An Implicit Form of Krasulina's k-PCA Update without the Orthonormality Constraint

Abstract

We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly QR-decomposition step by bypassing the orthonormality constraint. We show that the new update in fact corresponds to an online EM step applied to a probabilistic $k$-PCA model. The probabilistic view of the updates allows us to combine multiple models in a distributed setting. We show experimentally that the implicit Krasulina update yields superior convergence while being significantly faster. We also give strong evidence that the new update can benefit from parallelism and is more stable w.r.t. tuning of the learning rate.