Degree of adaptive approximation

DeVore, Ronald A.; Yu, Xiang

doi:10.1090/s0025-5718-1990-1035930-5

Cited by 37 publications

(23 citation statements)

References 9 publications

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“…In Section V, we consider a more sophisticated nonlinear approximation procedure based on a tree structure, keeping coefficients of through an adaptive selection procedure that imposes additional structural constraints on the set of retained coefficients. This strategy is related to adaptive spline approximation algorithms studied in [11], and will be particularly useful for coding purposes.…”

Section: Approximation Resultsmentioning

confidence: 99%

“…It means that distortion on the order of can be achieved by a nonlinear wavelet approximation, if and only if the function has roughly " derivatives in ," where (and as in the linear approximation case). A weaker result can be obtained directly, using the equivalence (see the Appendix for details) between the norm of and the norm of the wavelet coefficients (11) Thus, if and if is defined as the rearrangement of the in decreasing order of absolute value, then we obtain that (12) where is a constant, so that (13) More precisely, (13) holds if and only if the coefficients of are in the "weak space" defined by condition (12) or by the equivalent condition 2 (14) The associated function space is slightly larger than , and in contrast to Besov spaces, cannot be described by moduli of smoothness.…”

Section: Propositionmentioning

confidence: 99%

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On the importance of combining wavelet-based nonlinear approximation with coding strategies

Cohen

Daubechies

Guleryuz

et al. 2002

IEEE Trans. Inform. Theory

113

105

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This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based nonlinear approximation. These classes include locally smooth signals that have singularities, and provide a model for many signals encountered in practice, in particular for images. However, we also show that nonlinear approximation results do not always translate to efficient compression strategies in a rate-distortion sense. Based on this observation, we construct compression techniques and formulate the family of functions/stochastic processes for which they provide efficient descriptions in a rate-distortion sense. We show that this family invariably leads to Besov spaces, yielding a natural relationship among Besov smoothness, linear/nonlinear approximation order, and compression performance in a rate-distortion sense. The designed compression techniques show similarities to modern high-performance transform codecs, allowing us to establish relevant rate-distortion estimates and identify performance limits.

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Section: Approximation Resultsmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

On the importance of combining wavelet-based nonlinear approximation with coding strategies

Cohen

Daubechies

Guleryuz

et al. 2002

IEEE Trans. Inform. Theory

113

105

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show abstract

“…where |G| is the number of intervals in G. One can estimate in different ways a n (f ) p := a n (f, E) p := inf |G| 1/p ε, (3.4) where the infimum is taken over all ε > 0 such that |G| = |G(ε, E)| ≤ n. See Birman and Solomjak [2] and DeVore [12] for estimates for functions f in Sobolev spaces. Other estimates can be found in Rice [25], de Boor and Rice [6], and DeVore and Yu [18] and the references therein. We only mention the following two results.…”

Section: Adaptive Algorithmsmentioning

confidence: 99%

Modified Adaptive Algorithms

Hu¹,

Kopotun²,

Yu³

2000

SIAM J. Numer. Anal.

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It is well known that the adaptive algorithm is simple and easy to program but the results are not fully competitive with other nonlinear methods such as free knot spline approximation. We modify the algorithm to take full advantages of nonlinear approximation. The new algorithms have the same approximation order as other nonlinear methods, which is proved by characterizing their approximation spaces. One of our algorithms is implemented on the computer, with numerical results illustrated by figures and tables. ). 1013If there is no potential confusion, especially in the case I = [0, 1], the interval I will be omitted in the notation for the sake of simplicity. For example, L p stands for L p [0, 1] and B α for B α [0, 1]. If X 0 and X 1 are quasi-normed, complete, linear spaces continuously embedded in a Hausdorff space X, then the K-functional for all f ∈ X 0 + X 1 and t ≥ 0 is defined asThis can be generalized if we replace · X1 by a quasi-seminorm | · | X1 on X 1 : K(f, t, X 0 , X 1 ) := inf { f 0 X0 + t|f 1 | X1 | f = f 0 + f 1 , f 0 ∈ X 0 , f 1 ∈ X 1 } .

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“…This type of result works in a broader scale of settings than just the -Sobolev balls. A modern understanding of the underlying nonlinear approximation properties of this approach is developed in [28].…”

Section: B Trees and Data Compressionmentioning

confidence: 99%

Data compression and harmonic analysis

Donoho

Vetterli

DeVore

et al. 1998

IEEE Trans. Inform. Theory

355

231

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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon's R(D) theory in the case of Gaussian stationary processes, which says that transforming into a Fourier basis followed by block coding gives an optimal lossy compression technique; practical developments like transformbased image compression have been inspired by this result. In this paper we also discuss connections perhaps less familiar to the Information Theory community, growing out of the field of harmonic analysis. Recent harmonic analysis constructions, such as wavelet transforms and Gabor transforms, are essentially optimal transforms for transform coding in certain settings. Some of these transforms are under consideration for future compression standards. We discuss some of the lessons of harmonic analysis in this century. Typically, the problems and achievements of this field have involved goals that were not obviously related to practical data compression, and have used a language not immediately accessible to outsiders. Nevertheless, through an extensive generalization of what Shannon called the "sampling theorem," harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations. We explain why harmonic analysis has interacted with data compression, and we describe some interesting recent ideas in the field that may affect data compression in the future.

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Degree of adaptive approximation

Abstract: ABSTRACT. We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

Cited by 37 publications

References 9 publications

On the importance of combining wavelet-based nonlinear approximation with coding strategies

On the importance of combining wavelet-based nonlinear approximation with coding strategies

Modified Adaptive Algorithms

Data compression and harmonic analysis

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