Semidiscrete Optimal Transport: Difference between revisions
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In particular, we will examine semidiscrete optimal transport in the case of the dual problem. The general dual problem for continuous measures can be stated as | In particular, we will examine semidiscrete optimal transport in the case of the dual problem. The general dual problem for continuous measures can be stated as | ||
<math display="block"> \max_{(\varphi, \psi) \in R(c)} \Big\{ \int_X \varphi d\mu + \int_Y \psi d\nu \Big\} </math> | <math display="block"> \max_{(\varphi, \psi) \in R(c)} \Big\{ \int_X \varphi d\mu + \int_Y \psi d\nu \Big\} </math><ref name="Santambrogio"> | ||
where <math> \mu, \nu </math> denote probability measures on domains <math> X, Y </math> respectively, and <math> c(x,y) </math> is a cost function defined over <math> X \times Y </math>. <math> R(c) </math> denotes the set of possible dual potentials, and the condition <math> \varphi(x) + \psi(y) \leq c(x,y) </math> is satisfied. It should also be noted that <math> \mu </math> has a density such that <math> \mu = f(x)dx </math>. Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as | where <math> \mu, \nu </math> denote probability measures on domains <math> X, Y </math> respectively, and <math> c(x,y) </math> is a cost function defined over <math> X \times Y </math>. <math> R(c) </math> denotes the set of possible dual potentials, and the condition <math> \varphi(x) + \psi(y) \leq c(x,y) </math> is satisfied. It should also be noted that <math> \mu </math> has a density such that <math> \mu = f(x)dx </math>. Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as | ||
Revision as of 03:34, 6 June 2020
Semidiscrete optimal transport refers to situations in optimal transport where two input measures are considered, and one measure is a discrete measure and the other one is continuous. Hence, because only one of the two measures is discrete, we arrive at the appropriate name "semidiscrete."
Formulation of the semidiscrete dual problem
In particular, we will examine semidiscrete optimal transport in the case of the dual problem. The general dual problem for continuous measures can be stated as
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- ↑ F. Santambrogio, Optimal Transport in Applied Mathematics, Chapter 6.
- ↑ G. Peyré and M. Cuturi, Computational Optimal Transport, Chapter 5.
- ↑ Valentin H. and Schuhmacher D., Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case, Institute for Mathematical Stochastics, University of Goettingen.