Combinatorics Seminar-K-Nearest Neighbor Approximation Via the Friend-of-a-Friend Principle

Title: K-Nearest Neighbor Approximation Via the Friend-of-a-Friend Principle

Speaker: Richard W.R. Darling, Math Research Group, NSA
Date and time: Thursday, October 24, 2:15–3:15pm
Place: Phillips 110

Abstract: Suppose V is an n-element set where for each x\in V, the elements of V\setminus\{x\} are ranked by their similarity to x. The K-nearest neighbor graph is a directed graph including an arc from each x to the K points of V\setminus\{x\} most similar to x. Constructive approximation to this graph using far fewer than n^2 comparisons is important for the analysis of large high-dimensional data sets. K-Nearest Neighbor Descent is a parameter-free heuristic where a sequence of graph approximations is constructed, in which second neighbors in one approximation are proposed as neighbors in the next. We provide a rigorous justification for O(n \log n) complexity of a similar algorithm, using range queries, when applied to a homogeneous Poisson process in suitable dimension, but show that the basic algorithm fails to achieve subquadratic complexity on sets whose similarity rankings arise from a "generic" linear order on the \binom{n}{2} inter-point distances in a metric space.