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Replicant compression coding in Besov spaces

Published online by Cambridge University Press:  15 May 2003

Gérard Kerkyacharian  and
Dominique Picard
Gérard Kerkyacharian
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris VI et Université Paris VII, 16 rue de Clisson, 75013 Paris, France. Université Paris X – Nanterre, 200 avenue de la République, 92001 Nanterre Cedex, France; picard@math.jussieu.fr.
Dominique Picard
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris VI et Université Paris VII, 16 rue de Clisson, 75013 Paris, France.
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Abstract

We present here a new proof of the theorem ofBirman and Solomyak on the metric entropy of the unit ball of aBesov space $B^s_{\pi,q}$ on a regular domain of ${\mathbb R}^d.$ Theresult is: if s - d(1/π - 1/p) +> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s . This prooftakes advantage of the representation of such spaces on wavelet typebases and extends the result to more general spaces. The lower boundis a consequence of very simple probabilistic exponentialinequalities. To prove the upper bound, we provide a newuniversal coding based on a thresholding-quantizing procedure usingreplication.

Keywords

EntropycodingBesovspaceswavelet basesreplication.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 239 - 250
Copyright
© EDP Sciences, SMAI, 2003

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References

Assouad, P., Deux remarques sur l'estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 1021-1024.
Birgé, L., Sur un théorème de minimax et son application aux tests. Probab. Math. Statist. 3 (1984) 259-282.
Birgé, L. and Massart, P., An adaptative compression algorithm in Besov spaces. Constr. Approx. 16 (2000) 1-36. CrossRef
Birman, M.S. and Solomiak, M.Z., Piecewise-polynomial approximation of functions of the classes W p . Mat. Sbornik 73 (1967) 295-317. CrossRef
Cohen, A., DeVore, R. and Dahmen, W., Multiscale methods on bounded domains. Trans. AMS 352 (2000) 3651-3685. CrossRef
Cohen, A., Dahmen, W., Daubechies, I. and DeVore, R., Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal. 11 (2001) 192-226. CrossRef
T.A. Cover and J.A. Thomas, Element of Information Theory. Wiley Interscience (1991).
Delyon, B. and Juditski, A., On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 (1996) 215-228. CrossRef
DeVore, R., Kyriazis, R. and Wang, P., Multiscale characterization of Besov spaces on bounded domains. J. Approx. Theory 93 (1998) 273-292. CrossRef
R. DeVore, Nonlinear approximation. Cambridge University Press, Acta Numer. 7 (1998) 51-150.
R. DeVore and G. Lorentz, Constructive Approximation. Springer-Verlag, New York (1993).
Donoho, D.L., Unconditional bases and bit-level compression. Appl. Comput. Harmon. Anal. 3 (1996) 388-392. CrossRef
Hoeffding, W., Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-30. CrossRef
W. Härdle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelet, Approximation and Statistical Applications. Springer Verlag, New York, Lecture Notes in Statist. 129 (1998).
Kerkyacharian, G. and Picard, D., Thresholding algorithms, maxisets and well-concentrated bases, with discussion. Test 9 (2000) 283-345. CrossRef
G. Kerkyacharian and D. Picard, Minimax or maxisets? Bernoulli 8 (2002) 219-253.
G. Kerkyacharian and D. Picard, Entropy, Universal coding, Approximation and bases properties. Technical Report (2001).
Kerkyacharian, G. and Picard, D., Density Estimation by Kernel and Wavelets methods - Optimality of Besov spaces. Statist. Probab. Lett. 18 (1993) 327-336. CrossRef
Kolmogorov, A.N. and Tikhomirov, V.M., π-entropy and π-capacity. Uspekhi Mat. Nauk 14 (1959) 3-86. (Engl. Translation: Amer. Math. Soc. Transl. Ser. 2 17 , 277-364.)
Le Cam, L., Convergence of estimator under dimensionality restrictions. Ann. Statist. 1 (1973) 38-53. CrossRef
L. Le Cam, Metric dimension and statistical estimation, in Advances in mathematical sciences: CRM's 25 years. Montreal, PQ (1994) 303-311.
Lorentz, G.G., Metric entropy and approximation. Bull. Amer. Math. Soc. 72 (1966) 903-937. CrossRef
S.M. Nikolskii, Approximation of functions of several variables and imbedding theorems (Russian), Second Ed. Moskva, Nauka (1977). English translation of the first Ed., Berlin (1975).
V.V. Petrov, Limit Theorems of Probability Theory: Sequences of independent Random Variables. Oxford University Press (1995).
S.A. van de Geer, Empirical processes in M-estimation. Cambridge University Press (2000).