Replicant compression coding in Besov spaces

Kerkyacharian, Gérard; Picard, Dominique

doi:10.1051/ps:2003011

Kerkyacharian, Gérard ; Picard, Dominique

ESAIM: Probability and Statistics, Tome 7 (2003), pp. 239-250.

Résumé

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{π, q}^{s}$ on a regular domain of $ℝ^{d} .$ The result is: if $s - d {(1 / π - 1 / p)}_{+}$ $> 0,$ then the Kolmogorov metric entropy satisfies $H (ϵ) \sim ϵ^{- d / s}$ . This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.

MR Zbl

DOI : 10.1051/ps:2003011

Classification : 41A25, 41A46, 65F99, 65N12, 65N55
Mots clés : entropy, coding, Besov spaces, wavelet bases, replication

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     title = {Replicant compression coding in {Besov} spaces},
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     pages = {239--250},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2003},
     doi = {10.1051/ps:2003011},
     mrnumber = {1987788},
     zbl = {1031.41014},
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     url = {http://archive.numdam.org/articles/10.1051/ps:2003011/}
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Kerkyacharian, Gérard; Picard, Dominique. Replicant compression coding in Besov spaces. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 239-250. doi : 10.1051/ps:2003011. http://archive.numdam.org/articles/10.1051/ps:2003011/

Bibliographie
Cité par

[1] P. Assouad, Deux remarques sur l'estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 1021-1024. | Zbl

[2] L. Birgé, Sur un théorème de minimax et son application aux tests. Probab. Math. Statist. 3 (1984) 259-282. | MR | Zbl

[3] L. Birgé and P. Massart, An adaptative compression algorithm in Besov spaces. Constr. Approx. 16 (2000) 1-36. | MR | Zbl

[4] M.S. Birman and M.Z. Solomiak, Piecewise-polynomial approximation of functions of the classes $W_{p}$ . Mat. Sbornik 73 (1967) 295-317. | MR | Zbl

[5] A. Cohen, R. Devore and W. Dahmen, Multiscale methods on bounded domains. Trans. AMS 352 (2000) 3651-3685. | MR | Zbl

[6] A. Cohen, W. Dahmen, I. Daubechies and R. Devore, Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal. 11 (2001) 192-226. | MR | Zbl

[7] T.A. Cover and J.A. Thomas, Element of Information Theory. Wiley Interscience (1991). | MR | Zbl

[8] B. Delyon and A. Juditski, On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 (1996) 215-228. | MR | Zbl

[9] R. Devore, R. Kyriazis and P. Wang, Multiscale characterization of Besov spaces on bounded domains. J. Approx. Theory 93 (1998) 273-292. | MR | Zbl

[10] R. Devore, Nonlinear approximation. Cambridge University Press, Acta Numer. 7 (1998) 51-150. | MR | Zbl

[11] R. Devore and G. Lorentz, Constructive Approximation. Springer-Verlag, New York (1993). | MR | Zbl

[12] D.L. Donoho, Unconditional bases and bit-level compression. Appl. Comput. Harmon. Anal. 3 (1996) 388-392. | MR | Zbl

[13] W. Hoeffding, Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-30. | MR | Zbl

[14] W. Härdle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelet, Approximation and Statistical Applications. Springer Verlag, New York, Lecture Notes in Statist. 129 (1998). | MR | Zbl

[15] G. Kerkyacharian and D. Picard, Thresholding algorithms, maxisets and well-concentrated bases, with discussion. Test 9 (2000) 283-345. | MR | Zbl

[16] G. Kerkyacharian and D. Picard, Minimax or maxisets? Bernoulli 8 (2002) 219-253. | MR | Zbl

[17] G. Kerkyacharian and D. Picard, Entropy, Universal coding, Approximation and bases properties. Technical Report (2001). | Zbl

[18] G. Kerkyacharian and D. Picard, Density Estimation by Kernel and Wavelets methods - Optimality of Besov spaces. Statist. Probab. Lett. 18 (1993) 327-336. | Zbl

[19] A.N. Kolmogorov and V.M. Tikhomirov, $ϵ$ -entropy and $ϵ$ -capacity. Uspekhi Mat. Nauk 14 (1959) 3-86. (Engl. Translation: Amer. Math. Soc. Transl. Ser. 2 17, 277-364.) | MR | Zbl

[20] L. Le Cam, Convergence of estimator under dimensionality restrictions. Ann. Statist. 1 (1973) 38-53. | MR | Zbl

[21] L. Le Cam, Metric dimension and statistical estimation, in Advances in mathematical sciences: CRM's 25 years. Montreal, PQ (1994) 303-311. | Zbl

[22] G.G. Lorentz, Metric entropy and approximation. Bull. Amer. Math. Soc. 72 (1966) 903-937. | MR | Zbl

[23] 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). | MR | Zbl

[24] V.V. Petrov, Limit Theorems of Probability Theory: Sequences of independent Random Variables. Oxford University Press (1995). | MR | Zbl

[25] S.A. Van De Geer, Empirical processes in M-estimation. Cambridge University Press (2000).

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