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Vladimir G. Pestov

Publications and source records attributed to Vladimir G. Pestov.

At least 19 recordsLinked to original sources
arXiv2026

Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. III

We establish the last missing link allowing to describe those complete separable metric spaces $X$ in which the $k$ nearest neighbour classifier is universally consistent, both in combinatorial terms of dimension theory and via a fundamental property of real analysis. The following are equivalent: (1) The $k$-nearest neighbour classifier is universally consistent in $X$, (2) The strong Lebesgue--Besicovitch differentiation property holds in $X$ for every locally finite Borel measure, (3) $X$ is sigma-finite dimensional in the sense of Jun-Iti Nagata. The equivalence (2)$\iff$(3) was announced by Preiss (1983), while a detailed proof of the implication (3)$\Rightarrow$(2) has only appeared in Assouad and Quentin de Gromard (2006). The implication (2)$\Rightarrow$(1) was established by Cérou and Guyader (2006). We prove the implication (1)$\Rightarrow$(3). We further show that the weak (instead of strong) Lebesgue--Besicovitch property is insufficient for the consistency of the $k$-NN rule, as witnessed, for example, by the Heisenberg group (here we correct a wrong claim made in the previous article (Kumari and Pestov 2024)). A bit counter-intuitively, there is a metric on the real line uniformly equivalent to the usual distance but under which the $k$-NN classifier fails. Finally, another equivalent condition that can be added to the above is the Cover--Hart property: (4) the error of the $1$-nearest neighbour classifier is asymptotically at most twice as bad as the Bayes error.

cs.LG↗
arXiv2026

On finite-dimensional encoding/decoding theorems for neural operators

Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations, one employs finite-dimensional encoding/decoding theorems of the following kind: every continuous mapping $f$ between function spaces $E$ and $F$ is approximated in the topology of uniform convergence on compacta by continuous mappings factoring through two finite dimensional Banach spaces. Such a result is known (Kovachki et al., 2023) for $E,F$ being Banach spaces having the approximation property. We point out that the result needs no assumptions on $E,F$ whatsoever and remains true not only for all normed spaces, but for arbitrary locally convex spaces as well. At the same time, an analogous result for $C^k$-smooth mappings and the $C^k$ compact open topology, $k\geq 1$, holds if and only if the space $E$ has the approximation property. This analysis may be useful already because non-normable locally convex function spaces are common in the theory of differential equations, the main field of applications for the emerging theory.

math.FA↗
arXiv2024

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (that is, Sobolev class $H^1$) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

math.FA↗
arXiv2024

Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension. II

We continue to investigate the $k$ nearest neighbour ($k$-NN) learning rule in complete separable metric spaces. Thanks to the results of Cérou and Guyader (2006) and Preiss (1983), this rule is known to be universally consistent in every such metric space that is sigma-finite dimensional in the sense of Nagata. Here we show that the rule is strongly universally consistent in such spaces in the absence of ties. Under the tie-breaking strategy applied by Devroye, Györfi, Krzyżak, and Lugosi (1994) in the Euclidean setting, we manage to show the strong universal consistency in non-Archimedian metric spaces (that is, those of Nagata dimension zero). Combining the theorem of Cérou and Guyader with results of Assouad and Quentin de Gromard (2006), one deduces that the $k$-NN rule is universally consistent in metric spaces having finite dimension in the sense of de Groot. In particular, the $k$-NN rule is universally consistent in the Heisenberg group which is not sigma-finite dimensional in the sense of Nagata as follows from an example independently constructed by Korányi and Reimann (1995) and Sawyer and Wheeden (1992).

cs.LG↗
arXiv2023

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

math.GR↗
arXiv2021

On epimorphisms in some categories of infinite-dimensional Lie groups

Let $X$ be a smooth compact connected manifold. Let $G=\mbox{Diff}\, X$ be the group of diffeomorphisms of $X$, equipped with the $C^\infty$-topology, and let $H$ be the stabilizer of some point in $X$. Then the inclusion $H\to G$, which is a morphism of two regular Fréchet--Lie groups, is an epimorphism in the category of smooth Lie groups modelled on complete locally convex spaces. At the same time, in the latter category, epimorphisms between finite dimensional Lie groups have dense range. We also prove that if $G$ is a Banach--Lie group and $H$ is a proper closed subgroup, the inclusion $H\to G$ is not an epimorphism in the category of Hausdorff groups.

math.RT↗
arXiv2020

Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension

The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

math.MG↗
arXiv2020

A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals

We show that the $k$-nearest neighbour learning rule is universally consistent in a metric space $X$ if and only if it is universally consistent in every separable subspace of $X$ and the density of $X$ is less than every real-measurable cardinal. In particular, the $k$-NN classifier is universally consistent in every metric space whose separable subspaces are sigma-finite dimensional in the sense of Nagata and Preiss if and only if there are no real-valued measurable cardinals. The latter assumption is relatively consistent with ZFC, however the consistency of the existence of such cardinals cannot be proved within ZFC. Our results were inspired by an example sketched by Cérou and Guyader in 2006 at an intuitive level of rigour.

cs.LG↗
arXiv2019

Elementos da teoria de aprendizagem de máquina supervisionada

This is a set of lecture notes for an introductory course (advanced undergaduates or the 1st graduate course) on foundations of supervised machine learning (in Portuguese). The topics include: the geometry of the Hamming cube, concentration of measure, shattering and VC dimension, Glivenko-Cantelli classes, PAC learnability, universal consistency and the k-NN classifier in metric spaces, dimensionality reduction, universal approximation, sample compression. There are appendices on metric and normed spaces, measure theory, etc., making the notes self-contained. Este é um conjunto de notas de aula para um curso introdutório (curso de graduação avançado ou o 1o curso de pós) sobre fundamentos da aprendizagem de máquina supervisionada (em Português). Os tópicos incluem: a geometria do cubo de Hamming, concentração de medida, fragmentação e dimensão de Vapnik-Chervonenkis, classes de Glivenko-Cantelli, aprendizabilidade PAC, consistência universal e o classificador k-NN em espaços métricos, redução de dimensionalidade, aproximação universal, compressão amostral. Há apêndices sobre espaços métricos e normados, teoria de medida, etc., tornando as notas autosuficientes.

cs.LG↗
arXiv2017

On amenability and groups of measurable maps

We show that if $G$ is an amenable topological group, then the topological group $L^{0}(G)$ of strongly measurable maps from $([0,1],λ)$ into $G$ endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group $G$ is amenable if $L^{0}(G)$ is.

math.FA↗
arXiv2017

Amenability versus property (T) for non locally compact topological groups

For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T) admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property (T), and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property (T) and property (FH) in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property (T), having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property (T) is closed under arbitrary infinite products with the usual product topology.

math.GR↗
arXiv2017

A topological transformation group without non-trivial equivariant compactifications

There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to Megrelishvili, which is a metric fan equipped with a certain group of homeomorphisms. The question of existence of a topological transformation group with the property in the title was asked by Yu.M. Smirnov in the 1980s.

math.DS↗
arXiv2016

On topological groups admitting a base at identity indexed with $ω^ω$

A topological group $G$ is said to have a local $ω^ω$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $ω^ω$. In particular, every metrizable group is such, but the class of groups with a local $ω^ω$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-arichimedean ordered fields lead to natural families of non-metrizable groups with a local $ω^ω$-base which nevertheless are Baire topological spaces. More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $ω^ω$-base admits a local $ω^ω$-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $ω^ω$-base if and only if the finest uniformity of $X$ has a $ω^ω$-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $ω^ω$-base provided $X$ is separable and the finest uniformity of $X$ has a $ω^ω$-base.

math.GN↗
arXiv2015

Projectively universal countable metrizable groups

We prove that there exists a countable metrizable topological group $G$ such that every countable metrizable group is isomorphic to a quotient of $G$. The completion $H$ of $G$ is a Polish group such that every Polish group is isomorphic to a quotient of $H$.

math.GR↗
arXiv2015

On topological groups with an approximate fixed point property

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability.

math.GR↗
arXiv2013

PFMFind: a system for discovery of peptide homology and function

Protein Fragment Motif Finder (PFMFind) is a system that enables efficient discovery of relationships between short fragments of protein sequences using similarity search. It supports queries based on score matrices and PSSMs obtained through an iterative procedure similar to PSI-BLAST. PSSM construction is customisable through plugins written in Python. PFMFind consists of a GUI client, an algorithm using an index for fast similarity search and a relational database for storing search results and sequence annotations. It is written mostly in Python. All components communicate between themselves using TCP/IP sockets and can be located on different physical machines.

q-bio.QM↗
arXiv2012

An introduction to hyperlinear and sofic groups

This is an edited write-up of lecture notes of the 7-th Appalachian set theory workshop of the same title led by the first named author at the Cornell University on November 22, 2008. A draft version of the notes was prepared by the second named author. This presentation is largely complementary to the earlier survey by the first-named author (Hyperlinear and sofic groups: a brief guide, Bull. Symb. Logic 14 (2008), pp. 449-480; arXiv:0804.3968v8 [math.GR]).

math.GR↗