNN architectures, to formulate and solve GRIP. We study our approach on a number of representative problems that demonstrate the effectiveness of the framework.

Figures (5)

• Figure 1: The forward problem of graph density completion.
• Figure 2: A source graph on the left is diffused over time, with transition matrix ${\bf P}^k$ obtaining a target graph on the right. The goal of the inverse problem is to identify the source given the target.
• Figure 3: The graph transport problem. The four sampled paths of length $3$ on the right are obtained from the graph on the left. The data is the sum of the node properties.
• Figure 4: A known source graph on the left is diffused over time, with an unknown transition matrix ${\bf P}^k$ determined by the edge weights, obtaining the target graph on the right. The goal of the inverse problem is to identify the edge weights given the source and the target.
• Figure 5: A qualitative comparison of different methods on the property completion problem on ShapeNet. Grey points in Observed Data indicate masked/unseen points. Neural approaches consistently outperform classical regularizers.