On the Approximation of Matrix Products and Positive Definite Matrices

In this paper, we introduce and analyze new randomized and deterministic algorithms to approximate the product of two matrices. In addition we provide what is, to the best of our knowledge, the first relative error bound for the Nyström approximation of quadratic forms. While deriving the proofs of the results, we highlight several new connections between matrix products, the Nyström extension and Schur complements. In addition, we see that using a sampling procedure similar to the recently introduced notion of volume sampling [1] yields good provable bounds, though the reasons underlying the use of this sampling technique are rather different in our case.