## Nonlinear Approximation with Walsh Atoms (1997)

Venue: | Surface Fitting and Multiresolution Methods |

Citations: | 13 - 3 self |

### BibTeX

@ARTICLE{Villemoes97nonlinearapproximation,

author = {Lars F. Villemoes},

title = {Nonlinear Approximation with Walsh Atoms},

journal = {Surface Fitting and Multiresolution Methods},

year = {1997},

volume = {13},

pages = {329--336}

}

### OpenURL

### Abstract

. As a model for nonlinear approximation from a redundant set of time-frequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations of pairwise orthogonal atoms. We describe the effect of this restriction on approximation rates, and then discuss the performance of the greedy algorithm. In particular, a uniform geometric rate of convergence is shown to hold for the class of initial functions consisting of linear combinations of two atoms. x1. Introduction Given a dictionary D = fe g 2 of elements in a Hilbert space H the nonlinear approximation error of a given element f 2 H relative to D is E n (f) = inf g2\Sigma n kf \Gamma gk; (1:1) where \Sigma n denotes the set of linear combinations of n dictionary elements from D. If we have an algorithm that produces n term approximations to f , a fundamental question is then how ...

### Citations

147 |
Compression of wavelet decompositions
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- 1992
(Show Context)
Citation Context ...on the line, and D is a standard wavelet basis. Then not only is the approximation algorithm optimal, but the the decay of E n (f) can be characterized in terms of the L p -smoothness of f for p ! 2, =-=[3]-=-. However, for redundant dictionaries of time-frequency atoms like Gabor functions, local cosines, or wavelet packets, [1,6], there is no canonical approximation algorithm and the oscillatory nature o... |

142 |
Matching pursuit in a time-frequency dictionary
- Mallat, Zhang
- 1993
(Show Context)
Citation Context ... E n (f) can be characterized in terms of the L p -smoothness of f for p ! 2, [3]. However, for redundant dictionaries of time-frequency atoms like Gabor functions, local cosines, or wavelet packets, =-=[1,6]-=-, there is no canonical approximation algorithm and the oscillatory nature of the dictionary elements makes sparsity of representation unrelated to classical notions of smoothness. We will make no att... |

94 |
Some remarks on greedy algorithms
- DeVore, Temlyakov
- 1996
(Show Context)
Citation Context ...ieve results in this direction, we would need to consider an initial function which is a sum of a linear combination of K atoms and a small error. From the general analysis of the greedy algorithm in =-=[4]-=- it follows that if f = P c i w p i then kf n ks( P jc i j)n \Gamma1=6 . From Theorem 4.1 we see that if f is a linear combination of K atoms with kfk = 1, then f = P 1 n=0 g n , where g n = P fl(n+1)... |

28 |
Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation
- Herley, Xiong, et al.
- 1997
(Show Context)
Citation Context ...lent to finding the cheapest tiling of\Omega N relative to an additive cost. It is possible to perform this search in O(N log N) operations although the number of tilings grows exponentially with N , =-=[5,7]-=-. The main tool for this is Lemma 2.2 by which we can split the problem into 4 subproblems of size N=2. In the next recursive step we have, seemingly, 16 problems of size N=4, but only 12 of these are... |

10 |
Wavelets and adapted waveform analysis: A toolkit for signal processing and numerical analysis
- Coifman, Wickerhauser
- 1993
(Show Context)
Citation Context ... E n (f) can be characterized in terms of the L p -smoothness of f for p ! 2, [3]. However, for redundant dictionaries of time-frequency atoms like Gabor functions, local cosines, or wavelet packets, =-=[1,6]-=-, there is no canonical approximation algorithm and the oscillatory nature of the dictionary elements makes sparsity of representation unrelated to classical notions of smoothness. We will make no att... |

7 |
Villemoes, “A fast algorithm for adapted time-frequency tilings
- Thiele, F
- 1996
(Show Context)
Citation Context ...right half, d the lower half, and u the upper half, then ` w d w u ' = 1 p 2 ` 1 1 1 \Gamma1 '` w l w r ' : (2:5) It is possible to prove (2.4) by iterated use of (2.5). For more details, we refer to =-=[7]-=- and [8]. For any power of two N = 2 J , a subspace of L 2 (IR) can be defined by VN = spanfw p j p 2 AN g; (2:6) where AN consist of all tiles which are contained in the rectangle\Omega N = [0; 1[\Th... |

3 |
Adaptive greedy approximations
- Avellaneda, Davis, et al.
- 1997
(Show Context)
Citation Context ... would be remarkable, since it has been empirically observed for time-frequency dictionaries that the greedy algorithm works well in the beginning, and this cannot be explained by asymptotic results, =-=[2]-=-. The study of the performance on a clean linear combination of K atoms is the starting point for an analysis of the initial success. We will prove geometric convergence for K = 2. This simple case al... |

1 |
approximation with Walsh atoms, Constr. Approx. , (to appear
- Villemoes, Best
(Show Context)
Citation Context ...lf, d the lower half, and u the upper half, then ` w d w u ' = 1 p 2 ` 1 1 1 \Gamma1 '` w l w r ' : (2:5) It is possible to prove (2.4) by iterated use of (2.5). For more details, we refer to [7] and =-=[8]-=-. For any power of two N = 2 J , a subspace of L 2 (IR) can be defined by VN = spanfw p j p 2 AN g; (2:6) where AN consist of all tiles which are contained in the rectangle\Omega N = [0; 1[\Theta[0; N... |