Optimal transport for inverse solution validation

Problem:
We need an universal tool to compare the inverse solution to simulated source configuration, i.e. have a function f(stc_simuleted, stc_estimated) which would quantify how "close" estimated solution is to simulated one.

The existing way to compute the l2 norm between two stcs does not take into account spatial properties of sources (spatially close sources can still result in high l2 error norm.)

To solve solve this issue Dipole Localization Error (DLE) is often used, but it does not seem to me an universal solution, cause it would have issues with distributed or multiple sources configurations.

Possible solution:
During several discussions of this problem, the idea of using of optimal transport was proposed.
I tried to formalize it, substantially based on H. Janati et al. "Wasserstein regularization for sparse multi-task regression", 2018.
The main idea is to do optimal transport from estimated solution to the simulated one, and use the amount of "energy" needed for this transport as a similarity measure, i.e. smaller it is, better is the estimation.
Please consult the latex file for mathematical formulation of this criteria, and tell me what do think about it.
@agramfort, do you have a code to compute the unbalanced Wasserstein distance that you used in this arcticle?
OT_for_simulation_metric.pdf

Start by reading the attached OT_for_simulation_metric.pdf and reviewing the issue's description of the existing l2 comparison and proposed unbalanced Wasserstein metric. Determine the intended public function and validation cases for simulated and estimated source configurations; done should include an agreed formulation and a clear implementation and testing scope.