Rescaled Pure Greedy Algorithm for Hilbert and Banach Spaces

TL;DR

This paper introduces a simple modification to the Pure Greedy Algorithm that achieves optimal convergence rates for function approximation in Hilbert and Banach spaces, enhancing sparse representation methods.

Contribution

The paper presents a novel, straightforward modification to the Pure Greedy Algorithm that guarantees optimal convergence rates in both Hilbert and Banach spaces.

Findings

Achieves optimal convergence rates for sparse approximation

Works in both Hilbert and Banach spaces

Simplifies the implementation of greedy algorithms

Abstract

We show that a very simple modification of the Pure Greedy Algorithm for approximating functions by sparse sums from a dictionary in a Hilbert or more generally a Banach space has optimal convergence rates on the class of convex combinations of dictionary elements