Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizations

ArXi:2502.01397v3 Announce Type: replace-cross 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.