On Representing (Anti)Symmetric Functions
This work addresses a foundational challenge in representing permutation-invariant and anti-symmetric functions for applications in quantum physics and computer vision, though it appears incremental as it builds on existing neural network approaches.
The paper tackled the problem of approximating symmetric and anti-symmetric functions, which are important in quantum physics and computer vision, by deriving natural polynomial approximations for symmetric cases and approximations based on a generalized Slater determinant for anti-symmetric cases, and provided a complete universality proof for the Equivariant MultiLayer Perceptron.
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.