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

20 August 2019
Jacob D. Baron
R. Darling
ArXiv (abs)PDFHTML
Abstract

Suppose VVV is an nnn-element set where for each x∈Vx \in Vx∈V, the elements of V∖{x}V \setminus \{x\}V∖{x} are ranked by their similarity to xxx. The KKK-nearest neighbor graph is a directed graph including an arc from each xxx to the KKK points of V∖{x}V \setminus \{x\}V∖{x} most similar to xxx. Constructive approximation to this graph using far fewer than n2n^2n2 comparisons is important for the analysis of large high-dimensional data sets. KKK-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(nlog⁡n)O( n \log{n} )O(nlogn) 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)\binom{n}{2}(2n​) inter-point distances in a metric space.

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