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CNet introduces a C++/CUDA framework for complex-valued neural networks (CVNNs) leveraging Wirtinger calculus for gradient descent and a physics-native approach where networks are cascades of complex, often unitary, operations. This framework enables classification via Born-rule measurements and incorporates learnable complex convolutions via FFT-Hadamard products, achieving competitive or superior performance on language modeling tasks with reduced training time compared to real-valued counterparts. Studies on radio-modulation classification and coherent-diffraction imaging highlight areas where new complex-valued operators are needed, while demonstrating that the complex formulation inherently learns physically correct structures.
Complex-valued neural networks, powered by Wirtinger autodifferentiation and FFT-Hadamard convolutions, match or exceed real-valued FNet performance on language modeling with half the training steps.
CNet is a C++/CUDA framework for building and training deep complex-valued neural networks (CVNNs) and, more generally, for optimizing complex-valued functions by gradient descent with Wirtinger (CR-calculus) derivatives. It takes a physics-native stance: a network is a cascade of complex -- and often unitary (the DFT) -- operations acting on an amplitude vector, and classification is a Born-rule measurement $p_k = |z_k|^2 / \|z\|^2$ rather than a softmax over real logits. Every layer ships a CPU reference and a CUDA kernel checked against finite differences, and the computation graph is cloned across the batch for GPU execution. On top of the base layers we add signal-processing primitives that turn the identity conv(x,k) = IFFT(FFT(x) . FFT(k)) into a learnable complex convolutional network, together with a true-Adam optimizer and a reduced-memory inference mode. We report three studies. First, a fully complex-valued, FNet-style causal sequence model built on a new $O(N \log N)$ causal Fourier mixer -- a triangular-masked DFT evaluated by a Bluestein / chirp-z factorization: once properly tuned it matches or exceeds a parameter-matched real-valued causal FNet on character-level language modeling, reaching the real model's converged quality in under half the training steps. Second and third, bottleneck analyses on radio-modulation classification (RML2016.10a) and the Fourier phase problem of coherent-diffraction imaging, which isolate exactly where complex-valued networks still need new operators. Across all three the complex formulation provably learns the physically correct structure. Code: https://github.com/crasmarum/CNet