Approximate Near Neighbors for General Symmetric Norms
  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. 
Alexandr Andoni, Huy L. Nguyen, Aleksandar Nikolov, Ilya Razenshteyn,   Erik Waingarten
cs.DS cs.CG cs.LG math.MG