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## Statistical inference for Bures-Wasserstein barycenters

Annals of Applied Probability. 2021. Vol. 31. No. 3. P. 1264-1298.

n this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fr\'echet mean of some distribution $P$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$.

We allow a barycenter to be constrained to some affine subspace of $\mathbb{H}_{+}(d)$ and provide conditions ensuring its existence and uniqueness.

We also investigate convergence and concentration properties of an empirical counterpart of $Q_*$ in both Frobenius norm and Bures-Wasserstein distance, and explain, how obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.

Publication based on the results of:

Kroshnin A., Journal of Convex Analysis 2018 Vol. 25 No. 4 P. 1371-1395

We consider the space P(X) of probability measures on arbitrary Radon space X endowed with a transportation cost J(μ, ν) generated by a nonnegative continuous cost function. For a probability distribution on P(X) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law ...

Added: November 23, 2018

Guillaume Carlier, Katharina Eichinger, Kroshnin A., SIAM Journal on Mathematical Analysis 2021 Vol. 53 No. 5 P. 5880-5914

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [J. Bigot, E. Cazelles, and N. Papadakis, SIAM J. Math. Anal., 51 (2019), pp. 2261--2285] as a regularization of Wasserstein barycenters [M. Agueh and G. Carlier, SIAM J. Math. Anal., 43 (2011), pp. 904--924]. After characterizing these barycenters in terms of a system of Monge--Ampère ...

Added: October 27, 2021

Tolmachov K., Variance of Linear Statistic for Plancherel Young Diagrams / . 2012.

In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by π/4). We also give an example of a local configuration with linearly growing variance in a fixed regime and prove the central limit ...

Added: December 8, 2013

Kolesnikov A., Emanuel Milman, 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

Kolesnikov A., Elisabeth Werner, 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

Kroshnin A., Darina Dvinskikh, Dvurechensky P. et al., , in: Proceedings of Machine Learning Research. Vol. 97: International Conference on Machine Learning, 9-15 June 2019, Long Beach, California, USA.: PMLR, 2019.. P. 3530-3540.

We study the complexity of approximating the Wasserstein barycenter of m discrete measures, or histograms of size n, by contrasting two alternative approaches that use entropic regularization. The first approach is based on the Iterative Bregman Projections (IBP) algorithm for which our novel analysis gives a complexity bound proportional to $m n^2 / \epsilon^2$ to approximate the original non-regularized barycenter. ...

Added: June 11, 2019

Vladimir Spokoiny, Theory of Probability and Its Applications 2014 Vol. 58 No. 2 P. 314-323

In this paper we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimization problem. As an application ...

Added: October 22, 2014

Гуминов С. В., Dvurechensky P., Tupitsa N. et al., Proceedings of Machine Learning Research 2021 Vol. 139 P. 3886-3898

Alternating minimization (AM) procedures are practically efficient in many applications for solving convex and non-convex optimization problems. On the other hand, Nesterov's accelerated gradient is theoretically optimal first-order method for convex optimization. In this paper we combine AM and Nesterov's acceleration to propose an accelerated alternating minimization algorithm. We prove $1/k^2$ convergence rate in terms ...

Added: September 29, 2021

Kolesnikov A., Tikhonov S. Y., Regularity of the Monge-Ampère equation in Besov's space / Cornell University. Series math "arxiv.org". 2012. No. 1203.3457.

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 28, 2013

Annales de l'institut Henri Poincare (B) Probability and Statistics 2017 Vol. 53 No. 4 P. 1719-1746

Nous relions les inégalités de transport-entropie à l'étude des points critiques de la fonction. Cette approche conduit en particulier à une nouvelle preuve du résultat de Otto et Villani 2000 montrant que le droit logarithmique de Sobolev implique l'inégalité des transports chez Talagrand. ...

Added: June 7, 2018

Friedrich Goetze, Naumov A.A., Tikhomirov A., Bernoulli: a journal of mathematical statistics and probability 2017 Vol. 23 No. 4B P. 3067-3113

In this paper we consider the product of two independent random matrices X^(1) and X^(2). Assume that X_{jk}^{(q)},1\le j,k \le n,q=1,2,, are i.i.d. random variables with \EX_{jk}^{q}=0, VarX_{jk}^{(q)}=1/ Denote by s_1(W),…,s_n(W) the singular values of W:=n^{-1}X^(1)X^(2). We prove the central limit theorem for linear statistics of the squared singular values s_1^2(W),…,s_n^2(W) showing that the limiting variance depends on \kappa_4:=\E(X_{11}^{(1)})^4−3. ...

Added: April 28, 2018

Bobkov S., Ulyanov V. V., Theory of Probability and Its Applications 2022 Vol. 66 No. 4 P. 537-549

We give a short overview of the results related to the refined forms of the central limit theorem, with a focus on independent integer-valued random variables (r.v.'s). In the independent and non-identically distributed (non-i.i.d.) case, an approximation is then developed for the distribution of the sum by means of the Chebyshev--Edgeworth correction containing the moments ...

Added: February 22, 2022

Kelbert M., Igor Karpikov, Science and Business: Ways of Development 2018 Vol. 79 No. 1 P. 56-68

This article gives a brief summary on the main theoretical and practical results for the Scale functions. The article is organized in the following way: the first part describes the main theoretical concepts of Lévy processes, gives the formal definition and analytical properties of the Scale function. The second part describes the most significant practical ...

Added: April 5, 2018

Delon J., Salomon J., Sobolevski A., SIAM Journal of Discrete Mathematics 2012 Vol. 26 No. 2 P. 801-827

In this paper, we introduce a class of local indicators that enable us to compute efficiently optimal transport plans associated with arbitrary weighted distributions of N demands and M supplies in R in the case where the cost function is concave. Indeed, whereas this problem can be solved linearly when the cost is a convex ...

Added: May 30, 2012

Panov V., Limit theorems for sums of random variables with mixture distribution / Cornell University. Series arXiv "math". 2017. No. 1703.10463.

In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions ...

Added: March 31, 2017

Bufetov A. I., Sevak Mkrtchyan, Maria Scherbina et al., Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles / Cornell University. Series math "arxiv.org". 2013. No. 1301.0342.

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles. ...

Added: February 21, 2013

Zhivotovskiy N., Electronic Journal of Probability 2020 Vol. 25 P. 1-30

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector X∈RnX∈Rn with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of XX itself. Our results recover, in particular, the ...

Added: August 2, 2020

Nikitin Y. Y., Valentin Vladimirovich Petrov, Andrei Yurievich Zaitsev et al., Vestnik of the St. Petersburg University: Mathematics 2018 Vol. 51 No. 2 P. 201-232

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the ...

Added: October 1, 2019

X. Bressaud, Bufetov A.I., P. Hubert, Proceedings of the London Mathematical Society 2014 Vol. 109 No. 2 P. 483-522

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. The functions γ we consider are the corresponding eigenfunctions. In Theorem 1.1, we prove that the limit inferior of the ergodic sums (n,γ(x_0)+⋯+γ(x_{n−1})) n∈N is bounded for every point x in the phase space. In Theorem ...

Added: October 23, 2014

Gribkova N., Probability and Mathematical Statistics 2017 Vol. 37 No. 1 P. 101-118

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. (1982) – the first and, as far as we know, the single article where some results ...

Added: February 28, 2020

Litvin Y. V., Игорь Вячеслвович Абрамов, Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016

Furmanov K. K., I. M. Nikol'skii, Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016

Arzhantsev I., Journal of Lie Theory 2000 Vol. 10 No. 2 P. 345-357

Added: July 8, 2014

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017