The universal approximation theorem for complex-valued neural networks

• Published in: Applied and computational harmonic analysis Vol. 64; pp. 33 - 61
• Main Author: Voigtlaender, Felix
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2023
• Subjects: Complex-valued neural networks; Deep neural networks; Holomorphic functions; Polyharmonic functions; Universal approximation theorem; 30E10; Deep neural networks; 41A30; 41A63; 31A30; Complex-valued neural networks; Universal approximation theorem; 68T07; Holomorphic functions; Polyharmonic functions
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2022.12.002

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, nor an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function.