发表机构
University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出无特征间隙假设下流式PCA的Oja算法新分析，仅需二阶矩界实现近最优速率，并扩展至差分隐私应用，解决开放猜想。
AI 中文摘要
流式主成分分析（PCA）旨在对数据流进行单次遍历时恢复前导谱子空间。我们针对该问题最一般的无间隙变体，给出了对无处不在的Oja算法[Oja82]的新分析，其中对底层均值矩阵不作任何特征间隙假设，并辅以近乎匹配的下界。先前实现流式PCA近最优速率的工作要么需要间隙假设[JJK+16, HNWW21]，要么仅限于秩一更新[AZL17, Lia23]。我们的证明仅使用单个随机更新的二阶矩界，绕过了先前近最优分析所需的几乎必然界，以及相应的离线矩阵Bernstein界。我们还将结果扩展到Rayleigh商概念的近似PCA，解决了[JJK+16]的一个开放问题。作为主要应用，我们为次高斯数据提供了无间隙的差分隐私PCA保证，在对数因子范围内解决了[Bro26]的猜想1.1。
英文摘要
Streaming principal component analysis (PCA) seeks to recover a leading spectral subspace in a single pass over a data stream. We give a new analysis of the ubiquitous Oja's algorithm [Oja82] for the most general, gap-free variant of this problem, where no eigengap assumptions are made on the underlying mean matrix, complemented by a nearly-matching lower bound. Prior works achieving near-optimal rates for streaming PCA either required gap assumptions [JJK+16, HNWW21], or were limited to rank-one updates [AZL17, Lia23]. Our proof only uses a second moment bound on the individual stochastic updates, bypassing the almost sure bounds needed by prior near-optimal analyses, and the analogous offline matrix Bernstein bound. We also extend our result to a Rayleigh quotient notion of approximate PCA, addressing an open question of [JJK+16]. As our main application, we give gap-free differentially private PCA guarantees for sub-Gaussian data, settling Conjecture 1.1 of [Bro26] up to logarithmic factors.
