I'm not familiar with the R[x,y,...]^G notation, but I'll assume it refers to the group of polynomials that are invariant under rotation of their inputs. I'm not sure whether the rotations of discretely sampled images really do form a group, since intuitively two 45° rotations lose information compared to a 90° rotation, but maybe you can fix that by assuming the right kind of periodicity.

Even assuming that rotations aren't lossy, I get at best a reduction in the number of parameters by a factor of √(number of variables), by fixing the rotation of a set of variables (representing sample point) so that one of them lies on a specific axis. In other words, this reduces the exponent by 1/2, which is still not small enough to make even second-degree polynomials feasible.

However, that doesn't mean I think symmetry priors like this are useless, so if you can point out further literature on this topic, that would be great! (It might also help me understand how exponentiating a group by another makes sense.)