Puchkin N., Noskov F., Spokoiny V., Bernoulli: a journal of mathematical statistics and probability 2025 Vol. 31 No. 2 P. 1664–1691
We study properties of a sample covariance estimate $\widehat \Sigma$ given a finite sample of $n$ i.i.d. centered random elements in $\mathbb{R}^d$ with the covariance matrix $\Sigma$. We derive dimension-free bounds on the squared Frobenius norm of $(\widehat\Sigma - \Sigma)$ under reasonable assumptions. For instance, we show that $\smash{\|\widehat\Sigma - \Sigma\|_{\rm F}^2}$ differs from its ...
Added: February 12, 2025
Durmus A., Moulines E., Naumov A. et al., / Series arXiv "math". 2023.
In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein-type inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. Our proof technique is, as far as we know, ...
Added: June 18, 2023
Kolesnikov A., Zimin A., Sandomirskiy F. et al., / Series Theoretical Economics "arxiv.org". 2022. No. 2203.06837.
We consider the problem of revenue-maximizing Bayesian auction design with several i.i.d. bidders and several items. We show that the auction-design problem can be reduced to the problem of continuous optimal transportation introduced by Beckmann. We establish the strong duality between the two problems and demonstrate the existence of solutions. We then develop a new ...
Added: April 10, 2022
Kolesnikov A., Werner E., Advances in Mathematics 2022 Vol. 396 Article 108110
Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke–Santaló inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a “pointwise Prékopa–Leindler inequality” and show a monotonicity property of the multimarginal Blaschke–Santaó functional. ...
Added: December 4, 2021
Gladkov N., Kolesnikov A., Zimin A., Journal of Mathematical Analysis and Applications 2022 Vol. 506 No. 2 Article 125666
The multistochastic Monge–Kantorovich problem on the product X=∏i=1nXi of n spaces is a generalization of the multimarginal Monge–Kantorovich problem. For a given integer number 1≤k<n we consider the minimization problem ∫cdπ→inf on the space of measures with fixed projections onto every Xi1×…×Xik for arbitrary set of k indices {i1,…,ik}⊂{1,…,n}. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual ...
Added: December 4, 2021
Zimin A., Gladkov N., / Series arXiv "math". 2018.
We construct an explicit solution for the multimarginal transportation problem on the unit cube [0,1]3 with the cost function xyz and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to ...
Added: October 10, 2018
Gladkov N., Kolesnikov A., Zimin A., / Series arXiv "math". 2018.
The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], ...
Added: July 31, 2018
Kolesnikov A., Milman E., Annals of Probability 2018 Vol. 46 No. 6 P. 3578–3615
What is the optimal way to cut a convex bounded domain K in Euclidean space (Rn,|⋅|) into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lov\'asz and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of n) in the surface area, ...
Added: March 26, 2018
Bogachev V., Калинин А. Н., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 463 № 4 С. 383–386
Установлены точные условия равенства минимумов в задачах Монжа и Канторовича ...
Added: November 15, 2017
Bogachev V., Калинин А. Н., Popova S., Записки научных семинаров ПОМИ РАН 2017 Т. 457 С. 53–73
Статья посвящена исследованию условий, при которых задачи Монжа и Канторовича с непрерывной функцией стоимости на произведении двух вполне регулярных пространств и двумя заданными безатомическими радоновскими мерами-проекциями на эти пространства имеют совпадающие значения соответствующих инфимумов. ...
Added: November 1, 2017
Klartag B., Kolesnikov A., / Series math "arxiv.org". 2016.
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the ``hyperbolic" toric K\"ahler-Einstein equation $e^{\Phi} = \det D^2 \Phi$ on proper convex cones. We prove a generalization of this ...
Added: April 14, 2016
Zaev D., Записки научных семинаров ПОМИ РАН 2015 Т. 437 С. 100–130
Пусть X – польское топологическое пространство. P(X) – множество вероятностных борелевских мер на нем, T:X→X – гомеоморфизм. Мы доказываем, что для симплекса Dom⊆P инвариантных относительно T мер значение метрики Канторовича на Dom можно полностью восстановить, зная только ее значения на крайних точках. Этот факт тесно связан со следующим результатом: инвариантный оптимальный транспортный план может быть представлен как смесь инвариантных оптимальных транспортных планов между крайними точками ...
Added: March 9, 2016
Zaev D., Математические заметки 2015 Т. 98 № 5 С. 664–683
В работе рассматривается задача Монжа–Канторовича с дополнительным ограничением: допустимый транспортный план должен обращаться в нуль на некотором фиксированном подпространстве функций. Различный выбор подпространств порождает различные дополнительные условия на транспортные планы. Наши основные результаты сформулированы в общем виде и распространяются на ряд важных частных случаев. В том числе, они верны для задачи Монжа–Канторовича, решаемой в классе инвариантных ...
Added: March 9, 2016
Kolesnikov A., Bulletin des Sciences Mathematiques 2014 Vol. 138 No. 2 P. 165–198
Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν , 2) T has the form T=ϕ∇ϕ|∇ϕ| , where ϕ is a function with convex sublevel sets. We prove a change of variables formula for T . We also establish Sobolev estimates for ϕ ...
Added: February 24, 2016
Kolesnikov A., Lysenko N. Y., / Series arXiv "math". 2015.
We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to minimum of a finite number n of affine functions li satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of n products Ii×Ji, where {Ii}, {Ji} are partitions of the line into unions ...
Added: February 23, 2016
Kolesnikov A., Zaev D., / Series arXiv "math". 2015.
We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, ...
Added: February 23, 2016
Kolesnikov A., Theory of Probability and Its Applications 2013 Vol. 57 No. 2 P. 243–264
We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \, dx$ and $\nu=e^{-W} \, dx$ on ${\bf R}^d$. Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \, d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm, is controlled by the Fisher information ...
Added: December 23, 2015
Колесников А., Bulletin des Sciences Mathematiques 2014 Vol. 138 No. 2 P. 165–198
Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form <img />T=φ∇φ|∇φ|, where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and ...
Added: December 23, 2015
Bogachev V., Колесников А., Успехи математических наук 2012 Т. 67 № 5 С. 3–110
Дан обзор совеременного состояния исследований, связанных с задачами Монжа и Канторовича оптимальной транспортировки мер. ...
Added: February 26, 2014
Kudryavtseva O., Nagapetyan T., Kolesnikov A., Journal Mathematical Economics, Netherlands 2013 Vol. 49 P. 501–505
The famous Afriat’s theorem from the theory of revealed preferences establishes necessary and sufficient conditions for the existence of utility function for a given set of choices and prices. The result on the existence of a homogeneous utility function can be considered as a particular fact of the Monge–Kantorovich mass transportation theory. In this paper ...
Added: September 27, 2013
Kolesnikov A., Богачев В. И., Доклады Академии наук 2012 Т. 44 № 2 С. 131–136
Работа связана с изучением соболевской регулярности отображений
оптимальной транспортировки в бесконечномерных пространствах, наделенных гауссовской мерой. Найдены условия принадлежности соболевскому классу для таких отображений. Доказана формула замены переменных. ...
Added: February 19, 2013