We investigate theoretical and numerical properties of sparse sketching for both dense and sparse Linear Least Squares (LLS) problems. We show that, sketching with hashing matrices --- with one nonzero entry per column and of size proportional to the rank of the data matrix --- generates a subspace embedding with high probability, provided the given data matrix has low coherence; thus optimal residual values are approximately preserved when the LLS matrix has similarly important rows. We then show that using $s-$hashing matrices, with $s>1$ nonzero entries per column, satisfy similarly good sketching properties for a larger class of low coherence data matrices. Numerically, we introduce our solver Ski-LLS for solving generic dense or sparse LLS problems. Ski-LLS builds upon the successful strategies employed in the Blendenpik and LSRN solvers, that use sketching to calculate a preconditioner before applying the iterative LLS solver LSQR. Ski-LLS significantly improves upon these sketching solvers by judiciously using sparse hashing sketching while also allowing rank-deficiency of input; furthermore, when the data matrix is sparse, Ski-LLS also applies a sparse factorization to the sketched input. Extensive numerical experiments show Ski-LLS is also competitive with other state-of-the-art direct and preconditioned iterative solvers for sparse LLS, and outperforms them in the significantly over-determined regime.
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- Numerical Analysis Group Internal Seminar