Improved Distributed Principal Component Analysis by Maria-Florina Balcan, Vandana Kanchanapally, Yingyu Liang, David Woodruff

We study the distributed computing setting in which there are multiple servers, each holding a set of points, who wish to compute functions on the union of their point sets. A key task in this setting is Principal Component Analysis (PCA), in which the servers would like to compute a low dimensional subspace capturing as much of the variance of the union of their point sets as possible. Given a procedure for approximate PCA, one can use it to approximately solve problems such as $k$-means clustering and low rank approximation. The essential properties of an approximate distributed PCA algorithm are its communication cost and computational efficiency for a given desired accuracy in downstream applications. We give new algorithms and analyses for distributed PCA which lead to improved communication and computational costs for $k$-means clustering and related problems. Our empirical study on real world data shows a speedup of orders of magnitude, preserving communication with only a negligible degradation in solution quality. Some of these techniques we develop, such as a general transformation from a constant success probability subspace embedding to a high success probability subspace embedding with a dimension and sparsity independent of the success probability, may be of independent interest.

related paper:

This paper proposes a distributed PCA algorithm, with the theoretical guarantee that any good approximation solution on the projected data for k-means clustering is also a good approximation on the original data, while the projected dimension required is independent of the original dimension. When combined with the distributed coreset-based clustering approach in [3], this leads to an algorithm in which the number of vectors communicated is independent of the size and the dimension of the original data. Our experiment results demonstrate the effectiveness of the algorithm.

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