Oja's Algorithm for Streaming Sparse PCA

Syamantak Kumar, Purnamrita Sarkar

Keywords: Streaming PCA, Oja's Algorithm, Sparse PCA, Support Recovery, Entrywise Bounds

TL;DR: We give a O(d) space, O(nd) time algorithm for Streaming Sparse PCA via a novel analysis of Oja's Algorithm followed by thresholding

Abstract:

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, *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.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 11592