Which Physical Symmetries Can Be Imposed by Averaging?

Averaging a trained model over a compact symmetry group makes it invariant, and when the data share the symmetry the averaged model is no worse than the original under a convex loss. Many symmetries of physical systems, such as translations, Lorentz boosts and changes of units, form non-compact groups, so the average must be restricted to a bounded set of group elements, usually a ball around the identity. We study what this truncation preserves. For connected Lie groups, ball averages become invariant for every bounded model if and only if the volume of balls grows polynomially. Under exponential growth, as for the Lorentz group, they do not, and the average is governed by the values of the model at the cutoff. In addition, no data distribution is invariant under a non-compact group that moves inputs arbitrarily far, as translations and boosts do. The guarantee that averaging does not increase the risk is then lost, as an exact counterexample and a trained top-quark tagger show. Amenability decides neither question. It decides only whether some other sequence of sets makes the average invariant, and we construct such sets for the affine and similarity groups.