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Статті в журналах з теми "Entropic optimal transport"
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Altschuler, Jason M., Jonathan Niles-Weed, and Austin J. Stromme. "Asymptotics for Semidiscrete Entropic Optimal Transport." SIAM Journal on Mathematical Analysis 54, no. 2 (March 14, 2022): 1718–41. http://dx.doi.org/10.1137/21m1440165.
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Keriven, Nicolas. "Entropic Optimal Transport on Random Graphs." SIAM Journal on Mathematics of Data Science 5, no. 4 (November 29, 2023): 1028–50. http://dx.doi.org/10.1137/22m1518281.
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Amari, Shun-ichi, Ryo Karakida, Masafumi Oizumi, and Marco Cuturi. "Information Geometry for Regularized Optimal Transport and Barycenters of Patterns." Neural Computation 31, no. 5 (May 2019): 827–48. http://dx.doi.org/10.1162/neco_a_01178.
Повний текст джерелаАнотація:
We propose a new divergence on the manifold of probability distributions, building on the entropic regularization of optimal transportation problems. As Cuturi ( 2013 ) showed, regularizing the optimal transport problem with an entropic term is known to bring several computational benefits. However, because of that regularization, the resulting approximation of the optimal transport cost does not define a proper distance or divergence between probability distributions. We recently tried to introduce a family of divergences connecting the Wasserstein distance and the Kullback-Leibler divergence from an information geometry point of view (see Amari, Karakida, & Oizumi, 2018 ). However, that proposal was not able to retain key intuitive aspects of the Wasserstein geometry, such as translation invariance, which plays a key role when used in the more general problem of computing optimal transport barycenters. The divergence we propose in this work is able to retain such properties and admits an intuitive interpretation.
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Rigollet, Philippe, and Jonathan Weed. "Entropic optimal transport is maximum-likelihood deconvolution." Comptes Rendus Mathematique 356, no. 11-12 (November 2018): 1228–35. http://dx.doi.org/10.1016/j.crma.2018.10.010.
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Clason, Christian, Dirk A. Lorenz, Hinrich Mahler, and Benedikt Wirth. "Entropic regularization of continuous optimal transport problems." Journal of Mathematical Analysis and Applications 494, no. 1 (February 2021): 124432. http://dx.doi.org/10.1016/j.jmaa.2020.124432.
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Junge, Oliver, Daniel Matthes, and Bernhard Schmitzer. "Entropic transfer operators." Nonlinearity 37, no. 6 (April 16, 2024): 065004. http://dx.doi.org/10.1088/1361-6544/ad247a.
Повний текст джерелаАнотація:
Abstract We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.
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Amid, Ehsan, Frank Nielsen, Richard Nock, and Manfred K. Warmuth. "Optimal Transport with Tempered Exponential Measures." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 10 (March 24, 2024): 10838–46. http://dx.doi.org/10.1609/aaai.v38i10.28957.
Повний текст джерелаАнотація:
In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, ``a-la-Kantorovich'', which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, ``a-la-Sinkhorn-Cuturi'', which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.
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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.
Повний текст джерелаАнотація:
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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Ito, Kaito, and Kenji Kashima. "Entropic model predictive optimal transport over dynamical systems." Automatica 152 (June 2023): 110980. http://dx.doi.org/10.1016/j.automatica.2023.110980.
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Lo, Chi-Jen (Roger), Mahesh K. Marina, Nishanth Sastry, Kai Xu, Saeed Fadaei, and Yong Li. "Shrinking VOD Traffic via Rényi-Entropic Optimal Transport." Proceedings of the ACM on Measurement and Analysis of Computing Systems 8, no. 1 (February 16, 2024): 1–34. http://dx.doi.org/10.1145/3639033.
Повний текст джерелаАнотація:
In response to the exponential surge in Internet Video on Demand (VOD) traffic, numerous research endeavors have concentrated on optimizing and enhancing infrastructure efficiency. In contrast, this paper explores whether users' demand patterns can be shaped to reduce the pressure on infrastructure. Our main idea is to design a mechanism that alters the distribution of user requests to another distribution which is much more cache-efficient, but still remains 'close enough' (in the sense of cost) to fulfil each individual user's preference. To quantify the cache footprint of VOD traffic, we propose a novel application of Rényi entropy as its proxy, capturing the 'richness' (the number of distinct videos or cache size) and the 'evenness' (the relative popularity of video accesses) of the on-demand video distribution. We then demonstrate how to decrease this metric by formulating a problem drawing on the mathematical theory of optimal transport (OT). Additionally, we establish a key equivalence theorem: minimizing Rényi entropy corresponds to maximizing soft cache hit ratio (SCHR) --- a variant of cache hit ratio allowing similarity-based video substitutions. Evaluation on a real-world, city-scale video viewing dataset reveals a remarkable 83% reduction in cache size (associated with VOD caching traffic). Crucially, in alignment with the above-mentioned equivalence theorem, our approach yields a significant uplift to SCHR, achieving close to 100%.
Дисертації з теми "Entropic optimal transport"
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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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Thurin, Gauthier. "Quantiles multivariés et transport optimal régularisé." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0262.
Повний текст джерелаАнотація:
L’objet d’intérêt principal de cette thèse est la fonction quantile de Monge- Kantorovich. On s’intéresse d’abord à la question cruciale de son estimation, qui revient à résoudre un problème de transport optimal. En particulier, on tente de tirer profit de la connaissance a priori de la loi de référence, une information additionnelle par rapport aux algorithmes usuels, qui nous permet de paramétrer les potentiels de transport par leur série de Fourier. Ce faisant, la régularisation entropique du transport optimal permet deux avantages : la construction d’un algorithme efficace et convergent pour résoudre la version semi-duale de notre problème, et l’obtention d’une fonction quantile empirique lisse et monotone. Ces considérations sont ensuite étendues à l’étude de données sphériques, en remplaçant les séries de Fourier par des harmoniques sphériques, et en généralisant la carte entropique à ce cadre non-euclidien. Le second objectif de cette thèse est de définir de nouvelles notions de superquantiles et d’expected shortfalls multivariés, pour compléter l’information fournie par les quantiles. Ces fonctions caractérisent la loi d’un vecteur aléatoire, ainsi que la convergence en loi, sous certaines hypothèses, et trouvent des applications directes en analyse de risque multivarié, pour étendre les mesures de risque classiques de Value-at-Risk et Conditional-Value-at-RiskThis thesis is concerned with the study of the Monge-Kantorovich quantile function. We first address the crucial question of its estimation, which amounts to solve an optimal transport problem. In particular, we try to take advantage of the knowledge of the reference distribution, that represents additional information compared with the usual algorithms, and which allows us to parameterize the transport potentials by their Fourier series. Doing so, entropic regularization provides two advantages: to build an efficient and convergent algorithm for solving the semi-dual version of our problem, and to obtain a smooth and monotonic empirical quantile function. These considerations are then extended to the study of spherical data, by replacing the Fourier series with spherical harmonics, and by generalizing the entropic map to this non-Euclidean setting. The second main purpose of this thesis is to define new notions of multivariate superquantiles and expected shortfalls, to complement the information provided by the quantiles. These functions characterize the law of a random vector, as well as convergence in distribution under certain assumptions, and have direct applications in multivariate risk analysis, to extend the traditional risk measures of Value-at-Risk and Conditional-Value-at-Risk
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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.
Повний текст джерелаАнотація:
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.
Повний текст джерелаАнотація:
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
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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.
Повний текст джерелаАнотація:
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.
Повний текст джерелаАнотація:
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.
Повний текст джерелаАнотація:
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.
Повний текст джерелаАнотація:
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.
Повний текст джерелаАнотація:
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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Hillion, Erwan. "Analyse et géométrie dans les espaces métriques mesurés : inégalités de Borell-Brascamp-Lieb et conjecture de Olkin-Shepp." Toulouse 3, 2010. http://thesesups.ups-tlse.fr/1592/.
Повний текст джерелаАнотація:
Les travaux menés durant cette thèse sont basés sur la théorie des espaces de longueurs mesurés à courbure de Ricci uniformément minorée initiée par Sturm, Lott et Villani, utilisant de profonds résultats venant de la théorie du transport optimal. Dans une première partie, nous étudions deux familles d'inégalités fonctionnelles, dites de Prékopa-Leindler et de Borell-Brascamp-Lieb, et montrons qu'elles permettent de donner une définition alternative aux bornes sur la courbure de Ricci, satisfaisant un cahier des charges similaire à celui rempli par la condition CD(K,N) de Sturm, Lott et Villani. La seconde partie est consacrée à la recherche d'une généralisation de la définition de Sturm-Lott-Villani au cadre des espaces discrets. Un accent particulier est mis sur le problème de la translation de mesures de probabilité sur un graphe linéaire, et à l'étude de la convexité de l'entropie le long d'une telle translation. L'expression d'une telle translation sous forme d'un convolution binomiale a permis d'éclairer sous un nouvel angle une conjecture formulée par Olkin et Shepp, relative à l'entropie des sommes de Bernoulli indépendantes, et de la démontrer dans un cas particulierThe work done during this PhD thesis is based on the theory of Ricci curvature bounds in measured length spaces, developed by Sturm, Lott and Villani, using deep results coming from the optimal transportation theory. In a first part, we study two families of functional inequalities, called Prékopa-Leindler and Borell-Brascamp-Lieb inequalities, and show that they allows us to give an alternate definition to Ricci curvature bounds, satisfying a "wishlist" similar to the one fulfilled by the Sturm-Lott- Villani condition CD(K,N). The second part is about a possible generalization of Sturm-Lott-Villani definition in a discrete setting. We emphasise the case of the translation of probability measures on a linear graph, and study the convexity of entropy along such a translation. The expression of this translation as a binomial convolution enlightens a conjecture stated by Olkin and Shepp about the entropy of sums of idependent Bernoulli random variables, for which we give a partial proof
Частини книг з теми "Entropic optimal transport"
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Stromme, Austin J. "Minimum Intrinsic Dimension Scaling for Entropic Optimal Transport." In Advances in Intelligent Systems and Computing, 491–99. Cham: Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-65993-5_60.
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Portinale, Lorenzo. "Entropic Regularised Optimal Transport in a Noncommutative Setting." In Bolyai Society Mathematical Studies, 241–61. Cham: Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-50466-2_5.
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Rioul, Olivier. "Optimal Transport to Rényi Entropies." In Lecture Notes in Computer Science, 143–50. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-68445-1_17.
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Yamaka, Woraphon. "Maximum Entropy Learning with Neural Networks." In Optimal Transport Statistics for Economics and Related Topics, 150–62. Cham: Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-35763-3_8.
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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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Carlen, Eric. "Dynamics and Quantum Optimal Transport: Three Lectures on Quantum Entropy and Quantum Markov Semigroups." In Bolyai Society Mathematical Studies, 29–89. Cham: Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-50466-2_2.
Повний текст джерелаТези доповідей конференцій з теми "Entropic optimal transport"
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Wang, Tao, and Ziv Goldfeld. "Neural Estimation of Entropic Optimal Transport." In 2024 IEEE International Symposium on Information Theory (ISIT), 2116–21. IEEE, 2024. http://dx.doi.org/10.1109/isit57864.2024.10619399.
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Reshetova, Daria, Wei-Ning Chen, and Ayfer Özgür. "Training Generative Models from Privatized Data via Entropic Optimal Transport." In 2024 IEEE International Symposium on Information Theory (ISIT), 605–10. IEEE, 2024. http://dx.doi.org/10.1109/isit57864.2024.10619114.
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Lo, Chi-Jen (Roger), Mahesh K. Marina, Nishanth Sastry, Kai Xu, Saeed Fadaei, and Yong Li. "Shrinking VOD Traffic via Rényi-Entropic Optimal Transport." In SIGMETRICS/PERFORMANCE '24: ACM SIGMETRICS/IFIP PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems. New York, NY, USA: ACM, 2024. http://dx.doi.org/10.1145/3652963.3655081.
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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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Yan, Yuguang, Wen Li, Hanrui Wu, Huaqing Min, Mingkui Tan, and Qingyao Wu. "Semi-Supervised Optimal Transport for Heterogeneous Domain Adaptation." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/412.
Повний текст джерелаАнотація:
Heterogeneous domain adaptation (HDA) aims to exploit knowledge from a heterogeneous source domain to improve the learning performance in a target domain. Since the feature spaces of the source and target domains are different, the transferring of knowledge is extremely difficult. In this paper, we propose a novel semi-supervised algorithm for HDA by exploiting the theory of optimal transport (OT), a powerful tool originally designed for aligning two different distributions. To match the samples between heterogeneous domains, we propose to preserve the semantic consistency between heterogeneous domains by incorporating label information into the entropic Gromov-Wasserstein discrepancy, which is a metric in OT for different metric spaces, resulting in a new semi-supervised scheme. Via the new scheme, the target and transported source samples with the same label are enforced to follow similar distributions. Lastly, based on the Kullback-Leibler metric, we develop an efficient algorithm to optimize the resultant problem. Comprehensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of our proposed method.
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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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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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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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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.
Повний текст джерелаАнотація:
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.