Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning
<|reference_start|>Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning: 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 low-rank Nystr\"om 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 Nystr\"om 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 well-conditioned except for $k$ outlying large singular values in $\tilde{O}(n^{2.065} + k^\omega)$ time, improving on a recent result of [Derezi\'nski, 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 state-of-the-art 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.<|reference_end|>
@article{dereziński2024faster,
    title={Faster Linear Systems and Matrix Norm Approximation via Multi-level
  Sketched Preconditioning},
    author={Micha{l} Derezi'nski, Christopher Musco and Jiaming Yang},
    journal={arXiv preprint arXiv:2405.05865},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.05865},
    primaryClass={cs.DS cs.LG cs.NA math.NA math.OC}
}