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Conventional Convolutional Neural Networks (CNNs), which are realized in spatial domain, exhibit high computational complexity. This results in high resource utilization and memory usage and makes them unsuitable for implementation in resource and energy-constrained embedded systems. A promising approach for low-complexity and high-speed solution is to apply CNN modeled in the spectral domain. One of the main challenges in this approach is the design of activation functions. Some of the proposed solutions perform activation functions in spatial domain, necessitating multiple and computationally expensive spatial-spectral domain switching. On the other hand, recent work on spectral activation functions resulted in very  computationally intensive solutions. This paper proposes a complex-valued activation function for spectral domain CNNs that only transmits input values that have positive-valued real or imaginary component. This activation function is computationally inexpensive in both forward and backward propagation and provides sufficient nonlinearity that ensures high classification accuracy. We apply this complex-valued activation function in a LeNet-5 architecture and achieve an accuracy gain of up to 7% for MNIST and 6% for Fashion MNIST dataset, while providing up to 79% and 85% faster inference times, respectively, over state-of-the-art activation functions for spectral domain.

A Low-complexity Complex-valued Activation Function for Fast and Accurate Spectral Domain Convolutional Neural Network