{"slug": "cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation", "title": "CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution", "summary": "CNet, a C++/CUDA framework for building and training complex-valued neural networks, was released on arXiv as paper 2610.08592v1 with code at github.com/crasmarum/CNet. The framework implements Wirtinger (CR-calculus) autodifferentiation, a Born-rule measurement p_k = |z_k|^2 / ||z||^2 in place of softmax, and FFT-Hadamard learnable complex convolutions, with every layer shipping a CPU reference and a CUDA kernel checked against finite differences. In a character-level language modeling study, 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 via Bluestein/chirp-z factorization) matched or exceeded a parameter-matched real-valued causal FNet and reached the real model's converged quality in under half the training steps.", "body_md": "arXiv:2610.08592v1 Announce Type: new \nAbstract: 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.\n  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.\n  Code: https://github.com/crasmarum/CNet", "url": "https://wpnews.pro/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation", "canonical_source": "https://www.machinebrief.com/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-z920", "published_at": "2026-10-07 04:00:00+00:00", "updated_at": "2026-10-07 05:48:14.479071+00:00", "lang": "en", "topics": ["machine-learning", "neural-networks", "ai-research", "developer-tools"], "entities": ["CNet", "arXiv", "CUDA", "Bluestein factorization", "chirp-z factorization", "RML2016.10a", "FNet", "github.com/crasmarum/CNet"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation", "markdown": "https://wpnews.pro/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation.md", "text": "https://wpnews.pro/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation.txt", "jsonld": "https://wpnews.pro/news/cnet-a-complex-valued-deep-learning-framework-with-wirtinger-autodifferentiation.jsonld"}}