Academic literature on the topic 'Entropy optimal transport'
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Journal articles on the topic "Entropy optimal transport"
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Tong, Qijun, and Kei Kobayashi. "Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions." Entropy 23, no. 3 (March 3, 2021): 302. http://dx.doi.org/10.3390/e23030302.
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The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and q-normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and q-normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate q-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.
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Bonafini, Mauro, and Bernhard Schmitzer. "Domain decomposition for entropy regularized optimal transport." Numerische Mathematik 149, no. 4 (November 19, 2021): 819–70. http://dx.doi.org/10.1007/s00211-021-01245-0.
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AbstractWe study Benamou’s domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback–Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient implementation, such as adaptive sparsity, a coarse-to-fine scheme and parallelization, paving the way to numerically solving large-scale optimal transport problems. We demonstrate efficient numerical performance for computing the Wasserstein-2 distance between 2D images and observe that, even without parallelization, domain decomposition compares favorably to applying a single efficient implementation of the Sinkhorn algorithm in terms of runtime, memory and solution quality.
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Bao, Han, and Shinsaku Sakaue. "Sparse Regularized Optimal Transport with Deformed q-Entropy." Entropy 24, no. 11 (November 10, 2022): 1634. http://dx.doi.org/10.3390/e24111634.
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Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy (q=1) and the squared 2-norm (q=0), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed.
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Essid, Montacer, Debra F. Laefer, and Esteban G. Tabak. "Adaptive optimal transport." Information and Inference: A Journal of the IMA 8, no. 4 (May 16, 2019): 789–816. http://dx.doi.org/10.1093/imaiai/iaz008.
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AbstractAn adaptive, adversarial methodology is developed for the optimal transport problem between two distributions $\mu $ and $\nu $, known only through a finite set of independent samples $(x_i)_{i=1..n}$ and $(y_j)_{j=1..m}$. The methodology automatically creates features that adapt to the data, thus avoiding reliance on a priori knowledge of the distributions underlying the data. Specifically, instead of a discrete point-by-point assignment, the new procedure seeks an optimal map $T(x)$ defined for all $x$, minimizing the Kullback–Leibler divergence between $(T(x_i))$ and the target $(y_j)$. The relative entropy is given a sample-based, variational characterization, thereby creating an adversarial setting: as one player seeks to push forward one distribution to the other, the second player develops features that focus on those areas where the two distributions fail to match. The procedure solves local problems that seek the optimal transfer between consecutive, intermediate distributions between $\mu $ and $\nu $. As a result, maps of arbitrary complexity can be built by composing the simple maps used for each local problem. Displaced interpolation is used to guarantee global from local optimality. The procedure is illustrated through synthetic examples in one and two dimensions.
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PEYRÉ, GABRIEL, LÉNAÏC CHIZAT, FRANÇOIS-XAVIER VIALARD, and JUSTIN SOLOMON. "Quantum entropic regularization of matrix-valued optimal transport." European Journal of Applied Mathematics 30, no. 6 (September 28, 2017): 1079–102. http://dx.doi.org/10.1017/s0956792517000274.
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This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This “quantum” formulation of optimal transport (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycentres within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis.
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Santambrogio, Filippo. "Dealing with moment measures via entropy and optimal transport." Journal of Functional Analysis 271, no. 2 (July 2016): 418–36. http://dx.doi.org/10.1016/j.jfa.2016.04.009.
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Gentil, Ivan, Christian Léonard, and Luigia Ripani. "About the analogy between optimal transport and minimal entropy." Annales de la faculté des sciences de Toulouse Mathématiques 26, no. 3 (2017): 569–600. http://dx.doi.org/10.5802/afst.1546.
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Mihelich, M., D. Faranda, B. Dubrulle, and D. Paillard. "Statistical optimization for passive scalar transport: maximum entropy production versus maximum Kolmogorov–Sinai entropy." Nonlinear Processes in Geophysics 22, no. 2 (March 25, 2015): 187–96. http://dx.doi.org/10.5194/npg-22-187-2015.
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Abstract. We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov–Sinai entropy for a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov–Sinai entropy, seen as functions of a parameter f connected to the jump probability, admit a unique maximum denoted fmaxEP and fmaxKS. The behaviour of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this paper is that fmaxEP and fmaxKS have the same Taylor expansion at first order in the deviation from equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP(N) tends towards a non-zero value, while fmaxKS(N) tends to 0 when N goes to infinity. For values of N typical of those adopted by Paltridge and climatologists working on maximum entropy production (N ≈ 10–100), we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N* such that fmaxEP and fmaxKS coincide, at least up to a second-order parameter proportional to the non-equilibrium fluxes imposed to the boundaries. We find that the optimal resolution N* depends on the non-equilibrium fluxes, so that deeper convection should be represented on finer grids. This result points to the inadequacy of using a single grid for representing convection in climate and weather models. Moreover, the application of this principle to passive scalar transport parametrization is therefore expected to provide both the value of the optimal flux, and of the optimal number of degrees of freedom (resolution) to describe the system.
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Mihelich, M., D. Faranda, B. Dubrulle, and D. Paillard. "Statistical optimization for passive scalar transport: maximum entropy production vs. maximum Kolmogorov–Sinay entropy." Nonlinear Processes in Geophysics Discussions 1, no. 2 (November 18, 2014): 1691–713. http://dx.doi.org/10.5194/npgd-1-1691-2014.
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Abstract. We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov–Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov–Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at first order in the deviation of equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP(N) tends towards a non-zero value, while fmaxKS(N) tends to 0 when N goes to infinity. For values of N typical of that adopted by Paltridge and climatologists (N ≈ 10 ~ 100), we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N* such that fmaxEP and fmaxKS coincide, at least up to a second order parameter proportional to the non-equilibrium fluxes imposed to the boundaries. We find that the optimal resolution N* depends on the non equilibrium fluxes, so that deeper convection should be represented on finer grids. This result points to the inadequacy of using a single grid for representing convection in climate and weather models. Moreover, the application of this principle to passive scalar transport parametrization is therefore expected to provide both the value of the optimal flux, and of the optimal number of degrees of freedom (resolution) to describe the system.
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Islas, Carlos, Pablo Padilla, and Marco Antonio Prado. "Information Processing in the Brain as Optimal Entropy Transport: A Theoretical Approach." Entropy 22, no. 11 (October 29, 2020): 1231. http://dx.doi.org/10.3390/e22111231.
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We consider brain activity from an information theoretic perspective. We analyze the information processing in the brain, considering the optimality of Shannon entropy transport using the Monge–Kantorovich framework. It is proposed that some of these processes satisfy an optimal transport of informational entropy condition. This optimality condition allows us to derive an equation of the Monge–Ampère type for the information flow that accounts for the branching structure of neurons via the linearization of this equation. Based on this fact, we discuss a version of Murray’s law in this context.
Dissertations / Theses on the topic "Entropy optimal transport"
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Genevay, Aude. "Entropy-regularized Optimal Transport for Machine Learning." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLED002/document.
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Le Transport Optimal régularisé par l’Entropie (TOE) permet de définir les Divergences de Sinkhorn (DS), une nouvelle classe de distance entre mesures de probabilités basées sur le TOE. Celles-ci permettentd’interpolerentredeuxautresdistancesconnues: leTransport Optimal(TO)etl’EcartMoyenMaximal(EMM).LesDSpeuventêtre utilisées pour apprendre des modèles probabilistes avec de meilleures performances que les algorithmes existants pour une régularisation adéquate. Ceci est justifié par un théorème sur l’approximation des SDpardeséchantillons, prouvantqu’unerégularisationsusantepermet de se débarrasser de la malédiction de la dimension du TO, et l’on retrouve à l’infini le taux de convergence des EMM. Enfin, nous présentons de nouveaux algorithmes de résolution pour le TOE basés surl’optimisationstochastique‘en-ligne’qui,contrairementàl’étatde l’art, ne se restreignent pas aux mesures discrètes et s’adaptent bien aux problèmes de grande dimensionThis thesis proposes theoretical and numerical contributions to use Entropy-regularized Optimal Transport (EOT) for machine learning. We introduce Sinkhorn Divergences (SD), a class of discrepancies betweenprobabilitymeasuresbasedonEOTwhichinterpolatesbetween two other well-known discrepancies: Optimal Transport (OT) and Maximum Mean Discrepancies (MMD). We develop an ecient numerical method to use SD for density fitting tasks, showing that a suitable choice of regularization can improve performance over existing methods. We derive a sample complexity theorem for SD which proves that choosing a large enough regularization parameter allows to break the curse of dimensionality from OT, and recover asymptotic ratessimilartoMMD.Weproposeandanalyzestochasticoptimization solvers for EOT, which yield online methods that can cope with arbitrary measures and are well suited to large scale problems, contrarily to existing discrete batch solvers
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Baradat, Aymeric. "Transport optimal incompressible : dépendance aux données et régularisation entropique." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLX016/document.
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Cette thèse porte sur le problème de transport optimal incompressible, un problème introduit par Brenier à la fin des années 80 dans le but de décrire l’évolution d’un fluide incompressible et non-visqueux de façon lagrangienne, ou autrement dit en fixant l’état initial et l’état final de ce fluide, et en minimisant une certaine fonctionnelle sur un ensemble de dynamiques admissibles. Ce manuscrit contient deux parties.Dans la première, on étudie la dépendance du problème de transport optimal incompressible par rapport aux données. Plus précisément, on étudie la dépendance du champ de pression (le multiplicateur de Lagrange associé à la contrainte d’incompressibilité) par rapport à la mesure γ prescrivant l’état initial et l’état final du fluide. On montre au Chapitre 2 par des méthodes variationnelles que le gradient de la pression, en tant qu’élément d’un espace proche du dual des fonctions C^1, dépend de γ de façon hölderienne pour la distance de Monge-Kantorovic. En contrepartie, on montre au Chapitre 4 que pour tout r > 1, le champ de pression dans l'espace de Lebesgue L^r_t L^1_x ne peut pas être une fonction lipschitzienne de γ. Ce résultat est lié au caractère mal-posé de l’équation d’Euler cinétique, une équation cinétique s’interprétant comme l’équation d’optimalité dans le problème de transport optimal incompressible, à laquelle le Chapitre 3 de cette thèse est dédié.Dans une seconde partie, on s’intéresse à la régularisation entropique du problème de transport optimal incompressible, introduit par Arnaudon, Cruzeiro, Léonard et Zambrini en 2017 sous le nom de problème de Brödinger. On prouve au Chapitre 5 que comme dans le cas du transport optimal incompressible, on peut associer à toute solution un champ scalaire de pression agissant comme multiplicateur de Lagrange pour la contrainte d’incompressibilité. On montre ensuite au Chapitre 6 que lorsque le paramètre de régularisation tend vers zéro, le problème de Brödinger converge vers le problème de transport optimal incompressible au sens de la Γ-convergence, et avec convergence des champs de pression. Ce dernier chapitre est issu d'un travail effectué en collaboration avec L. MonsaingeonThis thesis focuses on Incompressible Optimal Transport, a minimization problem introduced by Brenier in the late 80's, aiming at describing the evolution of an incompressible and inviscid fluid in a Lagrangian way , i.e. by prescribing the state of the fluid at the initial and final times and by minimizing some functional among the set of admissible dynamics. This text is divided into two parts.In the first part, we study the dependence of this optimization problem with respect to the data. More precisely, we analyse the dependence of the pressure field, the Lagrange multiplier corresponding to the incompressibility constraint, with respect to the endpoint conditions, described by a probability measure γ determining the state of the fluid at the initial and final times. We show in Chapter 2 by purely variational methods that the gradient of the pressure field, as an element of a space that is close to the dual of C^1, is a Hölder continuous function of γ for the Monge-Kantorovic distance. On the other hand, we prove in Chapter 4 that for all r>1 the pressure field, as an element of L^r_t L^1_x, cannot be a Lipschitz continuous function of γ for the Monge-Kantorovic distance. This last statement is linked to an ill-posedness result proved in Chapter 3 for the so-called kinetic Euler equation, a kinetic PDE interpreted as the optimality equation of the Incompressible Optimal Transport problem.In the second part, we are interested in the entropic regularization of the Incompressible Optimal Transport problem: the so-called Brödinger problem, introduced by Arnaudon, Cruzeiro, Léonard and Zambrini in 2017. On the one hand, we prove in Chapter 5 that similarly to what happens in the Incompressible Optimal Transport case, to a solution always corresponds a scalar pressure field acting as the Lagrange multiplier for the incompressibility constraint. On the other hand, we prove in Chapter 6 that when the diffusivity coefficient tends to zero, the Brödinger problem converges towards the Incompressible Optimal Transport problem in the sense of Gamma-convergence, and with convergence of the pressure fields. The results of Chapter 6 come from a joint work with L. Monsaingeon
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Ripani, Luigia. "Le problème de Schrödinger et ses liens avec le transport optimal et les inégalités fonctionnelles." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1274/document.
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Au cours des 20 dernières années, la théorie du transport optimal s’est revelée être un outil efficace pour étudier le comportement asymptotique dans le cas des équations de diffusion, pour prouver des inégalités fonctionnelles et pour étendre des propriétés géométriques dans des espaces extrêmement généraux comme des espaces métriques mesurés, etc. La condition de courbure-dimension de la théorie Bakry-Emery apparaît comme la pierre angulaire de ces applications. Il suffit de penser au cas le plus simple et le plus important de la distance quadratique de Wasserstein W2 : la contraction du flux de chaleur en W2 caractérise les bornes inférieures uniformes pour la courbure de Ricci ; l’inégalité de Talagrand du transport, comparant W2 à l’entropie relative est impliquée et implique, par l’inégalité HWI, l’inégalité log-Sobolev ; les géodésiques de McCann dans l’espace de Wasserstein (P2(Rn),W2) permettent de prouver des propriétés fonctionnelles importantes comme la convexité, et des inégalités fonctionnelles standards telles que l’isopérymétrie, des propriétés de concentration de mesure, l’inégalité de Prékopa-Leindler et ainsi de suite. Néanmoins, le manque de régularité des plans minimisation nécessite des arguments d’analyse non lisse. Le problème de Schrödinger est un problème de minimisation de l’entropie avec des contraintes marginales et un processus de référence fixes. À partir de la théorie des grandes déviations, lorsque le processus de référence est le mouvement Brownien, sa valeur minimale A converge vers W2 lorsque la température est nulle. Les interpolations entropiques, solutions du problème de Schrödinger, sont caractérisées en termes de semigroupes de Markov, ce qui implique naturellement les calculs Γ2 et la condition de courbure-dimension. Datant des années 1930 et négligé pendant des décennies, le problème de Schrodinger connaît depuis ces dernières années une popularité croissante dans différents domaines, grâce à sa relation avec le transport optimal, à la regularité de ses solutions, et à d’autres propriétés performantes dans des calculs numériques. Le but de ce travail est double. D’abord, nous étudions certaines analogies entre le problème de Schrödinger et le transport optimal fournissant de nouvelles preuves de la formulation duale de Kantorovich et de celle, dynamique, de Benamou-Brenier pour le coût entropique A. Puis, en tant qu’application de ces connexions, nous dérivons certaines propriétés et inégalités fonctionnelles sous des conditions de courbure-dimension. En particulier, nous prouvons la concavité de l’entropie exponentielle le long des interpolations entropiques sous la condition de courbure-dimension CD(0, n) et la régularité du coût entropique le long du flot de la chaleur. Nous donnons également différentes preuves de l’inégalité variationnelle évolutionnaire pour A et de la contraction du flux de la chaleur en A, en retrouvant comme cas limite, les résultats classiques en W2, sous CD(κ,∞) et CD(0, n). Enfin, nous proposons une preuve simple de la propriété de concentration gaussienne via le problème de Schrödinger comme alternative aux arguments classiques tel que l’argument de Marton basé sur le transport optimalIn the past 20 years the optimal transport theory revealed to be an efficient tool to study the asymptotic behavior for diffusion equations, to prove functional inequalities, to extend geometrical properties in extremely general spaces like metric measure spaces, etc. The curvature-dimension of the Bakry-Émery theory appears as the cornerstone of those applications. Just think to the easier and most important case of the quadratic Wasserstein distance W2: contraction of the heat flow in W2 characterizes uniform lower bounds for the Ricci curvature; the transport Talagrand inequality, comparing W2 to the relative entropy is implied and implies via the HWI inequality the log-Sobolev inequality; McCann geodesics in the Wasserstein space (P2(Rn),W2) allow to prove important functional properties like convexity, and standard functional inequalities, such as isoperimetry, measure concentration properties, the Prékopa Leindler inequality and so on. However the lack of regularity of optimal maps, requires non-smooth analysis arguments. The Schrödinger problem is an entropy minimization problem with marginal constraints and a fixed reference process. From the Large deviation theory, when the reference process is driven by the Brownian motion, its minimal value A converges to W2 when the temperature goes to zero. The entropic interpolations, solutions of the Schrödinger problem, are characterized in terms of Markov semigroups, hence computation along them naturally involves Γ2 computations and the curvature-dimension condition. Dating back to the 1930s, and neglected for decades, the Schrödinger problem recently enjoys an increasing popularity in different fields, thanks to this relation to optimal transport, smoothness of solutions and other well performing properties in numerical computations. The aim of this work is twofold. First we study some analogy between the Schrödinger problem and optimal transport providing new proofs of the dual Kantorovich and the dynamic Benamou-Brenier formulations for the entropic cost A. Secondly, as an application of these connections we derive some functional properties and inequalities under curvature-dimensions conditions. In particular, we prove the concavity of the exponential entropy along entropic interpolations under the curvature-dimension condition CD(0, n) and regularity of the entropic cost along the heat flow. We also give different proofs the Evolutionary Variational Inequality for A and contraction of the heat flow in A, recovering as a limit case the classical results in W2, under CD(κ,∞) and also in the flat dimensional case. Finally we propose an easy proof of the Gaussian concentration property via the Schrödinger problem as an alternative to classical arguments as the Marton argument which is based on optimal transport
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Chizat, Lénaïc. "Transport optimal de mesures positives : modèles, méthodes numériques, applications." Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLED063/document.
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L'objet de cette thèse est d'étendre le cadre théorique et les méthodes numériques du transport optimal à des objets plus généraux que des mesures de probabilité. En premier lieu, nous définissons des modèles de transport optimal entre mesures positives suivant deux approches, interpolation et couplage de mesures, dont nous montrons l'équivalence. De ces modèles découle une généralisation des métriques de Wasserstein. Dans une seconde partie, nous développons des méthodes numériques pour résoudre les deux formulations et étudions en particulier une nouvelle famille d'algorithmes de "scaling", s'appliquant à une grande variété de problèmes. La troisième partie contient des illustrations ainsi que l'étude théorique et numérique, d'un flot de gradient de type Hele-Shaw dans l'espace des mesures. Pour les mesures à valeurs matricielles, nous proposons aussi un modèle de transport optimal qui permet un bon arbitrage entre fidélité géométrique et efficacité algorithmiqueThis thesis generalizes optimal transport beyond the classical "balanced" setting of probability distributions. We define unbalanced optimal transport models between nonnegative measures, based either on the notion of interpolation or the notion of coupling of measures. We show relationships between these approaches. One of the outcomes of this framework is a generalization of the p-Wasserstein metrics. Secondly, we build numerical methods to solve interpolation and coupling-based models. We study, in particular, a new family of scaling algorithms that generalize Sinkhorn's algorithm. The third part deals with applications. It contains a theoretical and numerical study of a Hele-Shaw type gradient flow in the space of nonnegative measures. It also adresses the case of measures taking values in the cone of positive semi-definite matrices, for which we introduce a model that achieves a balance between geometrical accuracy and algorithmic efficiency
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DOLDI, ALESSANDRO. "EQUILIBRIUM, SYSTEMIC RISK MEASURES AND OPTIMAL TRANSPORT: A CONVEX DUALITY APPROACH." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/812668.
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This Thesis focuses on two main topics. Firstly, we introduce and analyze the novel concept of Systemic Optimal Risk Transfer Equilibrium (SORTE), and we progressively generalize it (i) to a multivariate setup and (ii) to a dynamic (conditional) setting. Additionally we investigate its relation to a recently introduced concept of Systemic Risk Measures (SRM). We present Conditional Systemic Risk Measures and study their properties, dual representation and possible interpretations of the associated allocations as equilibria in the sense of SORTE. On a parallel line of work, we develop a duality for the Entropy Martingale Optimal Transport problem and provide applications to problems of nonlinear pricing-hedging. The mathematical techniques we exploit are mainly borrowed from functional and convex analysis, as well as probability theory. More specifically, apart from a wide range of classical results from functional analysis, we extensively rely on Fenchel-Moreau-Rockafellar type conjugacy results, Minimax Theorems, theory of Orlicz spaces, compactness results in the spirit of Komlós Theorem. At the same time, mathematical results concerning utility maximization theory (existence of optima for primal and dual problems, just to mention an example) and optimal transport theory are widely exploited.
The notion of SORTE is inspired by the Bühlmann's classical Equilibrium Risk Exchange (H. Bühlmann, "The general economic premium principle", Astin Bulletin, 1984). In both the Bühlmann and the SORTE definition, each agent is behaving rationally by maximizing his/her expected utility given a budget constraint. The two approaches differ by the budget constraints. In Bühlmann's definition the vector that assigns the budget constraint is given a priori. In the SORTE approach, on the contrary, the budget constraint is endogenously determined by solving a systemic utility maximization problem. SORTE gives priority to the systemic aspects of the problem, in order to first optimize the overall systemic performance, rather than to individual rationality. Single agents' preferences are, however, taken into account by the presence of individual optimization problems. The two aspects are simultaneously considered via an optimization problem for a value function given by summation of single agents' utilities. After providing a financial and theoretical justification for this new idea, in this research sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of such a SORTE are presented.
Once laid the theoretical foundation for the newly introduced SORTE, this Thesis proceeds in extending such a notion to the case when the value function to be optimized has two components, one being the sum of the single agents' utility functions, as in the aforementioned case of SORTE, the other consisting of a truly systemic component. This marks the progress from SORTE to Multivariate Systemic Optimal Risk Transfer Equilibrium (mSORTE). Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general setting allows us to introduce and study a Nash Equilibrium property of the optimizers. Existence, uniqueness, Pareto optimality and the Nash Equilibrium property of the newly defined mSORTE are proved in this Thesis. Additionally, it is shown how mSORTE is in fact a proper generalization, and covers both from the conceptual and the mathematical point of view the notion of SORTE.
Proceeding further in the analysis, the relations between the concepts of mSORTE and SRM are investigated in this work. The notion of SRM we start from was introduced in the papers "A unified approach to systemic risk measures via acceptance sets" (Math. Finance, 2019) and "On fairness of systemic risk measures" (Finance Stoch., 2020) by F. Biagini, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis. SRM of Biagini et al. are generalized in this Thesis to a dynamic (namely conditional) setting, adding also a systemic, multivariate term in the threshold functions that Biagini et al. consider in their papers. The dynamic version of mSORTE is introduced, and it is proved that the optimal allocations of dynamic SRM, together with the corresponding fair pricing measures, yield a dynamic mSORTE. This in particular remains true if conditioning is taken with respect to the trivial sigma algebra, which is tantamount to working in the non-dynamic setting covered in Biagini et al. for SRM, and in the previous parts of our work for mSORTE. The case of exponential utility functions is thoroughly examined, and the explicit formulas we obtain for this specific choice of threshold functions allow for providing a time consistency property for allocations, dynamic SRM and dynamic mSORTE.
The last part of this Thesis is devoted to a conceptually separate topic. Nonetheless, a clear mathematical link between the previous work and the one we are to describe is established by the use of common techniques. A duality between a novel Entropy Martingale Optimal Transport (EMOT) problem (D) and an associated optimization problem (P) is developed. In (D) the approach taken in Liero et al. (M. Liero, A. Mielke, and G. Savaré, "Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures", Inventiones mathematicae, 2018) serves as a basis for adding the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al.. The Problem (D) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive penalization terms D, which may not have a divergence formulation. In Problem (P) the objective functional, associated via Fenchel conjugacy to the terms D, is not any more linear, as in Optimal Transport or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a non linear subhedging value. Our results in this Thesis establish a novel nonlinear robust pricing-hedging duality in financial mathematics, which covers a wide range of known robust results in its generality.
The research for this Thesis resulted in the production of the following works: F. Biagini, A. Doldi, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis, "Systemic optimal risk transfer equilibrium", Mathematics and Financial Economics, 2021; A. Doldi and M. Frittelli, "Multivariate Systemic Optimal Risk Transfer Equilibrium", Preprint: arXiv:1912.12226, 2019; A. Doldi and M. Frittelli, "Conditional Systemic Risk Measures", Preprint: arXiv:2010.11515, 2020; A. Doldi and M. Frittelli, "Entropy Martingale Optimal Transport and Nonlinear Pricing-Hedging Duality", Preprint: arXiv:2005.12572, 2020.
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Telci, Ilker Tonguc. "Optimal water quality management in surface water systems and energy recovery in water distribution networks." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/45861.
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Two of the most important environmental challenges in the 21st century are to protect the quality of fresh water resources and to utilize renewable energy sources to lower greenhouse gas emissions. This study contributes to the solution of the first challenge by providing methodologies for optimal design of real-time water quality monitoring systems and interpretation of data supplied by the monitoring system to identify potential pollution sources in river networks. In this study, the optimal river water quality monitoring network design aspect of the overall monitoring program is addressed by a novel methodology for the analysis of this problem. In this analysis, the locations of sampling sites are determined such that the contaminant detection time is minimized for the river network while achieving maximum reliability for the monitoring system performance. The data collected from these monitoring stations can be used to identify contamination source locations. This study suggests a methodology that utilizes a classification routine which associates the observations on a contaminant spill with one or more of the candidate spill locations in the river network. This approach consists of a training step followed by a sequential elimination of the candidate spill locations which lead to the identification of potential spill locations. In order to contribute the solution of the second environmental challenge, this study suggests utilizing available excess energy in water distribution systems by providing a methodology for optimal design of energy recovery systems. The energy recovery in water distribution systems is possible by using micro hydroelectric turbines to harvest available excess energy inevitably produced to satisfy consumer demands and to maintain adequate pressures. In this study, an optimization approach for the design of energy recovery systems in water distribution networks is proposed. This methodology is based on finding the best locations for micro hydroelectric plants in the network to recover the excess energy. Due to the unsteady nature of flow in water distribution networks, the proposed methodology also determines optimum operation schedules for the micro turbines.
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DE, PONTI NICOLÒ. "Optimal transport: entropic regularizations, geometry and diffusion PDEs." Doctoral thesis, Università degli studi di Pavia, 2019. http://hdl.handle.net/11571/1292130.
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Muzellec, Boris. "Leveraging regularization, projections and elliptical distributions in optimal transport." Electronic Thesis or Diss., Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAG009.
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Pouvoir manipuler et de comparer de mesures de probabilité est essentiel pour de nombreuses applications en apprentissage automatique. Le transport optimal (TO) définit des divergences entre distributions fondées sur la géométrie des espaces sous-jacents : partant d'une fonction de coût définie sur l'espace dans lequel elles sont supportées, le TO consiste à trouver un couplage entre les deux mesures qui soit optimal par rapport à ce coût. Par son ancrage géométrique, le TO est particulièrement bien adapté au machine learning, et fait l'objet d'une riche théorie mathématique. En dépit de ces avantages, l'emploi du TO pour les sciences des données a longtemps été limité par les difficultés mathématiques et computationnelles liées au problème d'optimisation sous-jacent. Pour contourner ce problème, une approche consiste à se concentrer sur des cas particuliers admettant des solutions en forme close, ou pouvant se résoudre efficacement. En particulier, le TO entre mesures elliptiques constitue l'un des rares cas pour lesquels le TO admet une forme close, définissant la géométrie de Bures-Wasserstein (BW). Cette thèse s'appuie tout particulièrement sur la géométrie de BW, dans le but de l'utiliser comme outil de base pour des applications en sciences des données. Pour ce faire, nous considérons des situations dans lesquelles la géométrie de BW est tantôt utilisée comme un outil pour l'apprentissage de représentations, étendue à partir de projections sur des sous-espaces, ou régularisée par un terme entropique. Dans une première contribution, la géométrie de BW est utilisée pour définir des plongements sous la forme de distributions elliptiques, étendant la représentation classique sous forme de vecteurs de R^d. Dans une deuxième contribution, nous prouvons l'existence de transports qui extrapolent des applications restreintes à des projections en faible dimension, et montrons que ces plans "sous-espace optimaux" admettent des formes closes dans le cas de mesures gaussiennes. La troisième contribution de cette thèse consiste à obtenir des formes closes pour le transport entropique entre des mesures gaussiennes non-normalisées, qui constituent les premières expressions non triviales pour le transport entropique. Finalement, dans une dernière contribution nous utilisons le transport entropique pour imputer des données manquantes de manière non-paramétrique, tout en préservant les distributions sous-jacentesComparing and matching probability distributions is a crucial in numerous machine learning (ML) algorithms. Optimal transport (OT) defines divergences between distributions that are grounded on geometry: starting from a cost function on the underlying space, OT consists in finding a mapping or coupling between both measures that is optimal with respect to that cost. The fact that OT is deeply grounded in geometry makes it particularly well suited to ML. Further, OT is the object of a rich mathematical theory. Despite those advantages, the applications of OT in data sciences have long been hindered by the mathematical and computational complexities of the underlying optimization problem. To circumvent these issues, one approach consists in focusing on particular cases that admit closed-form solutions or that can be efficiently solved. In particular, OT between elliptical distributions is one of the very few instances for which OT is available in closed form, defining the so-called Bures-Wasserstein (BW) geometry. This thesis builds extensively on the BW geometry, with the aim to use it as basic tool in data science applications. To do so, we consider settings in which it is alternatively employed as a basic tool for representation learning, enhanced using subspace projections, and smoothed further using entropic regularization. In a first contribution, the BW geometry is used to define embeddings as elliptical probability distributions, extending on the classical representation of data as vectors in R^d.In the second contribution, we prove the existence of transportation maps and plans that extrapolate maps restricted to lower-dimensional projections, and show that subspace-optimal plans admit closed forms in the case of Gaussian measures.Our third contribution consists in deriving closed forms for entropic OT between Gaussian measures scaled with a varying total mass, which constitute the first non-trivial closed forms for entropic OT and provide the first continuous test case for the study of entropic OT. Finally, in a last contribution, entropic OT is leveraged to tackle missing data imputation in a non-parametric and distribution-preserving way
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Nenna, Luca. "Numerical Methods for Multi-Marginal Optimal Transportation." Thesis, Paris Sciences et Lettres (ComUE), 2016. http://www.theses.fr/2016PSLED017/document.
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Dans cette thèse, notre but est de donner un cadre numérique général pour approcher les solutions des problèmes du transport optimal (TO). L’idée générale est d’introduire une régularisation entropique du problème initial. Le problème régularisé correspond à minimiser une entropie relative par rapport à une mesure de référence donnée. En effet, cela équivaut à trouver la projection d’un couplage par rapport à la divergence de Kullback-Leibler. Cela nous permet d’utiliser l’algorithme de Bregman/Dykstra et de résoudre plusieurs problèmes variationnels liés au TO. Nous nous intéressons particulièrement à la résolution des problèmes du transport optimal multi-marges (TOMM) qui apparaissent dans le cadre de la dynamique des fluides (équations d’Euler incompressible à la Brenier) et de la physique quantique (la théorie de fonctionnelle de la densité ). Dans ces cas, nous montrons que la régularisation entropique joue un rôle plus important que de la simple stabilisation numérique. De plus, nous donnons des résultats concernant l’existence des transports optimaux (par exemple des transports fractals) pour le problème TOMMIn this thesis we aim at giving a general numerical framework to approximate solutions to optimal transport (OT) problems. The general idea is to introduce an entropic regularization of the initialproblems. The regularized problem corresponds to the minimization of a relative entropy with respect a given reference measure. Indeed, this is equivalent to find the projection of the joint coupling with respect the Kullback-Leibler divergence. This allows us to make use the Bregman/Dykstra’s algorithm and solve several variational problems related to OT. We are especially interested in solving multi-marginal optimal transport problems (MMOT) arising in Physics such as in Fluid Dynamics (e.g. incompressible Euler equations à la Brenier) and in Quantum Physics (e.g. Density Functional Theory). In these cases we show that the entropic regularization plays a more important role than a simple numerical stabilization. Moreover, we also give some important results concerning existence and characterization of optimal transport maps (e.g. fractal maps) for MMOT
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Tamanini, Luca. "Analysis and Geometry of RCD spaces via the Schrödinger problem." Thesis, Paris 10, 2017. http://www.theses.fr/2017PA100082/document.
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Le but principal de ce manuscrit est celui de présenter une nouvelle méthode d'interpolation entre des probabilités inspirée du problème de Schrödinger, problème de minimisation entropique ayant des liens très forts avec le transport optimal. À l'aide de solutions au problème de Schrödinger, nous obtenons un schéma d'approximation robuste jusqu'au deuxième ordre et différent de Brenier-McCann qui permet d'établir la formule de dérivation du deuxième ordre le long des géodésiques Wasserstein dans le cadre de espaces RCD* de dimension finie. Cette formule était inconnue même dans le cadre des espaces d'Alexandrov et nous en donnerons quelques applications. La démonstration utilise un ensemble remarquable de nouvelles propriétés pour les solutions au problème de Schrödinger dynamique :- une borne uniforme des densités le long des interpolations entropiques ;- la lipschitzianité uniforme des potentiels de Schrödinger ;- un contrôle L2 uniforme des accélérations. Ces outils sont indispensables pour explorer les informations géométriques encodées par les interpolations entropiques. Les techniques utilisées peuvent aussi être employées pour montrer que la solution visqueuse de l'équation d'Hamilton-Jacobi peut être récupérée à travers une méthode de « vanishing viscosity », comme dans le cas lisse.Dans tout le manuscrit, plusieurs remarques sur l'interprétation physique du problème de Schrödinger seront mises en lumière. Cela pourra aider le lecteur à mieux comprendre les motivations probabilistes et physiques du problème, ainsi qu'à les connecter avec la nature analytique et géométrique de la dissertationMain aim of this manuscript is to present a new interpolation technique for probability measures, which is strongly inspired by the Schrödinger problem, an entropy minimization problem deeply related to optimal transport. By means of the solutions to the Schrödinger problem, we build an efficient approximation scheme, robust up to the second order and different from Brenier-McCann's classical one. Such scheme allows us to prove the second order differentiation formula along geodesics in finite-dimensional RCD* spaces. This formula is new even in the context of Alexandrov spaces and we provide some applications.The proof relies on new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:- equiboundedness of the densities along the entropic interpolations,- equi-Lipschitz continuity of the Schrödinger potentials,- a uniform weighted L2 control of the Hessian of such potentials. These tools are very useful in the investigation of the geometric information encoded in entropic interpolations. The techniques used in this work can be also used to show that the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. Throughout the whole manuscript, several remarks on the physical interpretation of the Schrödinger problem are pointed out. Hopefully, this will allow the reader to better understand the physical and probabilistic motivations of the problem as well as to connect them with the analytical and geometric nature of the dissertation
Book chapters on the topic "Entropy optimal transport"
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Gu, Wen, Teng Zhang, and Hai Jin. "Entropy Weight Allocation: Positive-unlabeled Learning via Optimal Transport." In Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), 37–45. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2022. http://dx.doi.org/10.1137/1.9781611977172.5.
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Tarkhamtham, Payap, and Woraphon Yamaka. "A Generalized Maximum Renyi Entropy Approach in Kink Regression Model." In Credible Asset Allocation, Optimal Transport Methods, and Related Topics, 411–25. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-97273-8_28.
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Boslau, Madlen. "Consumer Attitudes toward RFID Usage." In Encyclopedia of Multimedia Technology and Networking, Second Edition, 247–53. IGI Global, 2009. http://dx.doi.org/10.4018/978-1-60566-014-1.ch034.
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The term RFID refers to radio frequency identification and describes transponders or tags that are attached to animate or inanimate objects and are automatically read by a network infrastructure or networked reading devices. Current solutions such as optical character recognition (OCR), bar codes, or smart card systems require manual data entry, scanning, or readout along the supply chain. These procedures are costly, timeconsuming, and inaccurate. RFID systems are seen as a potential solution to these constraints, by allowing non-line-of-sight reception of the coded data. Identification codes are stored on a tag that consists of a microchip and an attached antenna. Once the tag is within the reception area of a reader, the information is transmitted. A connected database is then able to decode the identification code and identify the object. Such network infrastructures should be able to capture, store, and deliver large amounts of data robustly and efficiently (Scharfeld, 2001). The applications of RFID in use today can be sorted into two groups of products: • The first group of products uses the RFID technology as a central feature. Examples are security and access control, vehicle immobilization systems, and highway toll passes (Inaba & Schuster, 2005). Future applications include rechargeable public transport tickets, implants holding critical medical data, or dog tags (Böhmer, Brück, & Rees, 2005). • The second group of products consists of those goods merely tagged with an RFID label instead of a bar code. Here, the tag simply substitutes the bar code as a carrier of product information for identification purposes. This seems sensible, as RFID tags display a number of characteristics that allow for faster, easier, more reliable, and superior identification. Once consumers are able to buy RFID tagged products, their attitude toward such tags is of central importance. Consumer acceptance of RFID tags may have severe consequences for all companies tagging their products with RFID.
Conference papers on the topic "Entropy optimal transport"
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Liu, Dong, Minh Thanh Vu, Saikat Chatterjee, and Lars K. Rasmussen. "Entropy-regularized Optimal Transport Generative Models." In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019. http://dx.doi.org/10.1109/icassp.2019.8682721.
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Coleman, Todd P., Justin Tantiongloc, Alexis Allegra, Diego Mesa, Dae Kang, and Marcela Mendoza. "Diffeomorphism learning via relative entropy constrained optimal transport." In 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2016. http://dx.doi.org/10.1109/globalsip.2016.7906057.
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Hundrieser, Shayan, Marcel Klatt, and Axel Munk. "Entropic Optimal Transport on Countable Spaces: Statistical Theory and Asymptotics." In Entropy 2021: The Scientific Tool of the 21st Century. Basel, Switzerland: MDPI, 2021. http://dx.doi.org/10.3390/entropy2021-09837.
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Elvander, Filip, Isabel Haasler, Andreas Jakobsson, and Johan Karlsson. "Non-coherent Sensor Fusion via Entropy Regularized Optimal Mass Transport." In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2019. http://dx.doi.org/10.1109/icassp.2019.8682186.
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Tannenbaum, Emmanuel, Tryphon Georgiou, and Allen Tannenbaum. "Signals and control aspects of optimal mass transport and the Boltzmann entropy." In 2010 49th IEEE Conference on Decision and Control (CDC). IEEE, 2010. http://dx.doi.org/10.1109/cdc.2010.5717821.
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Chen, Fuqiang, and Daniel Peter. "A misfit function based on entropy regularized optimal transport for full-waveform inversion." In SEG Technical Program Expanded Abstracts 2018. Society of Exploration Geophysicists, 2018. http://dx.doi.org/10.1190/segam2018-2995612.1.
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Cao, Nan, Teng Zhang, Xuanhua Shi, and Hai Jin. "Posistive-Unlabeled Learning via Optimal Transport and Margin Distribution." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/393.
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Positive-unlabeled (PU) learning deals with the circumstances where only a portion of positive instances are labeled, while the rest and all negative instances are unlabeled, and due to this confusion, the class prior can not be directly available. Existing PU learning methods usually estimate the class prior by training a nontraditional probabilistic classifier, which is prone to give an overestimation. Moreover, these methods learn the decision boundary by optimizing the minimum margin, which is not suitable in PU learning due to its sensitivity to label noise. In this paper, we enhance PU learning methods from the above two aspects. More specifically, we first explicitly learn a transformation from unlabeled data to positive data by entropy regularized optimal transport to achieve a much more precise estimation for class prior. Then we switch to optimizing the margin distribution, rather than the minimum margin, to obtain a label noise insensitive classifier. Extensive empirical studies on both synthetic and real-world data sets demonstrate the superiority of our proposed method.
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Kasagi, Nobuhide, Yosuke Hasegawa, Koji Fukagata, and Kaoru Iwamoto. "Control of Turbulent Transport: Less Friction and More Heat Transfer." In 2010 14th International Heat Transfer Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/ihtc14-23344.
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Because of the importance of fundamental knowledge on turbulent heat transfer for further decreasing entropy production and improving efficiency in various thermo-fluid systems, we revisit a classical issue whether enhancing heat transfer is possible with skin friction reduced or at least not increased as much as heat transfer. The answer that numerous previous studies suggest is quite pessimistic because the analogy concept of momentum and heat transport holds well in a wide range of flows. Nevertheless, the recent progress in analyzing turbulence mechanics and designing turbulence control offers a chance to develop a scheme for dissimilar momentum and heat transport. By reexamining the governing equations and boundary conditions for convective heat transfer, the basic strategies for achieving dissimilar control in turbulent flow is generally classified into two groups, i.e., one for the averaged quantities and the other for the turbulent fluctuating components. As a result, two different approaches are discussed presently. First, under three typical heating conditions, the contribution of turbulent transport to wall friction and heat transfer is mathematically formulated, and it is shown that the difference in how the local turbulent transport of momentum and that of heat contribute to the friction and heat transfer coefficients is a key to answer whether the dissimilar control is feasible. Such control is likely to be achieved when the weight distributions for the stress and flux in the derived relationships are different. Secondly, we introduce a more general methodology, i.e., the optimal control theory. The Fre´chet differentials obtained clearly show that the responses of velocity and scalar fields to a given control input are quite different due to the fact that the velocity is a divergence-free vector while the temperature is a conservative scalar. By exploiting this inherent difference, the dissimilar control can be achieved even in flows where the averaged momentum and heat transport equations have the same form.
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Adeyinka, O. B., and G. F. Naterer. "Towards Optical Measurement of Entropy Transport in Turbulent Flows." In 39th AIAA Thermophysics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2007. http://dx.doi.org/10.2514/6.2007-4052.
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Vlasak, Jiri, Michal Sojka, and Zdeněk Hanzálek. "Parallel Parking: Optimal Entry and Minimum Slot Dimensions." In 8th International Conference on Vehicle Technology and Intelligent Transport Systems. SCITEPRESS - Science and Technology Publications, 2022. http://dx.doi.org/10.5220/0011045600003191.
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