Parametric swish-cLogLog activation function for complex-valued CNN in PolSAR image interpretation

Activation functions serve as a foundational element in complex-valued convolutional neural networks (CV-CNNs), enabling the architectures to learn non-linear representations from data. A significant challenge in their design lies in effectively processing complex numbers while maintaining training efficiency. Many prevalent activation functions deal with the real and imaginary components separately, thereby neglecting the inherent correlation in complex-valued features. Although some sophisticated functions that operate on the amplitude and phase components can capture these correlations, they often suffer from high computational cost and unstable gradient calculations. To address the above issues, this paper introduces the parametric swish-cLogLog activation function (PSL²AF), which jointly processes the real and imaginary parts and adaptively adjusts their correlation through learnable parameters. This design is deliberately formulated to preserve the relationship within complex data while introducing a self-controlled non-linearity form. Furthermore, its relatively lower computational cost compared to amplitude-phase activation functions makes it suitable for large-scale models. There are two approaches regarding parameter sharing. One method shares parameters across all channels, called the shared-parametric swish-cLogLog activation function (SPSL²AF). The other method assigns independent parameters to each channel, called the exclusive-parametric swish-cLogLog activation function (EPSL²AF). To facilitate the training of CV-CNNs, we provide a detailed derivation of the backpropagation formula. The proposed activation functions are validated on two synthetic aperture radar (SAR) datasets for image classification, demonstrating superior classification accuracy and stronger robustness compared to existing mainstream complex activation functions.