Polytopes, lattices, and spherical codes for the nearest neighbor problem

Abstract

We study locality-sensitive hash methods for the nearest neighbor problem for the angular distance, focusing on the approach of first projecting down onto a low-dimensional subspace, and then partitioning the projected vectors according to Voronoi cells induced by a suitable spherical code. This approach generalizes and interpolates between the fast but suboptimal hyperplane hashing of Charikar [STOC'02] and the asymptotically optimal but practically often slower hash families of Andoni-Indyk [FOCS'06], Andoni-Indyk-Nguyen-Razenshteyn [SODA'14] and Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt [NIPS'15]. We set up a framework for analyzing the performance of any spherical code in this context, and we provide results for various codes from the literature, such as those related to regular polytopes and root lattices. Similar to hyperplane hashing, and unlike cross-polytope hashing, our analysis of collision probabilities and query exponents is exact and does not hide order terms which vanish only for large $d$, facilitating an easy parameter selection. For the two-dimensional case, we derive closed-form expressions for arbitrary spherical codes, and we show that the equilateral triangle is optimal, achieving a better performance than the two-dimensional analogues of hyperplane and cross-polytope hashing. In three and four dimensions, we numerically find that the tetrahedron, $5$-cell, and $16$-cell achieve the best query exponents, while in five or more dimensions orthoplices appear to outperform regular simplices, as well as the root lattice families $A_k$ and $D_k$. We argue that in higher dimensions, larger spherical codes will likely exist which will outperform orthoplices in theory, and we argue why using the $D_k$ root lattices will likely lead to better results in practice, due to a better trade-off between the asymptotic query exponent and the concrete costs of hashing.