On the Problem of $p_1^{-1}$ in Locality-Sensitive Hashing

A Locality-Sensitive Hash (LSH) function is called $(r,cr,p_1,p_2)$-sensitive, if two data-points with a distance less than $r$ collide with probability at least $p_1$ while data points with a distance greater than $cr$ collide with probability at most $p_2$. These functions form the basis of the successful Indyk-Motwani algorithm (STOC 1998) for nearest neighbour problems. In particular one may build a $c$-approximate nearest neighbour data structure with query time $\tilde O(n^\rho/p_1)$ where $\rho=\frac{\log1/p_1}{\log1/p_2}\in(0,1)$. That is, sub-linear time, as long as $p_1$ is not too small. This is significant since most high dimensional nearest neighbour problems suffer from the curse of dimensionality, and can't be solved exact, faster than a brute force linear-time scan of the database. Unfortunately, the best LSH functions tend to have very low collision probabilities, $p_1$ and $p_2$. Including the best functions for Cosine and Jaccard Similarity. This means that the $n^\rho/p_1$ query time of LSH is often not sub-linear after all, even for approximate nearest neighbours! In this paper, we improve the general Indyk-Motwani algorithm to reduce the query time of LSH to $\tilde O(n^\rho/p_1^{1-\rho})$ (and the space usage correspondingly.) Since $n^\rho p_1^{\rho-1} < n \Leftrightarrow p_1 > n^{-1}$, our algorithm always obtains sublinear query time, for any collision probabilities at least $1/n$. For $p_1$ and $p_2$ small enough, our improvement over all previous methods can be \emph{up to a factor $n$} in both query time and space. The improvement comes from a simple change to the Indyk-Motwani algorithm, which can easily be implemented in existing software packages.