Relaxation in Greedy Approximation

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of com...

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Vydáno v:Constructive approximation Ročník 28; číslo 1; s. 1 - 25
Hlavní autor: Temlyakov, V.N.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer-Verlag 01.06.2008
Témata:
Analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
Smooth Banach Space
Nonincreasing Sequence
Banach Space
Greedy Algorithm
Hilbert Space
ISSN:0176-4276, 1432-0940
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Shrnutí:We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational complexity. However, the WRGA has limited applicability. It converges only for elements of the closure of the convex hull of a dictionary. In this paper we study algorithms that combine good features of both algorithms, the WRGA and the WCGA. In the construction of such algorithms we use different forms of relaxation. First results on such algorithms have been obtained in a Hilbert space by A. Barron, A. Cohen, W. Dahmen, and R. DeVore. Their paper was a motivation for the research reported here.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-006-0652-5