is the loss you're talking about like fused gromov-wasserstein distance? we hit permutation invariance issues like what you're talking about in some atomistic

is the loss you're talking about like fused gromov-wasserstein distance? we hit permutation invariance issues like what you're talking about in some atomistic simulations because the atoms need to be permutable if you want to use the same model for chemistry as protein folding/docking, and the FGW algo from e.g. https://arxiv.org/pdf/1811.02834.pdf https://arxiv.org/pdf/1811.02834.pdf https://tvayer.github.io/materials/Titouan_Marseille_2019.pdf https://tvayer.github.io/materials/Titouan_Marseille_2019.pd... relaxes the invariance issue by adding a feature distance to the euclidean distance. higher order distance matrices are a neat trick, but blow up VRAM past 10-50k atoms, but if you did it in mixed precision with newer gpus it could scale damn far. problem is, the distance between distance matrices assumes the target and source items are matched, so you get into iterative closest point alignment, and pretty soon you're just reinventing RMSD it would be cool for molec stuffs to have fast permutation-invariant set based loss functions using transport theory, but this might be better handled with a model-free approach (just let the AI figure out the loss function itself)