  We consider the problem of selecting non-zero entries of a matrix $A$ in
order to produce a sparse sketch of it, $B$, that minimizes $\|A-B\|_2$. For
large $m \times n$ matrices, such that $n \gg m$ (for example, representing $n$
observations over $m$ attributes) we give sampling distributions that exhibit
four important properties. First, they have closed forms computable from
minimal information regarding $A$. Second, they allow sketching of matrices
whose non-zeros are presented to the algorithm in arbitrary order as a stream,
with $O(1)$ computation per non-zero. Third, the resulting sketch matrices are
not only sparse, but their non-zero entries are highly compressible. Lastly,
and most importantly, under mild assumptions, our distributions are provably
competitive with the optimal offline distribution. Note that the probabilities
in the optimal offline distribution may be complex functions of all the entries
in the matrix. Therefore, regardless of computational complexity, the optimal
distribution might be impossible to compute in the streaming model.
