The permutational invariance is directly built-in to PIPs whereas the procedure to build this invariance into the ACE basis is more complex. A somewhat related approach uses a basis of B-splines was proposed for an atom-centered expansion up to 3-b terms and applied to tungsten. This basis is fast to evaluate; however, it is not manifestly permutationally invariant. The method that is perhaps closest to one developed here is a successor to our PIPs approach and is termed aPIPs. The method was successful in terms of precision and speed relative to other ML MB approaches. However, it was not pursued further by those authors. Finally, we note that while the aPIPs, ACE and present MB-PIPs methods are polynomial linear regression methods, the former two belong to the class of underdetermined least-squares, i.e., more linear coefficients than data, whereas MB-PIPs (and all of our PIPs fits) are overdetermined, i.e., fewer linear coefficients than data. Another important difference is a formal one, namely aPIPs and ACE represent the total energy as the sum of atomic energies instead of the MB expansion given above.