Exponentially convergent stochastic k-PCA without variance reduction
This provides a more efficient solution for dimensionality reduction in streaming data scenarios, though it appears incremental as it builds on an existing method.
The authors tackled the problem of online k-PCA by generalizing Krasulina's method to matrices, showing that their algorithm achieves exponential convergence to the ground-truth principal subspace on low-rank data without needing variance reduction steps.
We present Matrix Krasulina, an algorithm for online k-PCA, by generalizing the classic Krasulina's method (Krasulina, 1969) from vector to matrix case. We show, both theoretically and empirically, that the algorithm naturally adapts to data low-rankness and converges exponentially fast to the ground-truth principal subspace. Notably, our result suggests that despite various recent efforts to accelerate the convergence of stochastic-gradient based methods by adding a O(n)-time variance reduction step, for the k-PCA problem, a truly online SGD variant suffices to achieve exponential convergence on intrinsically low-rank data.