Which Spatial Partition Trees are Adaptive to Intrinsic Dimension?

TL;DR

This paper investigates how various spatial partition trees, including k-d, dyadic, and PCA trees, adapt to the intrinsic low-dimensional structure of data through theory and experiments, impacting statistical tasks.

Contribution

It broadens understanding of the adaptivity of different spatial partition trees to intrinsic data dimensions, beyond random projection trees.

Findings

Random projection trees are adaptive to intrinsic dimension.

Other trees like k-d, dyadic, PCA also exhibit some adaptivity.

Implications for regression, quantization, and nearest neighbor search.

Abstract

Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees, dyadic trees, and PCA trees. Our motivation is to get a feel for (i) the kind of intrinsic low dimensional structure that can be empirically verified, (ii) the extent to which a spatial partition can exploit such structure, and (iii) the implications for standard statistical tasks such as regression, vector quantization, and nearest neighbor search.