GraIP: A Benchmarking Framework For Neural Graph Inverse Problems

TL;DR

GraIP reframes diverse graph learning tasks as graph inverse problems by introducing a learnable inverse map $I$ and a differentiable forward map $F$, enabling end-to-end optimization across domains such as causal discovery, neural relational inference, and data-driven rewiring. The framework is instantiated in multiple settings, illustrating how priors, discretization strategies, and gradient estimators influence performance and identifiability, and a benchmark suite is provided to enable cross-domain comparisons. Key findings show that while a strong baseline combines an MPNN-based inverse with an I-MLE-like discretization, larger graphs face substantial ill-posedness and gradient-estimation challenges, motivating exploration of continuous relaxations and generative inverse models. Overall, GraIP offers a principled, transferable foundation for building graph inverse solvers that can bridge combinatorial, causal, and dynamic graph problems, driving future research toward robust, cross-domain structure learning.

Abstract

A wide range of graph learning tasks, such as structure discovery, temporal graph analysis, and combinatorial optimization, focus on inferring graph structures from data, rather than making predictions on given graphs. However, the respective methods to solve such problems are often developed in an isolated, task-specific manner and thus lack a unifying theoretical foundation. Here, we provide a stepping stone towards the formation of such a foundation and further development by introducing the Neural Graph Inverse Problem (GraIP) conceptual framework, which formalizes and reframes a broad class of graph learning tasks as inverse problems. Unlike discriminative approaches that directly predict target variables from given graph inputs, the GraIP paradigm addresses inverse problems, i.e., it relies on observational data and aims to recover the underlying graph structure by reversing the forward process, such as message passing or network dynamics, that produced the observed outputs. We demonstrate the versatility of GraIP across various graph learning tasks, including rewiring, causal discovery, and neural relational inference. We also propose benchmark datasets and metrics for each GraIP domain considered, and characterize and empirically evaluate existing baseline methods used to solve them. Overall, our unifying perspective bridges seemingly disparate applications and provides a principled approach to structural learning in constrained and combinatorial settings while encouraging cross-pollination of existing methods across graph inverse problems.

Figures (5)

• Figure 1: Overview of the GraIP framework. The input graph $G$ with optional node features $\mathbold{X}$ and target $y$ is fed into the inverse map ${I^{(\mathbold{\theta})}}$. This produces an intermediate solution graph $\tilde{G}$ with optional node features $\tilde{\mathbold{X}}$. The forward map $F^{(\mathbold{\varphi})}$ uses the intermediate solution to produce output $\hat{y}$, which is compared with $y$ using distance measure $d$ to compute the loss. The specific instantiation of each component depends on the domain. We show examples based on gene regulatory network inference and causal discovery.
• Figure 2: Overview of NRI as a graph inverse problem.
• Figure 3: Data-driven rewiring as a graph inverse problem.
• Figure 4: Overview of CO as an instance of the GraIP framework, using MIS as an example.
• Figure 5: Overview of GRN inference as in instance of the GraIP framework.