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The aim of this paper is two-fold: firstly, to present subspace embeddingproperties for $s$-hashing sketching matrices, with $s\geq 1$, that are optimalin the projection dimension $m$ of the sketch, namely, $m=\mathcal{O}(d)$,where $d$ is the dimension of the subspace. A diverse set of results arepresented that address the case when the input matrix has sufficiently lowcoherence (thus removing the $\log^2 d$ factor dependence in $m$, in thelow-coherence result of Bourgain et al (2015) at the expense of a smallercoherence requirement); how this coherence changes with the number $s$ ofcolumn nonzeros (allowing a scaling of $\sqrt{s}$ of the coherence bound), oris reduced through suitable transformations (when considering hashed -- insteadof subsampled -- coherence reducing transformations such as randomisedHadamard). Secondly, we apply these general hashing sketching results to thespecial case of Linear Least Squares (LLS), and develop Ski-LLS, a genericsoftware package for these problems, that builds upon and improves theBlendenpik solver on dense input and the (sequential) LSRN performance onsparse problems. In addition to the hashing sketching improvements, we addsuitable linear algebra tools for rank-deficient and for sparse problems thatlead Ski-LLS to outperform not only sketching-based routines on randomlygenerated input, but also state of the art direct solver SPQR and iterativecode HSL on certain subsets of the sparse Florida matrix collection; namely, onleast squares problems that are significantly overdetermined, or moderatelysparse, or difficult.

Hashing embeddings of optimal dimension, with applications to linear least squares