Solving Sparse Linear Systems Faster than Matrix Multiplication
Abstract
Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear system $Ax=b$ is $\tilde{O}(n^\omega)$, where $\omega < 2.372864$ is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly$(n)$ condition number. In this paper, we present an algorithm that solves linear systems in sparse matrices asymptotically faster than matrix multiplication for any $\omega > 2$. This speedup holds for any input matrix $A$ with $o(n^{\omega 1}/\log(\kappa(A)))$ nonzeros, where $\kappa(A)$ is the condition number of $A$. For poly$(n)$conditioned matrices with $\tilde{O}(n)$ nonzeros, and the current value of $\omega$, the bit complexity of our algorithm to solve to within any $1/\text{poly}(n)$ error is $O(n^{2.331645})$. Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorizations. It is inspired by the algorithm of [Eberly et al. ISSAC `06 `07] for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anticoncentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semirandom matrices.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.10254
 Bibcode:
 2020arXiv200710254P
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Numerical Analysis