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Near Optimal Frequent Directions for Sketching Dense and Sparse Matrices

Zengfeng Huang; 20(56):1−23, 2019.


Given a large matrix $A\in\mathbb{R}^{n\times d}$, we consider the problem of computing a sketch matrix $B\in\mathbb{R}^{\ell\times d}$ which is significantly smaller than but still well approximates $A$. We consider the problems in the streaming model, where the algorithm can only make one pass over the input with limited working space, and we are interested in minimizing the covariance error $\|A^TA-B^TB\|_2.$ The popular Frequent Directions algorithm of \cite{liberty2013simple} and its variants achieve optimal space-error tradeoffs. However, whether the running time can be improved remains an unanswered question. In this paper, we almost settle the question by proving that the time complexity of this problem is equivalent to that of matrix multiplication up to lower order terms. Specifically, we provide new space-optimal algorithms with faster running times and also show that the running times of our algorithms can be improved if and only if the state-of-the-art running time of matrix multiplication can be improved significantly.

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