We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\varrho:\mathbb{CC}\to \mathbb{C}$ that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor $\mathbb{R}$-affine. This is a much larger class of functions than in the dual setting of arbitrary width and fixed depth. Unlike in the real case, the sufficient width differs significantly depending on the considered activation function. We show that a width of $2n+2m+5$ is always sufficient and that in general a width of $\max\{2n,2m\}$ is necessary. We prove, however, that a width of $n+m+4$ suffices for a rich subclass of the admissible activation functions. Here, $n$ and $m$ denote the input and output dimensions of the considered networks.

翻译：我们研究了具有有界宽度和任意深度的复值神经网络的通用性。在温和假设下，我们完整刻画了那些使得其关联网络具有通用性的激活函数 $\varrho:\mathbb{CC}\to \mathbb{C}$ 的性质，即这些网络能够在紧致域上以任意精度逼近连续函数。具体地，我们证明：深层窄复值网络是通用的，当且仅当它们的激活函数既非全纯、也非反全纯，亦非 $\mathbb{R}$-仿射函数。与任意宽度和固定深度这一对偶设定相比，此类函数类更为广泛。与实数情形不同，充分宽度的值显著依赖于所考虑的激活函数。我们证明宽度 $2n+2m+5$ 总是充分的，且一般情况下宽度 $\max\{2n,2m\}$ 是必要的。然而，对于可接受激活函数的一个丰富子类，我们证明宽度 $n+m+4$ 即足以满足要求。其中 $n$ 和 $m$ 分别表示所考虑网络的输入和输出维度。