Part of Advances in Neural Information Processing Systems 24 (NIPS 2011)
Dominique Tschopp, Suhas Diggavi, Payam Delgosha, Soheil Mohajer
This paper addresses the problem of finding the nearest neighbor (or one of the $R$-nearest neighbors) of a query object $q$ in a database of $n$ objects, when we can only use a comparison oracle. The comparison oracle, given two reference objects and a query object, returns the reference object most similar to the query object. The main problem we study is how to search the database for the nearest neighbor (NN) of a query, while minimizing the questions. The difficulty of this problem depends on properties of the underlying database. We show the importance of a characterization: \emph{combinatorial disorder} $D$ which defines approximate triangle inequalities on ranks. We present a lower bound of $\Omega(D\log \frac{n}{D}+D^2)$ average number of questions in the search phase for any randomized algorithm, which demonstrates the fundamental role of $D$ for worst case behavior. We develop a randomized scheme for NN retrieval in $O(D^3\log^2 n+ D\log^2 n \log\log n^{D^3})$ questions. The learning requires asking $O(n D^3\log^2 n+ D \log^2 n \log\log n^{D^3})$ questions and $O(n\log^2n/\log(2D))$ bits to store.