TY - JOUR
T1 - Logarithmic reduction of the level of randomness in some probabilistic geometric constructions
AU - Artstein-Avidan, S.
AU - Milman, V. D.
N1 - Funding Information:
✩ This research was partially supported by BSF grant 2002-006, the first named author was also supported by the National Science Foundation under agreement No. DMS-0111298. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. * Corresponding author. E-mail addresses: artstein@princeton.edu (S. Artstein-Avidan), milman@post.tau.ac.il (V.D. Milman).
PY - 2006/6/1
Y1 - 2006/6/1
N2 - Many of the surprising phenomena occurring in high dimensions are proved by use of probabilistic arguments, which show the existence of organized and regular structures but do not hint as to where exactly do these structures lie. It is an intriguing question whether some of them could be realized explicitly. In this paper we show that the amount of randomness used can be reduced significantly in many of these questions from asymptotic convex geometry, and most of the random steps can be substituted by completely explicit algorithmic steps. The main tool we use is random walks on expander graphs.
AB - Many of the surprising phenomena occurring in high dimensions are proved by use of probabilistic arguments, which show the existence of organized and regular structures but do not hint as to where exactly do these structures lie. It is an intriguing question whether some of them could be realized explicitly. In this paper we show that the amount of randomness used can be reduced significantly in many of these questions from asymptotic convex geometry, and most of the random steps can be substituted by completely explicit algorithmic steps. The main tool we use is random walks on expander graphs.
KW - Asymptotic geometric analysis
KW - Explicit constructions
KW - Randomness reduction
KW - Sections of ℓ
UR - http://www.scopus.com/inward/record.url?scp=33645900266&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2005.11.003
DO - 10.1016/j.jfa.2005.11.003
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AN - SCOPUS:33645900266
SN - 0022-1236
VL - 235
SP - 297
EP - 329
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -