Factorization-in-Loop: Proximal Fill-in Minimization for Sparse Matrix Reordering

The recent submission of 'Factorization-in-Loop: Proximal Fill-in Minimization for Sparse Matrix Reordering' to arXiv highlights a significant advancement in the field of computational mathematics, specifically in the optimization of LU factorization for large sparse matrices. Fill-ins, which are new nonzero elements that arise during this factorization process, can lead to increased memory usage and longer computation times, making their minimization crucial. The paper introduces a novel reordering network that minimizes the l1 norm of triangular factors to approximate the exact number of fill-ins, utilizing a graph encoder to effectively predict row or column node scores. This method addresses the NP-hard challenge of finding the optimal row or column permutation to minimize fill-ins, a problem that has previously lacked theoretical guarantees. By bridging the gap between predicted node scores and actual triangular factors through innovative reparameterization techniques, this resear…