Complex-Valued Neural Network

CVNNs process numbers in the form a + bi as atomic entities, preserving the magnitude and phase relationship without decoupling them into separate real-valued channels. This is critical for wireless signals where phase encodes directional information for beamforming and precoding. Real-valued networks force an artificial separation that can destroy the geometric structure of the signal, requiring the model to waste capacity relearning the relationship between I and Q branches.

CVNNs rely on Wirtinger calculus to perform backpropagation directly in the complex domain. Unlike real-valued calculus, a function of a complex variable is holomorphic only if it satisfies the Cauchy-Riemann equations. Wirtinger derivatives treat the complex variable and its conjugate as independent, enabling gradient-based optimization of non-holomorphic activation functions like modReLU and cardioid functions that operate on magnitude and phase separately.

A single complex-valued neuron with orthogonal decision boundaries can solve problems like the XOR classification that require a multi-layer real-valued network. The complex multiplication operation inherently performs rotation and scaling simultaneously:

• Rotation: Encoded by the phase of the weight
• Scaling: Encoded by the magnitude of the weight This gives a single complex neuron greater expressive power, often leading to shallower architectures for equivalent task performance.

The covariance structure of complex-valued networks imposes stronger regularization than real-valued dropout. CVNNs exhibit a natural tendency to learn coherent signal structures rather than overfitting to noise. In channel estimation tasks, this translates to robust performance at low Signal-to-Noise Ratio (SNR) regimes where real-valued networks often hallucinate non-physical channel components that violate the underlying electromagnetic propagation physics.

Standard real-valued activations like ReLU cannot be directly applied in the complex domain without violating differentiability. CVNNs use specialized activations that act on the magnitude while preserving the phase:

• modReLU: Applies ReLU to the magnitude, leaving phase unchanged
• zReLU: Passes only elements with phase in [0, π/2]
• Complex Cardioid: A holomorphic function with a heart-shaped output manifold These preserve the angular information critical for spatial processing in massive MIMO.

CVNNs eliminate the preprocessing bottleneck of converting complex IQ samples into real-valued spectrograms or feature vectors. The network ingests raw in-phase and quadrature samples directly, learning optimal representations end-to-end. This is particularly advantageous for CSI compression architectures like CsiNet, where the input is a complex-valued channel matrix. Real-valued alternatives must double the input dimension, squaring the computational cost of the first layer.