On the convergence of greedy algorithms for initial segments of the Haar basis

TL;DR

This paper proves that certain greedy algorithms for sparse approximation in finite-dimensional Banach spaces with Haar basis dictionaries terminate finitely, with iteration bounds depending on the initial segment length.

Contribution

It establishes finite termination and iteration bounds for the X-Greedy and Dual Greedy algorithms with Haar basis dictionaries in $L_p$ spaces.

Findings

Algorithms terminate after finitely many steps.

Number of iterations is bounded by a function of the initial segment length.

Results extend to a class of strictly monotone bases.

Abstract

We consider the $X$-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in $L_p[0,1]$ ($1 < p < \infty$) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.