### TCS+ talk: Thursday, September 17 — Richard Peng, Georgia Tech

The next TCS+ talk will take place this coming Thursday, September 17th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Richard Peng from Georgia Tech will speak about “Solving Sparse Linear Systems Faster than Matrix Multiplication” (abstract below).

You can reserve a spot as an individual or a group to join us live by signing up on the online form. Due to security concerns, registration is required to attend the interactive talk. (The link to the YouTube livestream will also be posted on our website on the day of the talk, so people who did not sign up will still be able to watch the talk live.) As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

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 $\text{poly}(n)$ 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 $\text{poly}(n)$-conditioned matrices with $O(n)$ nonzeros, and the current value of $\omega$, the bit complexity of our algorithm to solve to within any $1/\text{poly}(n)$ 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-Giesbrecht-Giorgi-Storjohann-Villard ISSAC 0607] 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.