Using a Non-Commutative Bernstein Bound to Approximate Some Matrix Algorithms in the Spectral Norm

We focus on row sampling based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and l2 regression. For A ∈ R (m points in d ≪ m dimensions), and appropriate row-sampling probabilities, which typically depend on the norms of the rows of the m × d left singular matrix of A (the leverage scores), we give row-sampling algorithms with linear (up to polylog factors) dependence on the stable rank of A. This result is achieved through the application of non-commutative Bernstein bounds.