What question did this study set out to answer?

The aim is to explore efficient algorithms for sparse approximation with a focus on greedy methods.

January 14, 2026

Brief introduction in greedy approximation

Key Points

The aim is to explore efficient algorithms for sparse approximation with a focus on greedy methods.
Survey of greedy algorithms relevant to sparse approximation
Discussion of convergence and rates of convergence
Highlighting theoretical guarantees of efficiency
Proofs related to important greedy algorithms
Listing of open problems in the field.
Greedy algorithms are shown to provide effective approximations in sparse representation.
Theoretical guarantees support their implementation in various applications.
Convergence results illustrate performance in data sciences and image processing.

Abstract

Sparse approximation is important in many applications because of the 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 the 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 a brief description of different kinds of greedy algorithms and results on their convergence and rate of convergence. Also, in §§ 14 and 15 we give some typical proofs of convergence and rate of convergence results for important greedy algorithms and in Section § 16 we list some open problems. Bibliography: 91 titles.

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