  The \emph{Sparse Johnson-Lindenstrauss Transform} of Kane and Nelson (SODA
2012) provides a linear dimensionality-reducing map $A \in \mathbb{R}^{m \times
u}$ in $\ell_2$ that preserves distances up to distortion of $1 + \varepsilon$
with probability $1 - \delta$, where $m = O(\varepsilon^{-2} \log 1/\delta)$
and each column of $A$ has $O(\varepsilon m)$ non-zero entries. The previous
analyses of the Sparse Johnson-Lindenstrauss Transform all assumed access to a
$\Omega(\log 1/\delta)$-wise independent hash function. The main contribution
of this paper is a more general analysis of the Sparse Johnson-Lindenstrauss
Transform with less assumptions on the hash function. We also show that the
\emph{Mixed Tabulation hash function} of Dahlgaard, Knudsen, Rotenberg, and
Thorup (FOCS 2015) satisfies the conditions of our analysis, thus giving us the
first analysis of a Sparse Johnson-Lindenstrauss Transform that works with a
practical hash function.
