Local Doubling Dimension of Point Sets

TL;DR

This paper introduces the t-restricted doubling dimension to measure local intrinsic dimension of point sets within a fixed scale, and presents algorithms for constructing hierarchical net-forests and approximate Cech complexes based on this concept.

Contribution

It defines the t-restricted doubling dimension and develops probabilistic algorithms for constructing net-forests and approximate complexes tailored to local scales.

Findings

The net-forest construction is probabilistic with high probability guarantees.

Complexity of the approximate Cech complex depends on local intrinsic dimension.

Algorithms efficiently compute local neighborhood information within fixed scales.

Abstract

We introduce the notion of t-restricted doubling dimension of a point set in Euclidean space as the local intrinsic dimension up to scale t. In many applications information is only relevant for a fixed range of scales. We present an algorithm to construct a hierarchical net-tree up to scale t which we denote as the net-forest. We present a method based on Locality Sensitive Hashing to compute all near neighbours of points within a certain distance. Our construction of the net-forest is probabilistic, and we guarantee that with high probability, the net-forest is supplemented with the correct neighbouring information. We apply our net-forest construction scheme to create an approximate Cech complex up to a fixed scale; and its complexity depends on the local intrinsic dimension up to that scale.