تقرير
Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning
| العنوان: | Faster Linear Systems and Matrix Norm Approximation via Multi-level Sketched Preconditioning |
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| المؤلفون: | Dereziński, Michał, Musco, Christopher, Yang, Jiaming |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Computer Science - Data Structures and Algorithms, Computer Science - Machine Learning, Mathematics - Numerical Analysis, Mathematics - Optimization and Control |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.05865 |
| رقم الأكسشن: | edsarx.2405.05865 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |