Multivariate optimal transport Optimal transport offers a way to find the optimal transport plan between 1 source distribution $P_r$ and 1 target distribution $P_\theta$, where $\boldsymbol\Gamma$ is the optimal transport plan and $\boldsymbol D$ is the distance matrix.

How or can the model instead be extended to the case where there are multiple source distributions, $P_r^1, P_r^2, \dots, P_r^k$  (or perhaps a weighted combination of them so that they collapse into a single distribution), from which we would like to move towards the target distribution $P_\theta$?
 A: In general, for the Kantorovich formualtion of the Optimal Transportation problem, we consider two probability spaces $(X,\mu)$ and $(Y,\nu)$ such that given a cost function $c: X\times Y\rightarrow [0,\infty)$ we look for the minimum total cost to transport the mass in $X$ to $Y$, i.e.
$$ \min_{\pi\in\Pi(\mu,\nu)} \iint_{X\times Y} c(x,y)d\pi(x,y) $$
where $\Pi(\mu,\nu)$ is the set of probability measures with marginals $\mu$ and $\nu$ (set of transport plans). Then in this setting we know that an optimal transport plan $\pi_{\text{opt}}$ exists if $X$ and $Y$ are Polish spaces (complete separable metric spaces) and if $c(x,y)$ is lower semi-continuous.
Considering now a multivariable case in which the source space is composed by $n\in\mathbb{N}$ measure spaces $(A_i,\mu_i)$ for all $1\leq i\leq n$, then we can define the cartesian product
$$ X=\prod_{i=1}^n A_n, \quad \mu = \bigotimes_{i=1}^n \mu_i. $$
On the one hand we need to make sure that $(X,\mu)$ is a probability space, that is,
$$ \int_{A_1\times...\times A_n} d(\mu_1\otimes...\otimes\mu_n) \equiv \prod_{i=1}^n\int_{A_i}d\mu_i = 1,$$
which in particular will happen if $(A_i,\mu_i)$ is a probability space for all $1\leq i\leq n$.
On the other hand we need all $A_i$ to be Polish spaces since the product of countably many Polish spaces is a Polish space. Therefore there will be an optimal transport plan $\pi_{\text{opt}}$ that takes the mass from $A_1\times ...\times A_n$ to $Y$ if the cost function $c:A_1\times ...\times A_n\times Y\rightarrow [0,\infty)$ is lower semi-continuous in the sense that given $(a_1,...,a_n,y)\in A_1\times ...\times A_n\times Y$, $\forall z<c(a_1,...,a_n,y)$ $\exists U\subset A_1\times...\times A_n\times Y$ neighbourhood of $(a_1,...,a_n,y)$ such that $z<c(x)$ for all $x\in U$.
