|
Metric entropy of high dimensional distributions
Authors:
Ron Blei, Fuchang Gao and Wenbo V. Li
Journal:
Proc. Amer. Math. Soc. 135 (2007), 4009-4018
MSC (2000):
Primary 60G15, 46B50.
Posted:
September 7, 2007
MathSciNet review:
2341952
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Let  be the collection of all  -dimensional probability distribution functions on ![$ [0,1]^d$](/proc/2007-135-12/S0002-9939-07-08935-6/gif-abstract0/img3.gif) ,  . The metric entropy of  under the ![$ L_2([0,1]^d)$](/proc/2007-135-12/S0002-9939-07-08935-6/gif-abstract0/img6.gif) norm is studied. The exact rate is obtained for  and bounds are given for  . Connections with small deviation probability for Brownian sheets under the sup-norm are established.
- 1.
S.
Artstein, V.
Milman, and S.
J. Szarek, Duality of metric entropy, Ann. of Math. (2)
159 (2004), no. 3, 1313–1328. MR 2113023
(2005h:47037), http://dx.doi.org/10.4007/annals.2004.159.1313
- 2.
S.
Artstein, V.
Milman, S.
Szarek, and N.
Tomczak-Jaegermann, On convexified packing and entropy
duality, Geom. Funct. Anal. 14 (2004), no. 5,
1134–1141. MR 2105957
(2005h:47038), http://dx.doi.org/10.1007/s00039-004-0486-3
- 3.
Bernd
Carl and Irmtraud
Stephani, Entropy, compactness and the approximation of
operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge
University Press, Cambridge, 1990. MR 1098497
(92e:47002)
- 4.
T.
Dunker, W.
Linde, T.
Kühn, and M.
A. Lifshits, Metric entropy of integration operators and small ball
probabilities for the Brownian sheet, J. Approx. Theory
101 (1999), no. 1, 63–77. MR 1724026
(2001d:60032), http://dx.doi.org/10.1006/jath.1999.3354
- 5.
D.
E. Edmunds and H.
Triebel, Function spaces, entropy numbers, differential
operators, Cambridge Tracts in Mathematics, vol. 120, Cambridge
University Press, Cambridge, 1996. MR 1410258
(97h:46045)
- 6.
Fuchang
Gao, Entropy of absolute convex hulls in Hilbert spaces, Bull.
London Math. Soc. 36 (2004), no. 4, 460–468. MR 2069008
(2005e:41071), http://dx.doi.org/10.1112/S0024609304003121
- 7.
Fuchang
Gao and Wenbo
V. Li, Small ball probabilities for the
Slepian Gaussian fields, Trans. Amer. Math.
Soc. 359 (2007), no. 3, 1339–1350. MR 2262853
(2008a:60093), http://dx.doi.org/10.1090/S0002-9947-06-03963-8
- 8.
A.
N. Kolmogorov and V.
M. Tihomirov, 𝜖-entropy and 𝜖-capacity of sets in
function spaces, Uspehi Mat. Nauk 14 (1959),
no. 2 (86), 3–86 (Russian). MR 0112032
(22 #2890)
- 9.
James
Kuelbs and Wenbo
V. Li, Metric entropy and the small ball problem for Gaussian
measures, J. Funct. Anal. 116 (1993), no. 1,
133–157. MR 1237989
(94j:60078), http://dx.doi.org/10.1006/jfan.1993.1107
- 10.
Wenbo
V. Li and Werner
Linde, Approximation, metric entropy and small ball estimates for
Gaussian measures, Ann. Probab. 27 (1999),
no. 3, 1556–1578. MR 1733160
(2001c:60059), http://dx.doi.org/10.1214/aop/1022677459
- 11.
W.
V. Li and Q.-M.
Shao, Gaussian processes: inequalities, small ball probabilities
and applications, Stochastic processes: theory and methods, Handbook
of Statist., vol. 19, North-Holland, Amsterdam, 2001,
pp. 533–597. MR
1861734, http://dx.doi.org/10.1016/S0169-7161(01)19019-X
- 12.
G.
G. Lorentz, Metric entropy and
approximation, Bull. Amer. Math. Soc. 72 (1966), 903–937.
MR
0203320 (34 #3173), http://dx.doi.org/10.1090/S0002-9904-1966-11586-0
- 13.
Michel
Talagrand, The small ball problem for the Brownian sheet, Ann.
Probab. 22 (1994), no. 3, 1331–1354. MR 1303647
(95k:60049)
- 14.
Aad
W. van der Vaart and Jon
A. Wellner, Weak convergence and empirical processes, Springer
Series in Statistics, Springer-Verlag, New York, 1996. With applications to
statistics. MR
1385671 (97g:60035)
- 1.
- Artstein, S., Milman, V., and Szarek, S. (2004). Duality of metric entropy, Ann. of Math. 159, 1313-1328. MR 2113023 (2005h:47037)
- 2.
- Artstein, S., Milman, V., Szarek, S. J., and Tomczak-Jaegermann, N. (2004). On Convexified Packing and Duality, Geom. and Funct. Anal. 14, 1134-1141. MR 2105957 (2005h:47038)
- 3.
- Carl, B. and Stephani, I. (1990). Entropy, Compactness and Approximation of Operators, Cambridge University Press, Cambridge. MR 1098497 (92e:47002)
- 4.
- Dunker, T., Kuhn, T., Lifshits, M. A., and Linde, W. (1999). Metric entropy of integration operators and small ball probabilities for the Brownian sheet, J. Approx. Theory 101, 63-77. MR 1724026 (2001d:60032)
- 5.
- EDMUNDS, D. E. and TRIEBEL, H. (1996). Function Spaces, Entropy Numbers and Differential Operators, Cambridge Univ. Press., Cambridge. MR 1410258 (97h:46045)
- 6.
- Gao, F. (2004). Entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), 460 - 468. MR 2069008 (2005e:41071)
- 7.
- Gao, F. and Li, W.V. (2007). Small ball probabilities for Slepian Gaussian fields, Trans. Amer. Math. Soc. 359, 1339-1350. MR 2262853
- 8.
- Kolmogorov, A.N. and Tihomirov, V.M. (1961).
 -entropy and  -capacity of sets in function spaces, Uspehi Mat. Nauk. 14 (1959), 3-86. English transl. in Amer. Math. Soc. Transl. 17, 277-364. MR 0112032 (22:2890)
- 9.
- Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116, 133-157. MR 1237989 (94j:60078)
- 10.
- Li, W.V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27, 1556-1578. MR 1733160 (2001c:60059)
- 11.
- Li, W.V. and Shao, Q.-M. (2001). Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., Vol. 19. 533-597. MR 1861734
- 12.
- Lorentz, G. (1966). Metric entropy and approximation, Bull. Amer. Math. Soc. 72, 903-937. MR 0203320 (34:3173)
- 13.
- Talagrand, M. (1994). The small ball problem for the Brownian sheet, Ann. Probab. 22, 1331-1354. MR 1303647 (95k:60049)
- 14.
- van der Vaart, A. and Wellner, J. Weak convergence and empirical processes. With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996. MR 1385671 (97g:60035)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
60G15,
46B50.
Retrieve articles in all journals
with MSC (2000):
60G15,
46B50.
Additional Information
Ron Blei
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
Email:
blei@math.uconn.edu
Fuchang Gao
Affiliation:
Department of Mathematics, University of Idaho, Moscow, Idaho 83844
Email:
fuchang@uidaho.edu
Wenbo V. Li
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email:
wli@math.udel.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08935-6
PII:
S 0002-9939(07)08935-6
Received by editor(s):
May 23, 2006
Received by editor(s) in revised form:
August 25, 2006, and September 19, 2006
Posted:
September 7, 2007
Additional Notes:
The first author was supported in part by NSF Grant DMS-0405855.
The second author was supported in part by NSF Grant DMS-0505805.
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
