Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points 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. 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：用于$d$维空间中$n$个数据点的流式主成分分析(PCA)的Oja算法，在$O(d)$空间和$O(nd)$时间以及单次遍历数据点的条件下，实现了与离线算法相同的正弦平方误差$O(r_{\mathsf{eff}}/n)$。此处$r_{\mathsf{eff}}$为有效秩（总体协方差矩阵$\Sigma$的迹与主特征值之比）。在此计算预算下，我们考虑稀疏PCA问题，其中$\Sigma$的主特征向量是$s$-稀疏的，且$r_{\mathsf{eff}}$可能很大。在此设置下，据我们所知，\textit{目前尚无已知的单遍算法}能够在$O(d)$空间和$O(nd)$时间内达到极小极大误差界，而无需强初始化条件或假设协方差矩阵的进一步结构（例如尖峰模型）。我们证明，一个简单的单遍处理流程——对Oja算法输出（Oja向量）进行阈值化——在某些正则性条件下，可以在$O(d)$空间和$O(nd)$时间内达到极小极大误差界。我们对未归一化Oja向量的分量进行了非平凡且新颖的分析，这涉及独立随机矩阵乘积在随机初始向量上的投影。这与先前对Oja算法和矩阵乘积的分析（均在$r_{\mathsf{eff}}$有界时进行）完全不同。