Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations
This identifies a fundamental limitation for applying GNNs to scientific computing tasks like matrix factorization, requiring architectural innovations beyond message-passing.
This position paper argues that message-passing GNNs are fundamentally incapable of approximating sparse triangular factorizations, demonstrating severe performance degradation with cosine similarity dropping below 0.6 compared to exact factorizations.
Graph Neural Networks (GNNs) have been proposed as a tool for learning sparse matrix preconditioners, which are key components in accelerating linear solvers. This position paper argues that message-passing GNNs are fundamentally incapable of approximating sparse triangular factorizations. We demonstrate that message-passing GNNs fundamentally fail to approximate sparse triangular factorizations for classes of matrices for which high-quality preconditioners exist but require non-local dependencies. To illustrate this, we construct a set of baselines using both synthetic matrices and real-world examples from the SuiteSparse collection. Across a range of GNN architectures, including Graph Attention Networks and Graph Transformers, we observe severe performance degradation compared to exact or K-optimal factorizations, with cosine similarity dropping below $0.6$ in key cases. Our theoretical and empirical results suggest that architectural innovations beyond message-passing are necessary for applying GNNs to scientific computing tasks such as matrix factorization. Experiments demonstrate that overcoming non-locality alone is insufficient. Tailored architectures are necessary to capture the required dependencies since even a completely non-local Graph Transformer fails to match the proposed baselines.