Oja's algorithm for streaming Principal Component Analysis (PCA) for $n$ datapoints in a $d$ dimensional space achieves the same sin-squared error $O(r_\mathsf{eff}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_\mathsf{eff}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $\Sigma$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $\Sigma$ is $s$-sparse, and $r_\mathsf{eff}$ can be large. In this setting, to our knowledge, \textit{there are no known single-pass algorithms} that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time as long as $r_\mathsf{eff}=O(n/\log n)$. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_\mathsf{eff}$ is bounded.

翻译：Oja算法针对$d$维空间中$n$个数据点的流式主成分分析（PCA），在$O(d)$空间复杂度、$O(nd)$时间复杂度和单次数据遍历的条件下，达到了与离线算法相同的正弦平方误差$O(r_\mathsf{eff}/n)$。此处$r_\mathsf{eff}$表示有效秩（总体协方差矩阵$\Sigma$的迹与主特征值之比）。在此计算资源约束下，我们研究稀疏PCA问题，其中$\Sigma$的主特征向量具有$s$-稀疏性，且$r_\mathsf{eff}$可能较大。在此设定下，据我们所知，\textit{目前尚无已知的单遍算法}能在$O(d)$空间和$O(nd)$时间内达到极小极大误差界，且既不要求强初始化条件，也不假设协方差矩阵具有额外结构（如尖峰模型）。我们证明，在$r_\mathsf{eff}=O(n/\log n)$的条件下，通过对Oja算法输出（Oja向量）进行阈值处理的简单单遍程序，可在某些正则性条件下以$O(d)$空间和$O(nd)$时间达到极小极大误差界。我们对未归一化Oja向量的分量进行了新颖且非平凡的分析，这涉及独立随机矩阵乘积在随机初始向量上的投影。该分析与以往$r_\mathsf{eff}$有界情形下对Oja算法及矩阵乘积的研究截然不同。