Localized sketching for matrix multiplication and ridge regression
This work addresses efficient computation in distributed and streaming settings for machine learning practitioners, offering a more memory- and computationally-friendly approach while maintaining performance, though it is incremental as it adapts existing methods to a localized structure.
The paper tackles the problem of sketched approximate matrix multiplication and ridge regression under localized sketching, where only part of the data matrix is available at any point, and shows that block diagonal sketching matrices achieve sample complexities of O(stable rank / ε^2) for matrix multiplication and O(stat. dim. / ε) for ridge regression, matching state-of-the-art bounds from global sketching.
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(stable rank / ε^2) and $O( stat. dim. ε)$ 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.