Abstract:
We prove that any real matrix $A$ contains a subset of at most $4k/\eps + 2k \log(k+1)$ rows whose span ``contains" a matrix of rank at most $k$ with error only $(1+\eps)$ times the error of the best rank-$k$ approximation of $A$. This leads to an algorithm to find such an approximation with complexity essentially $O(Mk/\eps)$, where $M$ is the number of nonzero entries of $A$. The algorithm maintains sparsity, and in the streaming model, it can be implemented using only $2(k+1)(\log(k+1)+1)$ passes over the input matrix. Previous algorithms for low-rank approximation use only one or two passes but obtain an additive approximation.