Lazy stochastic principal component analysis

Stochastic principal component analysis (SPCA) has become a popular dimensionality reduction strategy for large, high-dimensional datasets. We derive a simplified algorithm, called Lazy SPCA, which has reduced computational complexity and is better suited for large-scale distributed computation. We prove that SPCA and Lazy SPCA find the same approximations to the principal subspace, and that the pairwise distances between samples in the lower-dimensional space is invariant to whether SPCA is executed lazily or not. Empirical studies find downstream predictive performance to be identical for both methods, and superior to random projections, across a range of predictive models (linear regression, logistic lasso, and random forests). In our largest experiment with 4.6 million samples, Lazy SPCA reduced 43.7 hours of computation to 9.9 hours. Overall, Lazy SPCA relies exclusively on matrix multiplications, besides an operation on a small square matrix whose size depends only on the target dimensionality.Comment: To be published in: 2017 IEEE International Conference on Data Mining Workshops (ICDMW

Preconditioned Data Sparsification for Big Data with Applications to PCA and K-means

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.Comment: 28 pages, 10 figure