改进的噪声幂方法的间隙依赖性分析
An Improved Gap-Dependency Analysis of the Noisy Power Method
AI总结：
本文改进了噪声幂方法对谱间隙的依赖性，通过引入中间参数q，提升了样本复杂度和噪声容忍度的界限，应用于分布式隐私PCA和内存高效流PCA。
摘要：
我们考虑了在机器学习和统计中广泛应用的噪声幂方法，尤其是在资源受限下的主成分分析（PCA）中。现有分析显示噪声幂方法对输入数据矩阵的连续谱间隙(σ_k-σ_{k+1})存在不满意的依赖性，这可能非常小，从而限制了算法的应用。本文提出了一种新的噪声幂方法分析，实现了样本复杂度和噪声容忍度界限的改进依赖性。具体而言，我们将对(σ_k-σ_{k+1})的依赖性改进为对(σ_k-σ_{q+1})的依赖性，其中q是一个中间算法参数，可能远大于目标秩k。我们的证明基于对两个子空间接近性的新特征化，这不同于之前工作中分析的canonical angle特征化。最后，我们将改进的界限应用于分布式隐私PCA和内存高效的流PCA，并获得了优于现有文献结果的界限。
英文摘要：
We consider the noisy power method algorithm, which has wide applications in machine learning and statistics, especially those related to principal component analysis (PCA) under resource (communication, memory or privacy) constraints. Existing analysis of the noisy power method shows an unsatisfactory dependency over the "consecutive" spectral gap $(σ_k-σ_{k+1})$ of an input data matrix, which could be very small and hence limits the algorithm's applicability. In this paper, we present a new analysis of the noisy power method that achieves improved gap dependency for both sample complexity and noise tolerance bounds. More specifically, we improve the dependency over $(σ_k-σ_{k+1})$ to dependency over $(σ_k-σ_{q+1})$, where $q$ is an intermediate algorithm parameter and could be much larger than the target rank $k$. Our proofs are built upon a novel characterization of proximity between two subspaces that differ from canonical angle characterizations analyzed in previous works. Finally, we apply our improved bounds to distributed private PCA and memory-efficient streaming PCA and obtain bounds that are superior to existing results in the literature.
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。
