发表机构
University of Pennsylvania（宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对流式r-PCA，通过Oja算法解决一般秩的收敛与分布推断问题，建立了收敛速率、高斯近似及在线自助法，拓展了非凸随机近似的推断技术。
AI 中文摘要
我们通过Oja算法解决流式主成分分析（streaming PCA）中的两个开放问题：亚高斯数据下一般秩的算子范数收敛性，以及所得子空间估计量的分布推断。现有收敛分析（即使在秩1情形下）要么假设数据有界，要么存在非零余项，无法适配多项式消失的尾部谱；而现有分布结果仅局限于秩1情形。我们的收敛理论消除了这些余项，得到了精确速率。在密尖协方差 regime中，该速率与极小极大速率的差距仅为对数因子；更一般地，我们在温和的非退化条件下，于密尾和疏尾 regime中均证明了匹配的下界，差距仅为对数因子。该分析得到了Oja迭代的线性化，进而为一般秩子空间估计误差建立了高维高斯近似，带有显式极限协方差；我们还针对对齐差在凸集上建立了逐行高斯近似，将先前的秩1结果作为特例包含在内。为用于实际推断，我们开发了在线乘数自助法（online multiplier bootstrap）算法并证明其一致性。除流式PCA外，我们的技术还为非凸随机近似的高斯近似与自助法推断做出了贡献。
英文摘要
We address two open questions in streaming PCA via Oja's algorithm: sharp operator-norm convergence for general rank under sub-Gaussian data, and distributional inference for the resulting subspace estimator. Existing convergence analyses, even in the rank-one case, either assume bounded data or leave non-vanishing remainder terms that prevent adaptation to a polynomially vanishing tail spectrum, while existing distributional results are confined to the rank-one case. Our convergence theory removes these remainder terms and yields a sharp rate. In the dense-tail spiked covariance regime, this rate matches the minimax rate up to logarithmic factors. More generally, we prove a matching lower bound, up to logarithmic factors, across both dense-tail and sparse-tail regimes under a mild nondegeneracy condition. The analysis yields a linearization of Oja's iterates, which in turn enables a high-dimensional Gaussian approximation for the general-rank subspace estimation error with an explicit limiting covariance. We also establish a row-wise Gaussian approximation over convex sets for the aligned difference, recovering prior rank-one results as special cases. For practical inference, we develop an online multiplier bootstrap algorithm and prove its consistency. Beyond streaming PCA, our techniques contribute to Gaussian approximation and bootstrap inference for nonconvex stochastic approximation.
