Anisotropic k-Nearest Neighbor Search Using Covariance Quadtree

TL;DR

This paper introduces a covariance hyper-quadtree data structure for anisotropic k-nearest neighbor search using the Mahalanobis metric, which adapts to local data dimensionality for improved density estimation and fluid flow simulations.

Contribution

It presents a novel adaptive data structure that partitions space based on local principal components, enabling efficient anisotropic kNN search and density estimation.

Findings

Effective local dimensionality reduction via PCA

Improved density estimation in noisy data

Potential applications in anisotropic fluid simulations

Abstract

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.