On the Approximate Weak Chebyshev Greedy Algorithm in Uniformly Smooth Banach Spaces

TL;DR

This paper analyzes the convergence and performance of an approximate version of the Weak Chebyshev Greedy Algorithm in uniformly smooth Banach spaces, allowing for computational errors and perturbations.

Contribution

It introduces the AWCGA, a modified greedy algorithm that accounts for errors, and establishes necessary and sufficient conditions for its convergence.

Findings

Convergence conditions for AWCGA match those of WCGA when errors are in ℓ₁.

The rate of convergence is estimated for specific perturbations and errors.

In certain cases, AWCGA performs as well as the original WCGA.

Abstract

We study greedy approximation in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for any Banach space $X$ and a dictionary $\mathcal{D}$, and provides nonlinear $n$-term approximation with respect to $\mathcal{D}$. In this paper we study the Approximate Weak Chebyshev Greedy Algorithm (AWCGA) -- a modification of the WCGA that was introduces by V.N. Temlyakov. In the AWCGA we are allowed to calculate $n$-term approximation with a perturbation in computing the norming functional and a relative error in calculating the approximant. Such permission is natural for the numerical applications and simplifies realization of the algorithm. We obtain conditions that are necessary and sufficient for the convergence of the AWCGA for any element of $X$. In particular, we show that if perturbations and errors are from $\ell_1$ space then the conditions for the convergence of the AWCGA are the same as for the WCGA. For specifically chosen perturbations and errors we estimate the rate of convergence for any element $f$ from the closure of the convex hull of $\mathcal{D}$ and demonstrate that in special cases the AWCGA performs as well as the WCGA.