Sharp asymptotics of the functional quantization problem for Gaussian processes

Annals of Probability

Volumn 32, Issue 2, 2004, Pages 1574-1599

  Luschgy, Harald ^a   Pagès, Gilles ^b  

^a UNIVERSITY OF TRIER   (Germany)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Gaussian process; High resolution quantization; Product quantization; Shannon entropy

Indexed keywords

EID: 3042642304     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000324     Document Type: Article

References (24)

1. BERGER, T. (1971). Rate Distortion Theory. Prentice-Hall, Englewood Cliffs, NJ.
2. BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press.
3. BINIA, J. (1974). On the ε-entropy of certain Gaussian processes. IEEE Trans. Inform. Theory 20 190-196.
4. BRONSKI, J. C. (2003). Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theoret. Probab. 16 87-100.
5. DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
6. DEREICH, S. (2003). Small ball probabilities around random centers of Gaussian measures and applications to quantization. J. Theoret. Probab. 16 427-449.
7. DEREICH, S., FEHRINGER, F., MATOUSSI, A. and SCHEUTZOW, M. (2003). On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces. J. Theoret. Probab. 16 249-265.
8. DONOHO, D. L. (2000). Counting bits with Kolmogorov and Shannon. Technical Report 38, Stanford Univ.
9. FREEDMAN, D. (1999). On the Bernstein-von Mises theorem with infinite-dimensional parameters. Ann. Statist. 27 1119-1140.
10. 2-small balls. Ann. Probab. 31 1320-1337.
11. GERSHO, A. and GRAY, R. M. (1992). Vector Quantization and Signal Compression. Kluwer, Boston.
12. GRAF, S. and LUSCHGY, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
13. GRAF, S., LUSCHGY, H. and PAGES, G. (2003). Functional quantization and small ball probabilities for Gaussian processes. J. Theoret. Probab. 16 1047-1062.
14. GRAY, R. M. and NEUHOFF, D. L. (1998). Quantization. IEEE Trans. Inform. Theory 44 2325-2383.
15. IHARA, S. (1970). On ε-entropy of equivalent Gaussian processes. Nagoya Math. J. 37 121-130.
16. IHARA, S. (1993). Information Theory. World Scientific, Singapore.
17. KOLMOGOROV, A. N. (1956). On the Shannon theory of information transmission in the case of continuous signals. IRE Trans. Inform. Theory 2 102-108.
18. LUSCHGY, H. and PAGES, G. (2002). Functional quantization of Gaussian processes. J. Funct. Anal. 196 486-531.
19. PAPAGEORGIOU, A. and WASILKOWSKI, G. W. (1990). On the average complexity of multivariate problems. J. Complexity 6 1-23.
20. RITTER, K. (2000). Average-Case Anslysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, Berlin.
21. ROSENBLATT, M. (1963). Some results on the asymptotic behavior of eigenvalues for a class of integral equations with translation kernel. J. Math. Mech. 12 619-628.
22. ROSENFELD, A. and KAK, A. C. (1976). Digital Picture Processing. Academic Press, New York.
23. SHANNON, C. E. and WEAVER, W. (1949). The Mathematical Theory of Communication. Univ. Illinois Press, Urbana.
24. STEIN, M. L. (1999). Interpolation of Spatial Data. Springer, New York.