Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

By Filippo Santambrogio. 2015

894 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/5d3f82856178dca5b48d14a8259b66121145c97c

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Citations in the corpus

A new class of costs for optimal transport planning (2018, methodology): “It is known (see [16]) that the only optimal solution to the classical transport problem m = inf {∫ X×Y |y − x|2 γ (dx, dy) : γ ∈ (μ, ν) } (3.”
Deep relaxation: partial differential equations for optimizing deep neural networks (2017, methodology): “Tools from optimal transportation (Santambrogio, 2015) or functional inequalities (Bakry and ´Emery, 1985) imply that there is a threshold parameter related to the semiconvexity of f ( x ) for which the auxiliary SGD converges exponentially.”
Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, background): “Theorem 5 applies for a general D(x) and it is equivalent to the celebrated JKO functional (Jordan et al., 1997) in optimal transportation (Santambrogio, 2015; Villani, 2008) if the diffusion matrix is isotropic.”
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).”
Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming (2019, unknown): “…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).”
Statistical Aspects of Wasserstein Distances (2018, background): “In addition to the survey Rachev (1985), there are a number of books dedicated to optimal transport: Rachev & R¨uschendorf (1998), Villani (2003, 2008), Am-brosio & Gigli (2013), Santambrogio (2015), and the forthcoming Panaretos & Zemel (in review), leaning to the statistical side of the subject.”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, background): “…has a number of applications ranging from Geometry and Optimal Transport to Statistics and Data Science (Villani, 2003, 2008; Ambrosio et al., 2008; Santambrogio, 2015; Cuturi and Peyré, 2018) and the context of abstract metric spaces provides a unifying framework encompassing many non-standard…”