Greedy Approximation

@inproceedings{Temlyakov2011GreedyA,
  title={Greedy Approximation},
  author={Vladimir N. Temlyakov},
  year={2011}
}
This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks… 
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References

SHOWING 1-10 OF 98 REFERENCES
Greedy-Type Approximation in Banach Spaces and Applications
Abstract We continue to study the efficiency of approximation and convergence of greedy-type algorithms in uniformly smooth Banach spaces. Two greedy-type approximation methods, the Weak Chebyshev
Vector greedy algorithms
Greedy Algorithms with Regard to Multivariate Systems with Special Structure
Abstract. The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for
Greedy approximation and the multivariate Haar system
We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basisH inLp((0; 1)), 1 < p <1) a greedy type algorithm
Simultaneous greedy approximation in Banach spaces
Greedy Algorithms andM-Term Approximation with Regard to Redundant Dictionaries
We study the efficiency of greedy type algorithms with regard to redundant dictionaries in Hilbert space and we prove a general result which gives a sufficient condition on a dictionary to guarantee
Nonlinear Approximation in Finite-Dimensional Spaces
TLDR
This paper considers certain problems of nonlinear approximation which arise in image processing, this includes approximation using m terms from a dictionary of functions and greedy algorithms for approximation from such a dictionary.
Approximate Weak Greedy Algorithms
TLDR
A generalization of Temlyakov's weak greedy algorithm is presented, and a sufficient condition for norm convergence of the algorithm for an arbitrary dictionary in a Hilbert space is given.
On the mathematical foundations of learning
(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the
Rates of convex approximation in non-hilbert spaces
This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the
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