Locality Sensitive Hashing for Set-Queries, Motivated by Group Recommendations

Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let $s(x,y)$ be the similarity between two points $x$ and $y$. We define a similarity between a set $Q$ and a point $x$ by aggregating the similarities $s(p,x)$ for all $p\in Q$. For example, we can take $s(p,x)$ to be the angular similarity between $p$ and $x$ (i.e., $1-{\angle (x,p)}/{\pi}$), and aggregate by arithmetic or geometric averaging, or taking the lowest similarity. We develop locality sensitive hash families and data structures for a large set of such arithmetic and geometric averaging similarities, and analyze their collision probabilities. We also establish an analogous framework and hash families for distance functions. Specifically, we give a structure for the euclidean distance aggregated by either averaging or taking the maximum. We leverage SLSH to solve a geometric extension of the approximate near neighbors problem. In this version, we consider a metric for which the unit ball is an ellipsoid and its orientation is specified with the query. An important application that motivates our work is group recommendation systems. Such a system embeds movies and users in the same feature space, and the task of recommending a movie for a group to watch together, translates to a set-query $Q$ using an appropriate similarity.