Efficient matrix sketching over distributed data

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Conference Proceeding
Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, 2017, Part F127745 pp. 347 - 359
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© 2017 ACM. A sketch or synopsis of a large dataset captures vital properties of the original data while typically occupying much less space. In this paper, we consider the problem of computing a sketch of a massive data matrix A ϵ ℝn×d, which is distributed across a large number of s servers. Our goal is to output a matrix B ϵ ℝℓ × d which is significantly smaller than but still approximates A well in terms of covariance error, i.e., ||AT A - BT B||2. Here, for a matrix A, ||A||2 is the spectral norm of A, which is defined as the largest singular value of A. Following previous works, we call B a covariance sketch of A. We are mainly focused on minimizing the communication cost, which is arguably the most valuable resource in distributed computations. We show a gap between deterministic and randomized communication complexity for computing a covariance sketch. More specifically, we first prove a tight deterministic lower bound, then show how to bypass this lower bound using randomization. In Principle Component Analysis (PCA), the goal is to find a low-dimensional subspace that captures as much of the variance of a dataset as possible. Based on a well-known connection between covariance sketch and PCA, we give a new algorithm for distributed PCA with improved communication cost. Moreover, in our algorithms, each server only needs to make one pass over the data with limited working space.
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