Local Doubling Dimension of Point Sets

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 Čech complex up to a fixed scale; and its complexity depends on the local intrinsic dimension up to that scale.

Persistent homology for low-complexity models

“…The doubling dimension has been used as a measure of intrinsic dimension in topological data analysis, see [50,Chapter 5] and the references therein, and also [56,21]. To our knowledge, the relation of the Gaussian width to the doubling dimension of a metric space was first pointed out by Indyk and Naor [34], even though a close connection is apparent in work on suprema of Gaussian processes [61].…”

“…A common notion in the study of metric spaces is the doubling dimension [5,31]. This concept has been used to measure the intrinsic dimension of sets in topological data analysis, see [50,Chapter 5] or [56,21] for some examples.…”

“…The key is to approximate the point cloud X by a nested sequence of nets N α , in such a way that at each relevant level the spread of the net is constant. The approach was further extended in [21], who introduce a local version of the doubling dimension. It turns out that the doubling dimension is closely related to the Gaussian width, a fact pointed out in [34].…”

“…Note that we can alternatively represent an admissible sequence of partitions as a hierarchical tree, where each level approximates the data set more accurately. We suspect that proofs based on net-trees, such as those of the statements in [56,21], could be formulated in terms of the Gaussian width by associating to the point cloud (and all the associated nets) a stochastic process g x = g, x for x ∈ T .…”