A remark on covering
This work addresses a theoretical problem in functional analysis and approximation theory, offering a constructive approach to covering numbers, but it appears incremental as it builds on known techniques like incoherent dictionaries.
The paper tackles the problem of constructing efficient coverings of the unit ball in finite-dimensional Banach spaces, where existing volume-based methods only provide bounds without explicit constructions. It proposes using incoherent dictionaries to build good coverings, focusing on constructing coverings with radii close to one as a key step.
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.