A Stochastic PCA and SVD Algorithm with an Exponential Convergence Rate
This addresses efficiency issues in data analysis for practitioners, though it is incremental as it builds on existing variance-reduction techniques.
The paper tackles the problem of slow convergence or high computational cost in PCA and SVD algorithms by proposing VR-PCA, a stochastic algorithm that achieves exponential convergence to the optimal solution using cheap iterations.
We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis.