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- Vladimir Pestov
- 1999

This is an account of one man's view of the current perspective of theory of topological groups. We survey some recent developments which are, from our viewpoint, indicative of the future directions, concentrating on actions of topologi-cal groups on compacta, embeddings of topological groups, free topological groups, and 'massive' groups (such as groups of… (More)

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, uni-tary groups U (n) and symmetric groups S n , n ∈ N. Hyperlinear groups come from theory of operator algebras (Connes'… (More)

- Vladimir Pestov
- 2008

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isome-tries of Urysohn's universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary… (More)

We suggest that the curse of dimensionality affecting the similarity-based search in large datasets is a manifestation of the phenomenon of concentration of measure on high-dimensional structures. We prove that, under certain geometric assumptions on the query domain Ω and the dataset X, if Ω satisfies the so-called concentration property, then for most… (More)

We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness conditions, the errors of the ANOVA decompositions are of order O(n m/2) for indendent predictor variables and… (More)

- Vladimir Pestov
- 1997

This small survey of basic universal constructions related to the actions of topological groups on compacta is centred around a new result — an intrinsic description of extremely amenable topological groups (i.e., those having a fixed point in each compactum they act upon), solving a 1967 problem by Granirer. Another old problem whose solution (in the… (More)

We offer a theoretical validation of the curse of dimensionality in the pivot-based indexing of datasets for similarity search, by proving, in the framework of statistical learning, that in high dimensions no pivot-based indexing scheme can essentially outperform the linear scan. A study of the asymptotic performance of pivot-based indexing schemes is… (More)

- Vladimir Pestov
- 2005

We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups , exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik… (More)

- Thierry Giordano, Vladimir Pestov, Résumé
- 2001

— A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one.… (More)

—Exchangeable random variables form an important and well-studied generalization of i.i.d. variables, however simple examples show that no nontrivial concept or function classes are PAC learnable under general exchangeable data inputs X1, X2,. . .. Inspired by the work of Berti and Rigo on a Glivenko–Cantelli theorem for exchangeable inputs, we propose a… (More)