Projecting the surface measure of the sphere of ℓ                         n                         p

Assaf Naor; Dan Romik

doi:10.1016/S0246-0203(02)00008-0

Projecting the surface measure of the sphere of ℓ ⁿ _p

Assaf Naor, Dan Romik

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72 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	241-261
Number of pages	21
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	39
Issue number	2
DOIs	https://doi.org/10.1016/S0246-0203(02)00008-0
State	Published - 2003
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1016/S0246-0203(02)00008-0

Cite this

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title = " Projecting the surface measure of the sphere of ℓ n p ",

abstract = " 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.",

author = "Assaf Naor and Dan Romik",

note = "Funding Information: ∗Corresponding author. E-mail addresses: naor@math.huji.ac.il (A. Naor), romik@math.tau.ac.il (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.",

year = "2003",

doi = "10.1016/S0246-0203(02)00008-0",

language = "English (US)",

volume = "39",

pages = "241--261",

journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "2",

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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: naor@math.huji.ac.il (A. Naor), romik@math.tau.ac.il (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

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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 -

Projecting the surface measure of the sphere of ℓ ⁿ _p

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