14 research outputs found
Statistical inference for Bures--Wasserstein barycenters
In this work we introduce the concept of Bures--Wasserstein barycenter , that is essentially a Fréchet mean of some distribution supported on a subspace of positive semi-definite -dimensional Hermitian operators . We allow a barycenter to be constrained to some affine subspace of , and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of in both Frobenius norm and Bures--Wasserstein distance, and explain, how the obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics
Statistical inference for Bures-Wasserstein barycenters
In this work we introduce the concept of Bures-Wasserstein barycenter ,
that is essentially a Fr\'echet mean of some distribution
supported on a subspace of positive semi-definite Hermitian operators
. We allow a barycenter to be restricted to some affine
subspace of and provide conditions ensuring its existence
and uniqueness. We also investigate convergence and concentration properties of
an empirical counterpart of 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.Comment: 37 pages, 5 figure
Inexact Model: A Framework for Optimization and Variational Inequalities
In this paper we propose a general algorithmic framework for first-order
methods in optimization in a broad sense, including minimization problems,
saddle-point problems and variational inequalities. This framework allows to
obtain many known methods as a special case, the list including accelerated
gradient method, composite optimization methods, level-set methods, proximal
methods. The idea of the framework is based on constructing an inexact model of
the main problem component, i.e. objective function in optimization or operator
in variational inequalities. Besides reproducing known results, our framework
allows to construct new methods, which we illustrate by constructing a
universal method for variational inequalities with composite structure. This
method works for smooth and non-smooth problems with optimal complexity without
a priori knowledge of the problem smoothness. We also generalize our framework
for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page
On the complexity of approximating Wasserstein barycenter
We study the complexity of approximating Wassertein barycenter of discrete measures, or
histograms by contrasting two alternative approaches, both using entropic regularization. We provide
a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm
to approximate the original non-regularized barycenter. We also get the complexity bound for alternative
accelerated-gradient-descent-based approach and compare it with the bound obtained
for IBP. As a byproduct, we show that the regularization parameter in both approaches has to
be proportional to ", which causes instability of both algorithms when the desired accuracy is
high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as
a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also
consider the question of scalability of these algorithms using approaches from distributed optimization
and show that the first algorithm can be implemented in a centralized distributed setting
(master/slave), while the second one is amenable to a more general decentralized distributed
setting with an arbitrary network topology
On the complexity of approximating Wasserstein barycenter
We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology
On the Complexity of Approximating Wasserstein Barycenter
We study the complexity of approximating Wassertein barycenter of
discrete measures, or histograms of size by contrasting two alternative
approaches, both using 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 to
approximate the original non-regularized barycenter. Using an alternative
accelerated-gradient-descent-based approach, we obtain a complexity
proportional to . As a byproduct, we show that
the regularization parameter in both approaches has to be proportional to
, which causes instability of both algorithms when the desired
accuracy is high. To overcome this issue, we propose a novel proximal-IBP
algorithm, which can be seen as a proximal gradient method, which uses IBP on
each iteration to make a proximal step. We also consider the question of
scalability of these algorithms using approaches from distributed optimization
and show that the first algorithm can be implemented in a centralized
distributed setting (master/slave), while the second one is amenable to a more
general decentralized distributed setting with an arbitrary network topology.Comment: Corrected misprints. Added a reference to accelerated Iterative
Bregman Projections introduced in arXiv:1906.0362
Gradient methods for problems with inexact model of the objective
We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments
Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT
International audienceIn this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [5] as a regularization of Wasserstein barycenters [1]. After characterizing these barycenters in terms of a system of Monge-AmpĂšre equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters
Multiplier bootstrap for Bures--Wasserstein barycenters
Bures-Wasserstein barycenter is a popular and promising tool in analysis of complex data like graphs, images etc. In many applications the input data are random with an unknown distribution, and uncertainty quantification becomes a crucial issue. This paper offers an approach based on multiplier bootstrap to quantify the error of approximating the true Bures--Wasserstein barycenter Qâ by its empirical counterpart Qn. The main results state the bootstrap validity under general assumptions on the data generating distribution P and specifies the approximation rates for the case of sub-exponential P. The performance of the method is illustrated on synthetic data generated from the weighted stochastic block model