Optimal Transport: Old and New

By Cédric Villani. 2008

7,666 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Optimization & Training

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Citations in the corpus

Improved Training of Wasserstein GANs (2017, methodology): “The WGAN value function is constructed using the Kantorovich-Rubinstein duality [24] to obtain”
GAIT: A Geometric Approach to Information Theory (2019, methodology): “Although traditionally dominated by maximum likelihood estimation, a significant portion of research on parameter estimation has shifted towards methods based on optimal transport, such as the Wasserstein distance (Villani, 2008).”
Wasserstein GAN (2017, background): “Let us deﬁne Since X is compact, we know by the Kantorovich-Rubenstein duality [22] that there is an f ∈ F that attains the value Let us deﬁne X ∗ ( θ ) = { f ∈ F : V ( f, θ ) = W ( P r , P θ ) } .”
DeepMDP: Learning Continuous Latent Space Models for Representation Learning (2019, background): “Wasserstein Distance For two distribution P and Q defined on a metric space xχ, dy, the optimal transport problem (Villani, 2008) studies how to transform the probability mass of P into Q with the minimum cost, where the cost of a particle from point x to y comes from a metric dpx, yq.”
Towards Principled Methods for Training Generative Adversarial Networks (2017, background): “We refer the reader to Villani (2009) for an in-depth explanation of these ideas.”
Personalized Federated Learning: A Meta-Learning Approach (2020, background): “…need to show for any v ∈ R d with (cid:107) v (cid:107) ≤ 1 , we have Next, note that since p i and p both have bounded support, by Kantorovich–Rubinstein Duality Theorem Villani (2008), we have Using this result, to show (27), it suﬃces to show g ( z ) = v (cid:62) ∇ l ( z ; w ) is L Z -Lipschitz.”
Theoretical evidence for adversarial robustness through randomization: the case of the Exponential family (2019, background): “According to this interpretation, it seems quite natural to consider the total variation distance as a relaxation of the trivial distance on [0, 1] (see [43] for details).”
Certifying Some Distributional Robustness with Principled Adversarial Training (2017, background): “To make this rigorous, fix γ > 0, and consider the worst-case perturbation, typically called the transportation map or Monge map (Villani, 2009), Tγ(θ; z0) := argmax z∈Z {`(θ; z)− γc(z, z0)}.”
A Survey on Neural Architecture Search (2019, methodology): “The kernel value computation in this work is modeled as an optimal transport program (Villani, 2008).”
Conditional deep surrogate models for stochastic, high-dimensional, and multi-fidelity systems (2019, background): “and limitations of such models [32, 33, 34, 35, 36, 37], enhance the interpretability of their results [38, 39, 40], as well as establish formal connections with well studied topics in mathematics and statistics, including importance sampling [41, 42] and optimal transport [43, 44, 45].”
Co-Transport for Class-Incremental Learning (2021, background): “Optimal transport (OT): OT [34, 41] is first formulated to study the optimal transportation and resource allocation problem [18, 31].”
Flow Matching for Generative Modeling (2022, methodology): “One method of testing if a vector field v t generates a probability path p t is the continuity equation (Villani, 2009).”
Hyperparameter Selection for Imitation Learning (2021, methodology): “In particular, we compute the Wasserstein distance (Villani, 2008) between the distribution of states in the demonstrations and that of the agent3 (i.”
Sinkhorn Distances: Lightspeed Computation of Optimal Transport (2013, background): “Optimal transportation distances (Villani, 2009, § 6) – also known as Earth Mover’s following the seminal work of Rubner et al. (1997) and their application to computer vision – hold a special place among other distances in the probability simplex.”
Normalizing Flows for Probabilistic Modeling and Inference (2019, background): “Optimal transport and the Wasserstein metric (Villani, 2008) can also be formulated in terms of transformations of measures (‘transport of measures’)—also known as the Monge problem.”