  Permutation-invariant, -equivariant, and -covariant functions and
anti-symmetric functions are important in quantum physics, computer vision, and
other disciplines. Applications often require most or all of the following
properties: (a) a large class of such functions can be approximated, e.g. all
continuous function, (b) only the (anti)symmetric functions can be represented,
(c) a fast algorithm for computing the approximation, (d) the representation
itself is continuous or differentiable, (e) the architecture is suitable for
learning the function from data. (Anti)symmetric neural networks have recently
been developed and applied with great success. A few theoretical approximation
results have been proven, but many questions are still open, especially for
particles in more than one dimension and the anti-symmetric case, which this
work focusses on. More concretely, we derive natural polynomial approximations
in the symmetric case, and approximations based on a single generalized Slater
determinant in the anti-symmetric case. Unlike some previous super-exponential
and discontinuous approximations, these seem a more promising basis for future
tighter bounds. We provide a complete and explicit universality proof of the
Equivariant MultiLayer Perceptron, which implies universality of symmetric MLPs
and the FermiNet.
