Until recently, applications of neural networks in machine learning have almost exclusively relied on real-valued networks. It was recently observed, however, that complex-valued neural networks (CVNNs) exhibit superior performance in applications in which the input is naturally complex-valued, such as MRI fingerprinting. While the mathematical theory of real-valued networks has, by now, reached some level of maturity, this is far from true for complex-valued networks. In this paper, we analyze the expressivity of complex-valued networks by providing explicit quantitative error bounds for approximating $C^n$ functions on compact subsets of $\mathbb{C}^d$ by complex-valued neural networks that employ the modReLU activation function, given by $σ(z) = \mathrm{ReLU}(|z| - 1) \, \mathrm{sgn} (z)$, which is one of the most popular complex activation functions used in practice. We show that the derived approximation rates are optimal (up to log factors) in the class of modReLU networks with weights of moderate growth.
Universal approximation with complex-valued deep narrow neural networks
Mathematical Institute for Machine Learning and Data Science (MIDS), Catholic University of Eichstätt–Ingolstadt (KU)（艾希施泰特-英戈尔施塔特天主教大学 机器学习与数据科学数学研究所）
From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps
Shanghai Center for Mathematical Sciences（上海数学中心）；Department of Mathematics, Fudan University（复旦大学数学系）；Center for Applied Mathematics, Fudan University（复旦大学应用数学中心）；Alibaba Group（阿里巴巴集团）；Shanghai, China（（此处为城市信息，非机构，按要求不提取））；Hangzhou, China（（此处为城市信息，非机构，按要求不提取））；Fudan University（复旦大学）
证明了对每一个巴拿赫空间$Y$，从$n$维欧几里得空间到$Y$的函数的Besov空间与适当的具有等价范数的局部逼近空间一致.证明了$q$型索伯列夫空间连续嵌入同类型Besov空间的充要条件是$Y$有鞅余型$q$ .我们将其解释为Xu（1998）和Martínez，Torrea and Xu（2006）早期结果的推广。这两个结果结合起来给出了$Y$允许幂型凸性模模为$q$的等价范数的特征当且仅当弱可微函数具有多项式的良好局部逼近.
