Semidiscrete Optimal Transport: Difference between revisions
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<math display="block"> \max_{\varphi \in \R^m} \Big\{ \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\}. </math> | <math display="block"> \max_{\varphi \in \R^m} \Big\{ \int_X \varphi^c d\mu + \sum_j \varphi_j b_j \Big\}. </math> | ||
Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first is that <math> \varphi^c </math> denotes the c-transform of <math> \varphi </math>. The c-transform can be defined as <math> \varphi^c \ | Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first is that <math> \varphi^c </math> denotes the c-transform of <math> \varphi </math>. The c-transform can be defined as <math> \varphi^c \vcentcolon a </math>. |
Revision as of 04:46, 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
where denote probability measures on domains respectively, and is a cost function defined over . denotes the set of possible dual potentials, and the condition that is satisfied. Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as
Aside from using a discrete measure in place of what was originally a continuous one, there are a few other notable distinctions within this reformulation. The first is that denotes the c-transform of . The c-transform can be defined as Failed to parse (unknown function "\vcentcolon"): {\displaystyle \varphi^c \vcentcolon a } .