Sample complexity bounds for localized sketching

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Rakshith Sharma Srinivasa, Mark Davenport, Justin Romberg ;
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:3275-3284, 2020.

Abstract

We consider sketched approximate matrix multiplication and ridge regression in the novel setting of localized sketching, where at any given point, only part of the data matrix is available. This corresponds to a block diagonal structure on the sketching matrix. We show that, under mild conditions, block diagonal sketching matrices require only $O(\sr / \epsilon^2)$ and $O(\sd_{\lambda}/\epsilon)$ total sample complexity for matrix multiplication and ridge regression, respectively. This matches the state-of-the-art bounds that are obtained using global sketching matrices. The localized nature of sketching considered allows for different parts of the data matrix to be sketched independently and hence is more amenable to computation in distributed and streaming settings and results in a smaller memory and computational footprint.

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