Sparse approximation of individual functions

TL;DR

This paper investigates the asymptotic behavior of sparse approximation errors for individual functions, examining classical decay questions and improving greedy algorithm convergence bounds using prior iteration information.

Contribution

It introduces the concept of a posteriori bounds to enhance convergence rates of greedy algorithms in sparse approximation of individual functions.

Findings

Best sparse approximation errors can decay faster than the supremum over the class.

A posteriori bounds improve convergence rates of greedy algorithms.

Results apply to different asymptotic settings of approximation characteristics.

Abstract

Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding supremum over the function class? Second, we discuss sparse approximation by greedy type algorithms. We show that for any individual function from a given class we can improve the upper bound on the rate of convergence of the error of approximation by a greedy algorithm if we use some information from the previous iterations of the algorithm. We call bounds of this type a posteriori bounds.