  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 / \epsilon^2) and $O(
stat. dim. \epsilon)$ 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.
