On weak greedy algorithms

TL;DR

This paper extends weak greedy algorithms to a broader setting with a weakness sequence and proposes a new convergence framework focusing on subsets related to a dictionary.

Contribution

It generalizes existing results on weak greedy algorithms and introduces a novel convergence analysis for subsets linked to a dictionary.

Findings

Extended results to weak greedy algorithms with a weakness sequence

Formulated a new convergence setting for greedy algorithms

Focused on convergence within generalized octahedra associated with dictionaries

Abstract

The main goal of this paper is twofold. First, we extend some results known in the case of weak greedy algorithms with a scalar parameter to the case of weak greedy algorithms with a weakness sequence. Second, we formulate a new setting of the problem of convergence of greedy algorithms. Usually, we are interested in convergence of an algorithm for all elements of the space. We suggest to study convergence of an algorithm for a subset, which is a generalized octahedron associated with a given dictionary.