2021-03-22 | Richard Peng:Solving Sparse Linear Systems Faster than Matrix Multiplication



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-by-n linear system Ax=b is 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.

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))) non-zeros, where \kappa(A) is the condition number of A. For poly(n)-conditioned matrices with O(n) nonzeros, and the current value of \omega, the bit complexity of our algorithm to solve to within any 1/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-Giesbrecht-Giorgi-Storjohann-Villard ISSAC `06 `07] for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semi-random matrices.

Joint work with Santosh Vempala, manuscript at https://arxiv.org/abs/2007.10254.



322  10:00--11:00



Richard Peng is an assistant professor in the School of Computer Science at the Georgia Institute of Technology. His research interests are in the design, analysis, and implementation of efficient algorithms. These interests currently revolve around problems induced by practice that arise at the intersection of discrete, numerical, and randomized algorithms, and his representative results include linear systems solvers, max-flow/min-cut algorithms, and time/space efficient data structures for matchings, resistances, and matrices.

Richard received his BMath from Waterloo, PhD from CMU, and was a postdoc at MIT. Awards he received include the NSF Career Award, the 2011 Microsoft Research PhD Fellowship, the 2013 CMU SCS Distinguished Dissertation Award, and the 2021 SODA Best Paper Award. Richard is extensively involved with algorithmic problem solving outreach activities, including the North America Programming Camp and the DMOJ Online Judge.



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