K-Nearest Neighbor Approximation Via the Friend-of-a-Friend Principle
share
Suppose V is an n-element set where for each x ∈ V, the elements of V ∖{x} are ranked by their similarity to x. The K-nearest neighbor graph to the K points of V ∖{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 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 n2 inter-point distances in a metric space.
READ FULL TEXT