Given a fixed set of $n$ points in 2D (Earth Movers distance Prpblem), $P = \{p_1, p_2, ..., p_n\}$, I am trying to find the mass distribution $\bar{M}$ that minimizes the average 2-Wasserstein distance to a given set of mass distributions $\{M^1, M^2, ..., M^S\}$
My questions are:
- Is the optimal mass distribution $\bar{M}$ simply the mean of the mass distributions $ M^1, M^2, ..., M^S$?
- If not, how can we find $\bar{M}$?
- Are there any known algorithms or methods tailored for this problem?
Thank you in advance for any insights or references.