  Feature hashing and other random projection schemes are commonly used to
reduce the dimensionality of feature vectors. The goal is to efficiently
project a high-dimensional feature vector living in $\mathbb{R}^n$ into a much
lower-dimensional space $\mathbb{R}^m$, while approximately preserving
Euclidean norm. These schemes can be constructed using sparse random
projections, for example using a sparse Johnson-Lindenstrauss (JL) transform. A
line of work introduced by Weinberger et. al (ICML '09) analyzes the accuracy
of sparse JL with sparsity 1 on feature vectors with small
$\ell_\infty$-to-$\ell_2$ norm ratio. Recently, Freksen, Kamma, and Larsen
(NeurIPS '18) closed this line of work by proving a tight tradeoff between
$\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for sparse JL with sparsity
$1$.
  In this paper, we demonstrate the benefits of using sparsity $s$ greater than
$1$ in sparse JL on feature vectors. Our main result is a tight tradeoff
between $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for a general
sparsity $s$, that significantly generalizes the result of Freksen et. al. Our
result theoretically demonstrates that sparse JL with $s > 1$ can have
significantly better norm-preservation properties on feature vectors than
sparse JL with $s = 1$; we also empirically demonstrate this finding.
