65 research outputs found
Decentralized Algorithms for Wasserstein Barycenters
In dieser Arbeit beschĂ€ftigen wir uns mit dem Wasserstein Baryzentrumproblem diskreter WahrscheinlichkeitsmaĂe sowie mit dem population Wasserstein Baryzentrumproblem gegeben von a FrĂ©chet Mittelwerts von der rechnerischen und statistischen Seiten. Der statistische Fokus liegt auf der SchĂ€tzung der StichprobengröĂe von MaĂen zur Berechnung einer AnnĂ€herung des FrĂ©chet Mittelwerts (Baryzentrum) der WahrscheinlichkeitsmaĂe mit einer bestimmten Genauigkeit. FĂŒr empirische Risikominimierung (ERM) wird auch die Frage der Regularisierung untersucht zusammen mit dem Vorschlag einer neuen Regularisierung, die zu den besseren KomplexitĂ€tsgrenzen im Vergleich zur quadratischen Regularisierung beitrĂ€gt. Der Rechenfokus liegt auf der Entwicklung von dezentralen Algorithmen zurBerechnung von Wasserstein Baryzentrum: duale Algorithmen und Sattelpunktalgorithmen. Die Motivation fĂŒr duale Optimierungsmethoden ist geschlossene Formen fĂŒr die duale Formulierung von entropie-regulierten Wasserstein Distanz und ihren Derivaten, wĂ€hrend, die primale Formulierung nur in einigen FĂ€llen einen Ausdruck in geschlossener Form hat, z.B. fĂŒr GauĂsches MaĂ. AuĂerdem kann das duale Orakel, das den Gradienten der dualen Darstellung fĂŒr die entropie-regulierte Wasserstein Distanz zurĂŒckgibt, zu einem gĂŒnstigeren Preis berechnet werden als das primale Orakel, das den Gradienten der (entropie-regulierten) Wasserstein Distanz zurĂŒckgibt. Die Anzahl der dualen Orakel rufe ist in diesem Fall ebenfalls weniger, nĂ€mlich die Quadratwurzel der Anzahl der primalen Orakelrufe. Im Gegensatz zum primalen Zielfunktion, hat das duale Zielfunktion Lipschitz-stetig Gradient aufgrund der starken KonvexitĂ€t regulierter Wasserstein Distanz. AuĂerdem untersuchen wir die Sattelpunktformulierung des (nicht regulierten) Wasserstein Baryzentrum, die zum Bilinearsattelpunktproblem fĂŒhrt. Dieser Ansatz ermöglicht es uns auch, optimale KomplexitĂ€tsgrenzen zu erhalten, und kann einfach in einer dezentralen Weise prĂ€sentiert werden.In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures as well as the population Wasserstein barycenter problem given by a FrĂ©chet mean from computational and statistical sides. The statistical focus is estimating the sample size of measures needed to calculate an approximation of a FrĂ©chet mean (barycenter) of probability distributions with a given precision. For empirical risk minimization approaches, the question of the regularization is also studied along with proposing a new regularization which contributes to the better complexity bounds in comparison with the quadratic regularization. The computational focus is developing decentralized algorithms for calculating Wasserstein barycenters: dual algorithms and saddle point algorithms. The motivation for dual approaches is closed-forms for the dual formulation of entropy-regularized Wasserstein distances and their derivatives, whereas the primal formulation has a closed-form expression only in some cases, e.g., for Gaussian measures.Moreover, the dual oracle returning the gradient of the dual representation forentropy-regularized Wasserstein distance can be computed for a cheaper price in comparison with the primal oracle returning the gradient of the (entropy-regularized) Wasserstein distance. The number of dual oracle calls in this case will be also less, i.e., the square root of the number of primal oracle calls. Furthermore, in contrast to the primal objective, the dual objective has Lipschitz continuous gradient due to the strong convexity of regularized Wasserstein distances. Moreover, we study saddle-point formulation of the non-regularized Wasserstein barycenter problem which leads to the bilinear saddle-point problem. This approach also allows us to get optimal complexity bounds and it can be easily presented in a decentralized setup
Distributed optimization with quantization for computing Wasserstein barycenters
We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validate our algorithmic analysis
Decentralized Distributed Optimization for Saddle Point Problems
We consider distributed convex-concave saddle point problems over arbitrary
connected undirected networks and propose a decentralized distributed algorithm
for their solution. The local functions distributed across the nodes are
assumed to have global and local groups of variables. For the proposed
algorithm we prove non-asymptotic convergence rate estimates with explicit
dependence on the network characteristics. To supplement the convergence rate
analysis, we propose lower bounds for strongly-convex-strongly-concave and
convex-concave saddle-point problems over arbitrary connected undirected
networks. We illustrate the considered problem setting by a particular
application to distributed calculation of non-regularized Wasserstein
barycenters
Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport
In this work, we propose a novel method for quantifying distances between
Toeplitz structured covariance matrices. By exploiting the spectral
representation of Toeplitz matrices, the proposed distance measure is defined
based on an optimal mass transport problem in the spectral domain. This may
then be interpreted in the covariance domain, suggesting a natural way of
interpolating and extrapolating Toeplitz matrices, such that the positive
semi-definiteness and the Toeplitz structure of these matrices are preserved.
The proposed distance measure is also shown to be contractive with respect to
both additive and multiplicative noise, and thereby allows for a quantification
of the decreased distance between signals when these are corrupted by noise.
Finally, we illustrate how this approach can be used for several applications
in signal processing. In particular, we consider interpolation and
extrapolation of Toeplitz matrices, as well as clustering problems and tracking
of slowly varying stochastic processes
Dynamical Optimal Transport on Discrete Surfaces
We propose a technique for interpolating between probability distributions on
discrete surfaces, based on the theory of optimal transport. Unlike previous
attempts that use linear programming, our method is based on a dynamical
formulation of quadratic optimal transport proposed for flat domains by Benamou
and Brenier [2000], adapted to discrete surfaces. Our structure-preserving
construction yields a Riemannian metric on the (finite-dimensional) space of
probability distributions on a discrete surface, which translates the so-called
Otto calculus to discrete language. From a practical perspective, our technique
provides a smooth interpolation between distributions on discrete surfaces with
less diffusion than state-of-the-art algorithms involving entropic
regularization. Beyond interpolation, we show how our discrete notion of
optimal transport extends to other tasks, such as distribution-valued Dirichlet
problems and time integration of gradient flows
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