Quantifying Lie Group Learning with Local Symmetry Error

Despite increasing interest in using machine learning to discover symmetries, no quantitative measure has been proposed in order to compare the performance of different algorithms. Our proposal, both intuitively and theoretically grounded, is to compare Lie groups using a local symmetry error, based on the difference between their infinitesimal actions at any possible datapoint. Namely, we use a well-studied metric to compare the induced tangent spaces. We obtain an upper bound on this metric which is uniform across datapoints, under some conditions. We show that when one of the groups is a circle group, this bound is furthermore both tight and easily computable, thus globally characterizing the local errors. We demonstrate our proposal by quantitatively evaluating an existing algorithm. We note that our proposed metric has deficiencies in comparing tangent spaces of different dimensions, as well as distinct groups whose local actions are similar.