Localized sketching for matrix multiplication and ridge regression

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/ 2 ) and O(sd λ / ) 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.

Optimal Sampling Algorithms for Block Matrix Multiplication

“…c k and the last equality is based on Condition 2. Thus, combining (13) ( 15) and ( 16), by the Lindeberg-Feller central limit theorem [17,Proposition 2.27], we get (12).…”

“…They are easier to compute compared with (2) and (3). In addition, there are some other generalizations of the Ba-sicMatrixMultiplication algorithm [8,9] and some randomized algorithms for matrix multiplication based on random projection [10][11][12]. In particular, a block diagonal random projection method with different block sizes was developed in [12].…”