On fast bounded locality sensitive hashing

Abstract

In this paper, we examine the hash functions expressed as scalar products, i.e., $f(x)=<v,x>$, for some bounded random vector $v$. Such hash functions have numerous applications, but often there is a need to optimize the choice of the distribution of $v$. In the present work, we focus on so-called anti-concentration bounds, i.e. the upper bounds of $\mathbb{P}\left[|<v,x>| < \alpha \right]$. In many applications, $v$ is a vector of independent random variables with standard normal distribution. In such case, the distribution of $<v,x>$ is also normal and it is easy to approximate $\mathbb{P}\left[|<v,x>| < \alpha \right]$. Here, we consider two bounded distributions in the context of the anti-concentration bounds. Particularly, we analyze $v$ being a random vector from the unit ball in $l_{\infty}$ and $v$ being a random vector from the unit sphere in $l_{2}$. We show optimal up to a constant anti-concentration measures for functions $f(x)=<v,x>$. As a consequence of our research, we obtain new best results for \newline \textit{$c$-approximate nearest neighbors without false negatives} for $l_p$ in high dimensional space for all $p\in[1,\infty]$, for $c=\Omega(\max\{\sqrt{d},d^{1/p}\})$. These results improve over those presented in [16]. Finally, our paper reports progress on answering the open problem by Pagh~[17], who considered the nearest neighbor search without false negatives for the Hamming distance.