Background

\[W_p(C) = \left(\sum_{i=1}^n\sum_{j=1}^m C_{ij}^p \Gamma'_{ij}\right)^{1/p}\]

where \(\Gamma'\) is the optimal transport plan found by solving the following linear problem:

\begin{equation} \begin{aligned} \text{min}_{\ \Gamma} \quad & \sum_{i=1}^n\sum_{j=1}^m C_{ij}^p \Gamma_{ij} \\ \text{s.t.} \quad & \sum_{j=1}^m \Gamma_{ij} = \nu_i & \quad \forall i \\ \quad & \sum_{i=1}^n \Gamma_{ij} = \mu_j & \quad \forall j \\ \quad & \Gamma_{ij} \ge 0 & \quad \forall i, j \\ \end{aligned} \end{equation}

\begin{equation} \begin{aligned} \min_{\Theta} \quad & \left( \min_{\Gamma \in \mathcal{F}} \ \sum_{i=1}^n\sum_{j=1}^m C_{ij}^p \Gamma_{ij} \right)^{1/p} \\ \text{where} \quad & C = d\left( f_{\Theta}(E), Y \right) \label{eq:prob} \end{aligned} \end{equation}