scholarly journals Greedy Algorithms andM-Term Approximation with Regard to Redundant Dictionaries

1999 ◽  
Vol 98 (1) ◽  
pp. 117-145 ◽  
Author(s):  
V.N. Temlyakov
Keyword(s):  
Greedy Algorithms ◽  
Redundant Dictionaries ◽  
Term Approximation

Greedy approximation

Acta Numerica ◽  
2008 ◽  
Vol 17 ◽  
pp. 235-409 ◽  
Author(s):  
V. N. Temlyakov
Keyword(s):  
Nonlinear Approximation ◽  
Greedy Algorithms ◽  
Target Function ◽  
Large Data ◽  
Fundamental Principle ◽  
Sparse Representations ◽  
Data Sets ◽  
Greedy Approximation ◽  
Term Approximation ◽  
Redundant Systems

In this survey we discuss properties of specific methods of approximation that belong to a family of greedy approximation methods (greedy algorithms). It is now well understood that we need to study nonlinear sparse representations in order to significantly increase our ability to process (compress, denoise,etc.) large data sets. Sparse representations of a function are not only a powerful analytic tool but they are utilized in many application areas such as image/signal processing and numerical computation. The key to finding sparse representations is the concept ofm-term approximation of the target function by the elements of a given system of functions (dictionary). The fundamental question is how to construct good methods (algorithms) of approximation. Recent results have established that greedy-type algorithms are suitable methods of nonlinear approximation in bothm-term approximation with regard to bases, andm-term approximation with regard to redundant systems. It turns out that there is one fundamental principle that allows us to build good algorithms, both for arbitrary redundant systems and for very simple well-structured bases, such as the Haar basis. This principle is the use of a greedy step in searching for a new element to be added to a givenm-term approximant.

scholarly journals SPARSE APPROXIMATION AND RECOVERY BY GREEDY ALGORITHMS IN BANACH SPACES

2014 ◽  
Vol 2 ◽  
Author(s):  
V. N. TEMLYAKOV
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Greedy Algorithm ◽  
Matching Pursuit ◽  
Greedy Algorithms ◽  
Sparse Approximation ◽  
Trigonometric System ◽  
Space Setting ◽  
Redundant Dictionaries

AbstractWe study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty $.

Almost optimality of orthogonal super greedy algorithms for incoherent dictionaries

2017 ◽  
Vol 15 (04) ◽  
pp. 1750029 ◽  
Author(s):  
Chunfang Shao ◽  
Peixin Ye
Keyword(s):  
Hilbert Space ◽  
Greedy Algorithm ◽  
Greedy Algorithms ◽  
Absolute Value ◽  
Term Approximation

We investigate the efficiency of weak orthogonal super greedy algorithm (WOSGA) for [Formula: see text]-term approximation with respect to dictionaries which are [Formula: see text]-unconditional in arbitrary Hilbert space [Formula: see text] For an element [Formula: see text], let [Formula: see text] be the output of WOSGA after [Formula: see text] steps for some constant [Formula: see text]. We show that the residual [Formula: see text] can be bounded by a constant multiplying the error of best [Formula: see text]-term approximation to [Formula: see text] Moreover, we get an element [Formula: see text], through a simple postprocessing of [Formula: see text] by retaining its [Formula: see text] largest components in absolute value, which realizes near best [Formula: see text]-term approximation for [Formula: see text] Our results are obtained for dictionaries in [Formula: see text] which satisfies the weaker assumption than the RIP condition.

On Jackknifed Greedy Algorithms and Their Applications in NMR

2020 ◽  
Vol 84 (11) ◽  
pp. 1335-1340
Author(s):  
P. Kasprzak ◽  
K. Kazimierczuk ◽  
A. L. Shchukina
Keyword(s):  
Greedy Algorithms
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