A remark on covering

TL;DR

This paper explores a novel approach to constructing coverings of the unit ball in finite-dimensional Banach spaces using incoherent dictionaries, improving upon traditional volume comparison methods.

Contribution

It introduces a new method leveraging incoherent dictionaries to construct efficient coverings, especially focusing on initial coverings with large radii and iterative refinement.

Findings

Incoherent dictionaries enable the construction of better coverings.

The method provides a systematic way to refine coverings for smaller radii.

Traditional volume comparison techniques do not yield explicit coverings.

Abstract

We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of good coverings. Here we apply incoherent dictionaries for construction of good coverings. We use the following strategy. First, we build a good covering by balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We mostly concentrate on the first step of this strategy.