Towards Understanding Theoretical Advantages of Complex-Reaction Networks
This work addresses a foundational theoretical gap in machine learning by providing insights into the efficiency and optimization benefits of complex-valued networks, though it is incremental as it builds on existing interest in complex-valued architectures.
This paper tackles the problem of understanding the theoretical advantages of complex-valued neural networks over real-valued ones by introducing complex-reaction networks. It proves that these networks can approximate certain radial functions with a polynomial number of parameters, while real-valued networks require exponential parameters, and shows that their critical point set is a proper subset, potentially aiding in optimization.
Complex-valued neural networks have attracted increasing attention in recent years, while it remains open on the advantages of complex-valued neural networks in comparison with real-valued networks. This work takes one step on this direction by introducing the \emph{complex-reaction network} with fully-connected feed-forward architecture. We prove the universal approximation property for complex-reaction networks, and show that a class of radial functions can be approximated by a complex-reaction network using the polynomial number of parameters, whereas real-valued networks need at least exponential parameters to reach the same approximation level. For empirical risk minimization, our theoretical result shows that the critical point set of complex-reaction networks is a proper subset of that of real-valued networks, which may show some insights on finding the optimal solutions more easily for complex-reaction networks.