The universal approximation theorem for complex-valued neural networks
This work provides a foundational theoretical understanding for researchers and practitioners working with complex-valued neural networks, characterizing the architectural and activation function requirements for universal approximation.
This paper extends the universal approximation theorem to complex-valued neural networks, identifying the specific conditions on complex activation functions that enable these networks to approximate any continuous function on compact subsets of complex space. It reveals that the set of 'good' activation functions differs significantly between deep (at least two hidden layers) and shallow networks.
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 $σ: \mathbb{C} \to \mathbb{C}$ in which each neuron performs the operation $\mathbb{C}^N \to \mathbb{C}, z \mapsto σ(b + w^T z)$ with weights $w \in \mathbb{C}^N$ and a bias $b \in \mathbb{C}$, and with $σ$ applied componentwise. 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 $\mathbb{C}^d$ 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, or 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.