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arXiv · 2502.01397

Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations

Vladislav Trifonov·Ekaterina Muravleva·Ivan Oseledets
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Abstract

Graph Neural Networks (GNNs) have been proposed as a tool for learning sparse matrix preconditioners, which are key components in accelerating linear solvers. We present theoretical and empirical evidence that message-passing GNNs are fundamentally incapable of approximating 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 low cosine similarity ($\leq0.7$ in key cases) between predicted and reference factors. 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. Moreover, experiments demonstrate that overcoming non-locality alone is insufficient. Tailored architectures are necessary to capture the required dependencies since even a completely non-local Global Graph Transformer fails to match the proposed baselines.

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Vladislav Trifonov, Ekaterina Muravleva, Ivan Oseledets. 2025-02-03. Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations. https://arxiv.org/abs/2502.01397

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