Semidiscrete Optimal Transport

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."

Voronoi Cells. These partitions of the plane are Voronoi cells, which are used in the semidiscrete dual optimal transport problem. The cell structures are determined by the cost function used, here being the Euclidean metric.

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

$$\max _{(\varphi ,\psi )\in R(c)}{\Big \{}\int _{X}\varphi d\mu +\int _{Y}\psi d\nu {\Big \}}$$

where $\mu ,\nu$ denote probability measures on domains $X,Y$ respectively, and $c(x,y)$ is a cost function defined over $X\times Y$. $R(c)$ denotes the set of possible dual potentials, and the condition $\varphi (x)+\psi (y)\leq c(x,y)$ is satisfied. It should also be noted that $\mu$ has a density such that $\mu =f(x)dx$. Now, we would like to extend this notion of the dual problem to the semidiscrete case. Such a case can be reformulated as

$$\max _{\varphi \in \mathbb {R} ^{m}}{\Big \{}{\mathcal {E}}(\varphi )=\int _{X}\varphi ^{c}d\mu +\sum _{j}\varphi _{j}b_{j}{\Big \}}.$$

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 $\psi$ is replaced completely with $\varphi$. The second is that $\varphi ^{c}$ denotes the c-transform of $\varphi$. The c-transform can be defined as $\varphi ^{c}(x):=\min _{j}\{c(x,y_{j})-\varphi _{j}\}$.

Voronoi cells to find weights

Now, we will establish the notion of Voronoi cells. The Voronoi cells refer to a special subset of $X$, and the reason we are interested in such a subset is because we can use the Voronoi cells as a domain to find the weights $b_{j}$ that we established in our reformulation of the dual problem. In particular, if we denote the set of Voronoi cells as $V_{\varphi }(j)$, we can find our weights using the fact $b_{j}=\int _{V_{\varphi }(j)}f(x)dx$. Recall that $f(x)$ refers to a density of the measure $\mu$, i.e., $\mu =f(x)dx$. We define the Voronoi cells with

$$V_{\varphi }(j)={\Big \{}x\in X:\forall j'\neq j,{\frac {1}{2}}|x-y_{j}|^{2}-\varphi _{j}\leq {\frac {1}{2}}|x-y_{j'}|^{2}-\varphi _{j'}{\Big \}}.$$

We use the specific cost function $c(x,y)={\frac {1}{2}}|x-y|^{2}$ here. This is a special case and we may generalize to other cost functions if we desire. When we have this special case, the decomposition of our space $X$ is known as a "power diagram." Using our power diagram as a domain of integration, we can successfully find the weights $b_{j}$.

Finding the weights via the gradient

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