Anisotropic k-Nearest Neighbor Search Using Covariance Quadtree

Details Anisotropic k-Nearest Neighbor Search Using Covariance Quadtree Anisotropic k-Nearest Neighbor Search Using Covariance Quadtree

2011-08-31

arxiv.org/abs/1108.6304v1

Computer Science

Computer Vision and Pattern Recognition

Work presented at the Minisymposia of Computational Geometry in the joint events IX Argentinian Congress on Computational Mech

Scientific paper

We present a variant of the hyper-quadtree that divides a multidimensional space according to the hyperplanes associated to the principal components of the data in each hyperquadrant. Each of the $2^\lambda$ hyper-quadrants is a data partition in a $\lambda$-dimension subspace, whose intrinsic dimensionality $\lambda\leq d$ is reduced from the root dimensionality $d$ by the principal components analysis, which discards the irrelevant eigenvalues of the local covariance matrix. In the present method a component is irrelevant if its length is smaller than, or comparable to, the local inter-data spacing. Thus, the covariance hyper-quadtree is fully adaptive to the local dimensionality. The proposed data-structure is used to compute the anisotropic K nearest neighbors (kNN), supported by the Mahalanobis metric. As an application, we used the present k nearest neighbors method to perform density estimation over a noisy data distribution. Such estimation method can be further incorporated to the smoothed particle hydrodynamics, allowing computer simulations of anisotropic fluid flows.

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