Faster Linear Systems and Matrix Norm Approximation via Multilevel Sketched Preconditioning
Abstract
We present a new class of preconditioned iterative methods for solving linear systems of the form $Ax = b$. Our methods are based on constructing a lowrank Nyström approximation to $A$ using sparse random sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of $A$, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any $n\times n$ linear system that is wellconditioned except for $k$ outlying large singular values in $\tilde{O}(n^{2.065} + k^\omega)$ time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all $k \gtrsim n^{0.78}$. 2. We give the first $\tilde{O}(n^2 + {d_\lambda}^{\omega}$) time algorithm for solving a regularized linear system $(A + \lambda I)x = b$, where $A$ is positive semidefinite with effective dimension $d_\lambda$. This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten $p$norms and other matrix norms. For example, for the Schatten 1 (nuclear) norm, we give an algorithm that runs in $\tilde{O}(n^{2.11})$ time, improving on an $\tilde{O}(n^{2.18})$ method of [Musco et al., ITCS 2018]. Interestingly, previous stateoftheart algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.05865
 arXiv:
 arXiv:2405.05865
 Bibcode:
 2024arXiv240505865D
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control