I have a question about the root mean square error and Wasserstein distance on the paper https://arxiv.org/abs/2111.08736?context=stat. Consider two discrete probability distributions $P=\{P_i\}_{i=1}^n$ and $Q=\{Q_i\}_{i=1}^n$ so that $p_i\ge 0$ for all $i$, $q_j\ge 0$ for all $j$, and $\sum_i p_i=\sum_j q_j=1$.
In this paper, $i=1,...,m$ indexes a spatial partition of the region of ocean being studied in to m cells (and likewise for $j=1,...,n$) and $P_i$ gives the proportion of the Chlorophyll (or any other positive quantity the scalar field is representing) in the region that is in celli. In the special case that $i$ and $j$ index the same set of cells (such as $m=n$ pixels), one can define pixel-wise distances such as the root-mean-squared error,
$$RMSE(P,Q):=(\frac{1}{n}\sum_i (P_i-Q_i)^2)^{1/2}.
$$
and the 2-Wasserstein distance.
Question: Why "If $P$ and $Q$ do not exist on the same coordinates, they need to be reconciled (processed) to exist on the exact same cells in order to calculate RMSE. This requirement is not shared by the Wasserstein distance"?
