We improve the running times of algorithms for least squares regression and low-rank approximation to account for the sparsity of the input matrix. Namely, if nnz(A) denotes the number of non-zero entries of an input matrix A:- we show how to solve approximate least squares regression given an n x d matrix A in nnz(A) + poly(d log n) time- we show how to find an approximate best rank-k approximation of an n x n matrix in nnz(A) + n*poly(k log n) time. All approximations are relative error. Previous algorithms based on fast Johnson-Lindenstrauss transforms took at least ndlog d or nnz(A)*k time. We have implemented our algorithms, and preliminary results suggest the algorithms are competitive in practice.
Joint work with Ken Clarkson.