11 - Greedy Approximations

Summary

Abstract

In nonlinear approximation we seek ways to approximate complicated functions by simpler functions using methods that depend nonlinearly on the function being approximated. Recently, a particular kind of nonlinear approximation, namely greedy approximation has attracted a lot of attention in both theoretical and applied settings. Greedy type algorithms have proven to be very useful in various applications such as image compression, signal processing, design of neural networks, and the numerical solution of nonlinear partial differential equations. A theory of greedy approximation is now emerging. Some fundamental convergence results have already been established and many fundamental problems remain unsolved. In this survey we place emphasis on the study of the efficiency of greedy algorithms with regards to redundant systems (dictionaries). Redundancy, on the one hand, offers much promise for greater efficiency in terms of the rate of approximation. On the other hand, it gives rise to highly nontrivial theoretical and practical problems. We note that there is solid justification for the importance of redundant systems in both theoretical questions and practical applications. This survey is a continuation of the survey Temlyakov (2003a) on nonlinear approximations. Here we concentrate on more recent results on greedy approximation.

Introduction

In the last decade we have seen successes in the study of nonlinear approximation (see the surveys DeVore (1998) and Temlyakov (2003a)). This study was motivated by numerous applications. Nonlinear approximation is important in applications because of its efficiency.