Optimal Transport for Applied Mathematicians

By Filippo Santambrogio. 2015

2,226 citations recorded by the source corpus.

Subfields: Computational Mechanics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/adc6dcb0c9ed016fb804ed3572460d312962d8d3

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Citations in the corpus

Plugin estimation of smooth optimal transport maps (2021, background): “Optimal transport maps play a central role in the theory of optimal transport (Rachev and Rüschendorf, 1998; Villani, 2003; Santambrogio, 2015), and form an increasingly important methodology in statistics and machine learning (Kolouri et al., 2017; Panaretos and Zemel, 2019).”
Building Normalizing Flows with Stochastic Interpolants (2022, positive): “…density function ρ t ( x ) deﬁned as the push-forward of the base ρ 0 ( x ) by the map X t satisﬁes the continuity equation (see e.g. (Villani, 2009; Santambrogio, 2015) and Appendix A) ∂ t ρ t + ∇ · ( v t ρ t ) = 0 with ρ t =0 = ρ 0 and ρ t =1 = ρ 1 , (3) and the inference problem becomes to…”
Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems (2017, neutral): “17 [19] and strict convexity of x 7→ |x| %, W % % (η̄12, η) = ∫ 1 0 ∣∣∣12(F−1 η1 (p) + F−1 η2 (p))− F−1 η (p) ∣∣∣∣% dp ≤ 1 2 (∫ 1”
Joint distribution optimal transportation for domain adaptation (2017, background): “As such the Kantorovich relaxation can be seen as a generalization of the Monge problem, with less constraints on the existence and uniqueness of solutions [16].”
SGD without Replacement: Sharper Rates for General Smooth Convex Functions (2019, background): “The following result gives a fundamental characterization of Wasserstein distance (Santambrogio, 2015).”
Heterogeneous Few-Shot Model Rectification With Semantic Mapping (2020, background): “mization strategy [27], [28], [54], OT aligns distributions between sets [16], [55], and has been successfully applied in various machine learning fields with both its objective measure or the learned transportation plan.”
Vector quantile regression beyond correct specification (2016, methodology): “We refer to the textbooks [15], [16] and [14] for a presentation of optimal transport theory, and to [7] for a survey of applications to economics.”
Optimal Mass Transport: Signal processing and machine-learning applications (2017, background): “A more thorough history and background on the optimal mass transport problem can be found in Villani’s book Optimal Transport: Old and New [49] and Santambrogio’s book Optimal Transport for Applied Mathematicians [45].”
SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence (2020, methodology): “We refer readers to [Vil03,San15,San17] for introductory treatments of optimal transport, and to [AGS08,Vil09] for detailed treatments of Wasserstein gradient ﬂows.”
Deep Transfer Learning for Few-Shot SAR Image Classification (2019, background): “where α is a constant and β = (2(d + 1))−1 (see [32] for more details).”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “…distances have a long history and continue to attract interest from diverse fields in statistics, machine learning, and computer science, in particular image analysis; see for instance the monographs and reviews by Santambrogio (2015), Peyré and Cuturi (2019), and Panaretos and Zemel (2019).”
Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming (2019, positive): “Second, our framework for learning regulatory models assumes that trajectories are cell autonomous, but might be extended to incorporate intercellular interactions, such paracrine signaling, by using optimal transport for interacting particles (Ambrosio et al., 2008; Santambrogio, 2015) (Methods S1)”
GAN and VAE from an Optimal Transport Point of View (2017, background): “…is the cost measuring the deviation of a map ϕ : X → X to identity Such a cost is usually associated with the Monge formulation of optimal transport [10], whose original motivation was to ﬁnd an optimal map under that cost that would be able to push forward a given measure onto another[12, § 1.1].”