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We study the Wasserstein metric to measure distances between moleculesrepresented by the atom index dependent adjacency "Coulomb" matrix, used inkernel ridge regression based supervised learning. Resulting quantum machinelearning models exhibit improved training efficiency and result in smootherpredictions of molecular distortions. We first demonstrate smoothness for thecontinuous extraction of an atom from some organic molecule. Learning curves,quantifying the decay of the atomization energy's prediction error as afunction of training set size, have been obtained for tens of thousands oforganic molecules drawn from the QM9 data set. In comparison to conventionallyused metrics ($L_1$ and $L_2$ norm), our numerical results indicate systematicimprovement in terms of learning curve off-set for random as well as sorted (bynorms of row) atom indexing in Coulomb matrices. Our findings suggest that thismetric corresponds to a favorable similarity measure which introducesindex-invariance in any kernel based model relying on adjacency matrixrepresentations.

Wasserstein metric for improved QML with adjacency matrix representations