# ARC Colloquium: Richard Peng (Georgia Tech)

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• Date/Time:
• Monday August 31, 2020
11:00 am - 12:00 pm
• Location: Virtual via Bluejeans
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Summary Sentence: Solving Sparse Linear Systems Faster than Matrix Multiplication: Virtual via Bluejeans at 11:00am

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Algorithms & Randomness Center (ARC)

Richard Peng (Georgia Tech)

Monday, August 31, 2020

Virtual via Bluejeans - 11:00 am

Title: 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-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 et al. 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.

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