The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

翻译：幂方法是数据中通过低秩矩阵逼近提取主要主成分的基础工具之一。然而，当目标秩较大时，该过程涉及的矩阵乘法成本成为主要瓶颈。我们提出了一种利用快速草图技术加速幂方法的算法与理论框架——这是随机线性代数中的常用范式。该框架为奇异值分解、低秩分解及Nyström逼近提供了简单且可证明高效的方法，在基准问题上展现出强劲的数值性能。我们分析的核心创新在于引入正则化谱逼近——快速草图方法的这一特性在推广幂方法保证方面比传统论证更具灵活性。