Thresholded Oja does Sparse PCA?

We consider the problem of Sparse Principal Component Analysis (PCA) when the ratio $d/n \rightarrow c > 0$. There has been a lot of work on optimal rates on sparse PCA in the offline setting, where all the data is available for multiple passes. In contrast, when the population eigenvector is $s$-sparse, streaming algorithms that have $O(d)$ storage and $O(nd)$ time complexity either typically require strong initialization conditions or have a suboptimal error. We show that a simple algorithm that thresholds and renormalizes the output of Oja's algorithm (the Oja vector) obtains an optimal error rate. This is very surprising because, without thresholding, the Oja vector has a large error. Our analysis centers around bounding 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 nontrivial and novel since previous analyses of Oja's algorithm and matrix products have been done when the trace of the population covariance matrix is bounded while in our setting, this quantity can be as large as $n$.