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."

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 $\mu =f(x)dx$ has a density. 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. Now that we've defined $V_{\varphi }(j)$, we may use this as our domain of integration to find weights $b_{j}$.

A final thing to note is that another name for Voronoi cells is Laguerre cells.

Finding the weights via the gradient

Finding the weights via the above method is equivalent to maximizing ${\mathcal {E}}(\varphi )$, and we may do this by taking the partial derivatives of this function with respect to $\varphi _{j}$. Hence, because of such indexing, this is the same as taking the gradient of ${\mathcal {E}}(\varphi )$. In partial derivative form, we have

$${\frac {\partial {\mathcal {E}}}{\partial \varphi _{j}}}=-\int _{V_{\varphi }(j)}f(x)dx+b_{j}$$

,

and in gradient form, we have

$$\nabla {\mathcal {E}}(\varphi )_{j}=-\int _{V_{\varphi }(j)}f(x)dx+b_{j}.$$

Since $\nabla {\mathcal {E}}(\varphi )_{j}=0$ when it attains a maximum, we have the relation between the weights and the measure density that we established in the previous section. Note that the maximum is taken and not the minimum because our function ${\mathcal {E}}(\varphi )$ is a concave function. The discrete summation contained within this function is linear, but an infimum of a linear function is evaluated for the integration part, making the overall function concave.