TY - JOUR
T1 - Projecting the surface measure of the sphere of ℓ n p
AU - Naor, Assaf
AU - Romik, Dan
N1 - Funding Information:
∗Corresponding author. E-mail addresses: [email protected] (A. Naor), [email protected] (D. Romik). 1Present address: Theory Group, Microsoft Research, One Microsoft Way, Redmond WA, 98052-6399, USA. 2 Supported in part by the Bi-National Science Foundation Israel–USA and by the Clore Foundation. This work is part of a Ph.D. thesis being prepared under the supervision of Professor Joram Lindenstrauss. 3This work is part of a Ph.D. thesis being prepared under the supervision of Professor David Gilat.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2003
Y1 - 2003
N2 - We prove that the total variation distance between the cone measure and surface measure on the sphere of ℓ p n is bounded by a constant times 1/ √n. This is used to give a new proof of the fact that the coordinates of a random vector on the ℓ p n sphere are approximately independent with density proportional to exp(- t p ), a unification and generalization of two theorems of Diaconis and Freedman. Finally, we show in contrast that a projection of the surface measure of the ℓ p n sphere onto a random k-dimensional subspace is "close" to the k-dimensional Gaussian measure.
AB - We prove that the total variation distance between the cone measure and surface measure on the sphere of ℓ p n is bounded by a constant times 1/ √n. This is used to give a new proof of the fact that the coordinates of a random vector on the ℓ p n sphere are approximately independent with density proportional to exp(- t p ), a unification and generalization of two theorems of Diaconis and Freedman. Finally, we show in contrast that a projection of the surface measure of the ℓ p n sphere onto a random k-dimensional subspace is "close" to the k-dimensional Gaussian measure.
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U2 - 10.1016/S0246-0203(02)00008-0
DO - 10.1016/S0246-0203(02)00008-0
M3 - Article
AN - SCOPUS:0037334315
SN - 0246-0203
VL - 39
SP - 241
EP - 261
JO - Annales de l'institut Henri Poincare (B) Probability and Statistics
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
IS - 2
ER -