Brief introduction in greedy approximation

TL;DR

This paper provides an overview of greedy algorithms for sparse approximation, highlighting their efficiency, theoretical guarantees, and open problems, with applications in compressed sensing and data sciences.

Contribution

It offers a comprehensive survey of various greedy algorithms, their convergence properties, and includes proofs and open problems in the field.

Findings

Greedy algorithms are effective for sparse approximation.

Theoretical guarantees support their convergence.

Open problems suggest directions for future research.

Abstract

Sparse approximation is important in many applications because of concise form of an approximant and good accuracy guarantees. The theory of compressed sensing, which proved to be very useful in the image processing and data sciences, is based on the concept of sparsity. A fundamental issue of sparse approximation is the problem of construction of efficient algorithms, which provide good approximation. It turns out that greedy algorithms with respect to dictionaries are very good from this point of view. They are simple in implementation and there are well developed theoretical guarantees of their efficiency. This survey/tutorial paper contains brief description of different kinds of greedy algorithms and results on their convergence and rate of convergence. Also, Chapter IV gives some typical proofs of convergence and rate of convergence results for important greedy algorithms and Chapter V gives some open problems.