  We show that every symmetric normed space admits an efficient nearest
neighbor search data structure with doubly-logarithmic approximation.
Specifically, for every $n$, $d = n^{o(1)}$, and every $d$-dimensional
symmetric norm $\|\cdot\|$, there exists a data structure for
$\mathrm{poly}(\log \log n)$-approximate nearest neighbor search over
$\|\cdot\|$ for $n$-point datasets achieving $n^{o(1)}$ query time and
$n^{1+o(1)}$ space. The main technical ingredient of the algorithm is a
low-distortion embedding of a symmetric norm into a low-dimensional iterated
product of top-$k$ norms.
  We also show that our techniques cannot be extended to general norms.
