  We focus on \emph{row sampling} based approximations for matrix algorithms,
in particular matrix multipication, sparse matrix reconstruction, and
\math{\ell_2} regression. For \math{\matA\in\R^{m\times d}} (\math{m} points in
\math{d\ll m} dimensions), and appropriate row-sampling probabilities, which
typically depend on the norms of the rows of the \math{m\times d} left singular
matrix of \math{\matA} (the \emph{leverage scores}), we give row-sampling
algorithms with linear (up to polylog factors) dependence on the stable rank of
\math{\matA}. This result is achieved through the application of
non-commutative Bernstein bounds. Keywords: row-sampling; matrix
multiplication; matrix reconstruction; estimating spectral norm; linear
regression; randomized
