Low rank approximation and regression in input sparsity time

@inproceedings{Clarkson2013LowRA,
title={Low rank approximation and regression in input sparsity time},
author={K. Clarkson and D. Woodruff},
booktitle={STOC '13},
year={2013}
}

We design a new distribution over poly(r ε<sup>-1</sup>) x n matrices S so that for any fixed n x d matrix A of rank r, with probability at least 9/10, SAx<sub>2</sub> = (1 pm ε)Ax<sub>2</sub> simultaneously for all x ∈ R<sup>d</sup>. Such a matrix S is called a <i>subspace embedding</i>. Furthermore, SA can be computed in O(nnz(A)) + ~O(r<sup>2</sup>ε<sup>-2</sup>) time, where nnz(A) is the number of non-zero entries of A. This improves over all previous subspace embeddings, which required at… CONTINUE READING