LazySVD：即使更快的SVD分解也无需痛苦
LazySVD: Even Faster SVD Decomposition Yet Without Agonizing Pain
AI总结：
本文提出LazySVD框架，改进了k-SVD的突破性方法，实现了更快的无间隙方法，以及首个加速和随机方法，在特定参数范围内优于现有算法。
摘要：
本文研究了$k$-SVD，旨在获得矩阵$A$的第一个$k$个奇异向量。最近，$k$-SVD上发现了一些突破：Musco和Musco[1]利用块Krylov方法证明了首个无间隙收敛结果，Shamir[2]发现了首个方差缩减随机方法，Bhojanapalli等人[3]利用交替最小化提供了最快的$O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$-时间算法。本文提出了一种新的简单LazySVD框架以改进上述突破。该框架导致了一个更快的无间隙方法，优于[1]，并且首个加速和随机方法，优于[2]。在$O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$运行时间范围内，LazySVD在某些参数范围内优于[3]，甚至不使用交替最小化。
英文摘要：
We study $k$-SVD that is to obtain the first $k$ singular vectors of a matrix $A$. Recently, a few breakthroughs have been discovered on $k$-SVD: Musco and Musco [1] proved the first gap-free convergence result using the block Krylov method, Shamir [2] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [3] provided the fastest $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$-time algorithm using alternating minimization. In this paper, we put forward a new and simple LazySVD framework to improve the above breakthroughs. This framework leads to a faster gap-free method outperforming [1], and the first accelerated and stochastic method outperforming [2]. In the $O(\mathsf{nnz}(A) + \mathsf{poly}(1/\varepsilon))$ running-time regime, LazySVD outperforms [3] in certain parameter regimes without even using alternating minimization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
