?
Remarks on curvature in the transportation metric
Cornell University
,
2016.
Klartag B., Kolesnikov A.
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 theorem showing that for every $\Phi$ solving this equation on a proper convex domain $\Omega$ the corresponding metric measure space $(D^2 \Phi, e^{\Phi}dx)$ has a non-positive Bakry-{\'E}mery tensor. Modifying the Calabi's computations we obtain this result by applying tensorial maximum principle to the weighted Laplacian of the Bakry-{\'E}mery tensor. All of the computations are carried out in the generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of probability measures we prove a third-order uniform dimension-free a priori estimate in spirit of the second-order Caffarelli's contraction theorem.
Keywords: Monge-Kantorovich problemMonge-Ampere equationзадача Монжа-КанторовичаBakry-Emery tensorуравнение Монжа-Амператензор Бакри-Эмери
Publication based on the results of:
Kolesnikov A., Milman E., / Cornell University. Series arXiv "math". 2015.
Given a probability measure \mu supported on a convex subset \Omega of Euclidean space (\mathbb{R}^d,g_0), we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on (\Omega,g_0,\mu). To this end, we change the metric g_0 to a more general Riemannian one g, adapted in a certain sense to \mu, and perform our analysis on (\Omega,g,\mu). The types of metrics we consider are Hessian metrics (intimately related ...
Added: February 23, 2016
Kolesnikov A., Lysenko N. Y., / Cornell University. 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
Zimin A., Gladkov N., / Cornell University. 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
Kolesnikov A., / Cornell University. Series math "arxiv.org". 2012. No. 1201.2342.
We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the ...
Added: March 28, 2013
Kolesnikov A., Zaev D., / Cornell University. 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
Kolesnikov A., Werner E., Caglar U., / Cornell University. Series arXiv "math". 2021.
We further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities, new affine invariant entropy inequalities and new inequalities on functional affine surface area The functional inequalities lead to new affine invariant inequalities for convex bodies. Equality characterizations in these inequalities are related to a Monge Amp`ere differential ...
Added: December 4, 2021
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
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., 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
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
Kolesnikov A., Tikhonov S. Y., Calculus of Variations and Partial Differential Equations 2014 Vol. 49 No. 3-4 P. 1187-1197
Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 ...
Added: March 12, 2014
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
Smirnov E., Пенков И., Игнатьев М. В. et al., М. : ВИНИТИ РАН, 2018
Сборник трудов семинара по алгебре и геометрии Самарского государственного университета ...
Added: August 19, 2018
Kolesnikov A., Мильман Э., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 464 № 2 С. 136-140
It is well known that Poincarétype inequalities on Riemannian manifolds with measure satisfying the generalized Bakry–Émery condition can be obtained by using the Bochner–Lichnerowicz–Weitzenböck formula. In the case of manifolds with boundary, a suitable generalization is Reilly’s formula. New Poincaré type inequalities for manifolds with measure are obtained by systematically using this formula combined with ...
Added: February 23, 2016
Gladkov N., Zimin A., SIAM Journal on Mathematical Analysis 2020 Vol. 52 No. 4 P. 3666-3696
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 a nonconstant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique ...
Added: August 21, 2020
Kolesnikov A., Kudryavtseva O., Nagapetyan T., / Cornell University. Series math "arxiv.org". 2013.
The classical concept of the revealed preferences was introduced by P. Samuelson and studied by H.S. Houthakker, M. Richter, S. Afriat, H. Varian and many others. It was shown by Afriat that the so called SARP (or cyclically consistence) axiom is a necessary and sufficient condition for existence of an appropriate concave utility function for ...
Added: February 23, 2013
Zaev D., / Cornell University. Series math "arxiv.org". 2014.
We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal plans need to satisfy. Our main results are quite general and include several important examples. In particular, they include Monge-Kantorovich problems in the classes of ...
Added: May 14, 2014
Caglar U., Kolesnikov A., Werner E., Indiana University Mathematics Journal 2022 Vol. 71 No. 6 P. 2309-2333
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities ...
Added: June 23, 2023
Gladkov N., Kolesnikov A., Zimin A., / Cornell University. 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., Klartag B., / Cornell University. Series math "arxiv.org". 2013. No. 1402.2636.
We investigate the Brenier map \nabla \Phi between the uniform measures on two convex domains in \mathbb{R}^n or more generally, between two log-concave probability measures on \mathbb{R}^n. We show that the eigenvalues of the Hessian matrix D^2 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or ...
Added: March 12, 2014
Kolesnikov A., Мильман Э., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2016 Т. 470 № 2 С. 137-140
В работе получены оценки типа Пуанкаре для логарифмически вогнутой меры $\mu$ на
выпуклом множестве $\Omega$. Для этой цели $\Omega$ наделяется римановой метрикой $g$, в которой
риманово многообразие с мерой $(\Omega, g, \mu)$ имеет неотрицательный тензор Бакри-Эмери и,
как следствие, удовлетворяет неравенству Браскампа-Либа.
Рассмотрены несколько естественных классов метрик (гессиановы, конформные),
каждая из которых дает новые весовые неравенства типа Пуанкаре, Харди, логарифмического ...
Added: December 27, 2016
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
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