Tel-Aviv University
School of Mathematical Sciences
Department of Applied Mathematics

Applied Mathematics Seminar


Tuesday, January 4th, 2011

Schreiber 309, 15:10



Andrei Osipov

Yale University

A Randomized Approximate Nearest Neighbors Algorithm

Abstract:

We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {xj} in R^d, the algorithm attempts to find k nearest neighbors for each of xj, where k is a user-specified integer parameter. The algorithm is iterative, and its CPU time requirements are proportional to T ·N · (d· (log d) + k · (d + log k) · (log N)) + N · k² · (d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d+k). A byproduct of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {xj} for an arbitrary point x in R^d. The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d+log k)), and the memory requirements for the requisite data structure are of the order N · (d + k) + T · (d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme’s behavior for certain types of distributions of {xj}, and illustrate its performance via several numerical examples.

Joint work with Vladimir Rokhlin and Peter W. Jones.