Covering of high-dimensional sets

TL;DR

This paper investigates the problem of efficiently covering high-dimensional metric spaces, focusing on the challenges and methods relevant for dimensions $d \\geq 10$, where traditional small-dimensional approaches are ineffective.

Contribution

The paper analyzes the main issues in constructing efficient covering schemes for high-dimensional metric spaces, highlighting differences from small-dimensional cases.

Findings

High-dimensional covering schemes are fundamentally different from low-dimensional ones.

Traditional brute-force methods are ineffective for large dimensions.

Understanding covering in high dimensions is crucial for applications in data analysis and geometry.

Abstract

Let $(\mathcal{X},\rho)$ be a metric space and $\lambda$ be a Borel measure on this space defined on the $\sigma$-algebra generated by open subsets of $\mathcal{X}$; this measure $\lambda$ defines volumes of Borel subsets of $\mathcal{X}$. The principal case is where $\mathcal{X} = \mathbb{R}^d$, $\rho $ is the Euclidean metric, and $\lambda$ is the Lebesgue measure. In this article, we are not going to pay much attention to the case of small dimensions $d$ as the problem of construction of good covering schemes for small $d$ can be attacked by the brute-force optimization algorithms. On the contrary, for medium or large dimensions (say, $d\geq 10$), there is little chance of getting anything sensible without understanding the main issues related to construction of efficient covering designs.