Topics in Optimal Transportation

By Cédric Villani. 2003

5,319 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

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Citations in the corpus

Learning to Simulate Complex Physics with Graph Networks (2020, methodology): “We also investigated distributional metrics including optimal transport (OT) (Villani, 2003) (approximated by the Sinkhorn Algorithm (Cuturi, 2013)), and Maximum Mean Discrep-ancy (MMD) (Gretton et al., 2012).”
Sinkhorn Distances: Lightspeed Computation of Optimal Transport (2013, background): “The gluing lemma (Villani, 2003, Lemma 7.6) plays a crucial role to prove that optimal transportation distances are indeed distances.”
Wasserstein Auto-Encoders (2017, methodology): “A rich class of divergences between probability distributions is induced by the optimal transport (OT) problem [5].”
From optimal transport to generative modeling: the VEGAN cookbook (2017, background): “The optimal transport (OT) cost [5] is a way to measure a distance between probability distributions and provides a much weaker topology than many others, including f -divergences associated with the original GAN algorithms.”
Ricci curvature for metric-measure spaces via optimal transport (2004, background): “We refer to [10] and [41] for background material on length spaces and optimal transport, respectively.”
Entropies, convexity, and functional inequalities (2002, neutral): “py) H when Φ(x) = xlogx, cf. [ABC+00, Chap. 10]. Actually, S := −H is also known as Shannon Information in Information Theory or Boltzmann entropy in Kinetic Gases Theory, see for example [Lie75] and [Vil03]. Similarly, one can deﬁne the Shannon Φ-entropy of a discrete probability measure p1δx1 + ··”
Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group (2009, background): “For a detailed presentation, and more about optimal transport, see [30] or [31].”
Limit theorems for some branching measure-valued processes (2011, methodology): “But, If m1 = 1 n ∑n k=0 δxi andm2 = 1 n ∑n k=0 δyi are two discrete measures, where n ∈ N∗ andxi, yi ∈ F , we have the following matching representation [40]: W (p) dF (m1,m2) p = inf τ∈Sn 1 n n ∑”
Some Geometric Calculations on Wasserstein Space (2006, methodology): “We refer to [21] for background information on Wasserstein spaces.”
The first variation of the total mass of log-concave functions and related inequalities (2013, background): “1) is somehow more “flexible”, and finds many applications in different fields, such as convex geometry, probability, mass transportation; we refer the reader to [2, 3, 32] for more information on Prékopa-Leindler inequality, including proofs and bibliographical references.”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, positive): “18 of [30], that will be used as a building block.”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “A construction analogous to the one used in the proof of Proposition 1 in Yamada and Watanabe (1971) [also see Section 5.”
A Group-Theoretic Framework for Data Augmentation (2019, background): “, [68] for background and definitions needed): Definition 3.”
Fréchet means and Procrustes analysis in Wasserstein space (2017, methodology): “In the first case, by Theorem 4.14(iii) in Villani [74] there exist C1 (in fact, C2,α) convex potentials ϕi on Rd with tµ i µ = ∇ϕi, so that tµ i µ (x) is a singleton for all x ∈ Rd.”