On Representing (Anti)Symmetric Functions
On Representing (Anti)Symmetric Functions Marcus Hutter DeepMind & ANU http://www.hutter1.net/ 12 Jun 2020 Abstract 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 nat- ural 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 uni- versality proof of the Equivariant MultiLayer Perceptron, which implies universality of symmetric MLPs and the FermiNet. Contents 1 Introduction 2 2 Related Work 3 arXiv:2007.15298v1 [cs.NE] 30 Jul 2020 3 One-Dimensional Symmetry 4 4 One-Dimensional AntiSymmetry 8 5 d-dimensional (Anti)Symmetry 11 6 Universality of Neural Networks 13 7 Universal Equivariant Networks 15 8 (Anti)Symmetric Networks 18 9 Discussion 19 A List of Notation 22 Keywords Neural network, approximation, universality, Slater determinant, Vandermonde matrix, equivariance, symmetry, anti-symmetry, symmetric polynomials, polarized basis, multilayer perceptron, continuity, smoothness, ..
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