Improved approximate near neighbor search without false negatives for $l_2$

Abstract: We present a new algorithm for the $c$--approximate nearest neighbor search without false negatives for $l_2^d$. We enhance the dimension reduction method presented in \cite{wygos_red} and combine it with the standard results of Indyk and Motwani~\cite{motwani}. We present an efficient algorithm with Las Vegas guaranties for any $c>1$. This improves over the previous results, which require $c=\omega(\log\log{n})$ \cite{wygos_red}, where $n$ is the number of the input points. Moreover, we improve both the query time and the pre-processing time. Our algorithm is tunable, which allows for different compromises between the query and the pre-processing times. In order to illustrate this flexibility, we present two variants of the algorithm. The "efficient query" variant involves the query time of $O(d^2)$ and the polynomial pre-processing time. The "efficient pre-processing" variant involves the pre-processing time equal to $O(d^{\omega-1} n)$ and the query time sub-linear in $n$, where $\omega$ is the exponent in the complexity of the fast matrix multiplication. In addition, we introduce batch versions of the mentioned algorithms, where the queries come in batches of size $d$. In this case, the amortized query time of the "efficient query" algorithm is reduced to $O(d^{\omega -1})$.