A User’s Guide to Optimal Transport

By Luigi Ambrosio, Nicola Gigli. 2013

512 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/78d9abde1d253c5144d6a3245534f8bc5bc7f06a

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “The situation concerning these two different notions of gradient flows (EVI and EDE) in abstract metric spaces has been clarified by Savaré (in an unpublished note, but the proof can also be found in [2]), who showed that • All curveswhich are gradient flows in theEVI sense also satisfy theEDEcondit”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “1), (Qt(φ))t∈[0,2] is uniformly bounded in each L(X,m).”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “Let μ := 1/2(λ|[−3,−2] + λ|[2,3]), ν := 1/6(λ|[−4,−1] + λ|[1,4]).”
Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$CD(K,∞)-spaces (2012, methodology): “The distance between γ1, γ2 ∈ C([0, 1];X) is given by d∗(γ1, γ2) = max{d(γ1 t , γ 2 t ) : 0 ≤ t ≤ 1}.”
Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below (2013, unknown): “We refer to [23] and [1] for an overview of the subject.”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, background): “proof Without loss of generality and up to replace d2 with d2 ∧ 1 we can, and will, assume that d2 is a bounded distance so that the distance W2 metrizes the narrow convergence on the space P(M2) = P2(M2) (see for instance Chapter 2 of [6] or Chapter 6 of [56] for the relevant definitions and proper”
An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “General references for these topics are [52] and [2].”
Entropic regularization of continuous optimal transport problems (2019, background): “For example it is known that the problem has a solution π and that the support of π is contained in the so-called c -superdifferential of a c -concave function on Ω 1 , see [1, Thm. 1.13].”
Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “We assume that the reader is familiar with the basic notions of measure theory and optimal transport on abstract metric spaces (standard references are [3], [5] and [36]).”
An introduction to sampling via measure transport (2016, background): “Generalizations of this result accounting for diﬀerent cost functions and spaces can be found in [19, 2, 27, 11].”
Statistical inference for Bures–Wasserstein barycenters (2019, methodology): “For more details on general class of optimal transportation distances we recommend excellent monographs Ambrosio and Gigli (2013); Villani (2009).”
Bayesian Learning with Wasserstein Barycenters (2018, background): “See Ambrosio and Gigli [2013], Ambrosio et al. [2004], Villani [2003, 2008], McCann and Guillen [2011] for a thorough introduction to optimal transport.”
Monotone martingale transport plans and Skorokhod embedding (2017, unknown): “We refer to [28, 29, 1] for recent accounts of the theory.”