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"> \ | <math display="block"> \max_{(\varphi, \psi) \in \nR} \Big\{ \int_X \varphi d\mu + \int_Y \psi d\nu : \varphi(x) + \psi(y) \leq c(x,y) \Big\} </math> | ||
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>. 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>. Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as | ||
<math display="block"> \max \Big\{ \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\} </math> | <math display="block"> \max \Big\{ \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\} </math> | ||
Revision as of 04:34, 2 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max_{(\varphi, \psi) \in \nR} \Big\{ \int_X \varphi d\mu + \int_Y \psi d\nu : \varphi(x) + \psi(y) \leq c(x,y) \Big\} }
where denote probability measures on domains respectively, and is a cost function defined over . Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as
