TR06-042 Authors: Amit Deshpande, Santosh Vempala

Publication: 21st March 2006 14:40

Downloads: 2919

Keywords:

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.