Adaptive Sampling and Fast Low-Rank Matrix Approximation

We prove that any real matrix A contains a subset of at most 4k/e+ 2k log(k+1) rows whose span “contains” a matrix of rank at most k with error only (1+e) times the error of the best rank-k approximation of A. We complement it with an almost matching lower bound by constructing matrices where the span of any k/2e rows does not “contain” a relative (1+e)-approximation of rank k. Our existence result leads to an algorithm that finds such rank-k approximation in time $ O \left( M \left( \frac{k}{\epsilon} + k^{2} \log k \right) + (m+n) \left( \frac{k^{2}}{\epsilon^{2}} + \frac{k^{3} \log k}{\epsilon} + k^{4} \log^{2} k \right) \right), $ i.e., essentially O(Mk/e), where M is the number of nonzero entries of A. The algorithm maintains sparsity, and in the streaming model [12,14,15], it can be implemented using only 2(k+1)(log(k+1)+1) passes over the input matrix and $O \left( \min \{ m, n \} (\frac{k}{\epsilon} + k^{2} \log k) \right)$ additional space. Previous algorithms for low-rank approximation use only one or two passes but obtain an additive approximation.