Brief introduction in greedy approximation
V. Temlyakov
TL;DR
The work surveys greedy approximation with respect to dictionaries in Banach spaces, emphasizing how the modulus of smoothness $\\rho(u)$ governs convergence and rate across a broad family of algorithms. It develops a unified framework (WBGA) that encompasses WCGA, WGAFR, and RWRGA, establishing convergence and $\\,e(\\tau,p)$-type rates, and extends to stability under noise and approximate computations. The text then analyzes Lebesgue-type inequalities and dictionary properties (e.g., incoherence, Nikol\'skii-type conditions) to yield near-best $m$-term performance, with applications to bilinear and multilinear approximation via the Schmidt dictionary and tensor-product dictionaries. Overall, it combines abstract functional-analytic tools with concrete algorithmic constructions to advance sparse recovery in flexible settings and under general dictionary structures.
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.