# Complex-valued Neural Networks with Non-parametric Activation Functions

arXiv: https://arxiv.org/abs/1802.08026 · HTML: https://mlsanity.com/paper/1802.08026/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2018.

## Built on by (most cited first, 19 of 19)

- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [How important are activation functions in regression and classification? A survey, performance comparison, and future directions](https://mlsanity.com/paper/2209.02681/) (2022)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces](https://mlsanity.com/paper/2303.07115/) (2023)
- [Neural network is heterogeneous: Phase matters more](https://mlsanity.com/paper/2111.02014/) (2021)
- [Co-domain Symmetry for Complex-Valued Deep Learning](https://mlsanity.com/paper/2112.01525/) (2021)
- [A Hybrid Complex-valued Neural Network Framework with Applications to Electroencephalogram (EEG)](https://mlsanity.com/paper/2207.14799/) (2022)
- [Steinmetz Neural Networks for Complex-Valued Data](https://mlsanity.com/paper/2409.10075/) (2024)
- [Impact of PolSAR pre-processing and balancing methods on complex-valued neural networks segmentation tasks](https://mlsanity.com/paper/2210.17419/) (2022)
- [Improving Quaternion Neural Networks with Quaternionic Activation Functions](https://mlsanity.com/paper/2406.16481/) (2024)
- [Scaling Continuous Kernels with Sparse Fourier Domain Learning](https://mlsanity.com/paper/2409.09875/) (2024)
- [Towards White-Box Deep Wireless Sensing](https://mlsanity.com/paper/2507.21799/) (2025)
- [Characterizing 4-string contact interaction using machine learning](https://mlsanity.com/paper/2211.09129/) (2022)
- [Sophisticated deep learning with on-chip optical diffractive tensor processing](https://mlsanity.com/paper/2212.09975/) (2022)
- [Bayesian optimization for sparse neural networks with trainable activation functions](https://mlsanity.com/paper/2304.04455/) (2023)
- [On the Computational Complexities of Complex-valued Neural Networks](https://mlsanity.com/paper/2310.13075/) (2023)
- [LCB-CV-UNet: Enhanced Detector for High Dynamic Range Radar Signals](https://mlsanity.com/paper/2505.23454/) (2025)
- [Novel Complex-Valued Hopfield Neural Networks with Phase and Magnitude Quantization](https://mlsanity.com/paper/2507.00461/) (2025)

## Abstract (arXiv)

Complex-valued neural networks (CVNNs) are a powerful modeling tool for domains where data can be naturally interpreted in terms of complex numbers. However, several analytical properties of the complex domain (e.g., holomorphicity) make the design of CVNNs a more challenging task than their real counterpart. In this paper, we consider the problem of flexible activation functions (AFs) in the complex domain, i.e., AFs endowed with sufficient degrees of freedom to adapt their shape given the training data. While this problem has received considerable attention in the real case, a very limited literature exists for CVNNs, where most activation functions are generally developed in a split fashion (i.e., by considering the real and imaginary parts of the activation separately) or with simple phase-amplitude techniques. Leveraging over the recently proposed kernel activation functions (KAFs), and related advances in the design of complex-valued kernels, we propose the first fully complex, non-parametric activation function for CVNNs, which is based on a kernel expansion with a fixed dictionary that can be implemented efficiently on vectorized hardware. Several experiments on common use cases, including prediction and channel equalization, validate our proposal when compared to real-valued neural networks and CVNNs with fixed activation functions.