# CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution

arXiv: https://arxiv.org/abs/2610.08592 · HTML: https://mlsanity.com/paper/2610.08592/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 6 references that are in the map.
- First posted on arXiv 6 Oct 2026.

## Builds on (most cited first, 4 of 4)

- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [Global Filter Networks for Image Classification](https://mlsanity.com/paper/2107.00645/) (2021)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [On the Equivalence of Convolutional and Hadamard Networks using DFT](https://mlsanity.com/paper/1810.11650/) (2018)

## Abstract (arXiv)

CNet is a C++/CUDA framework for deep complex-valued neural networks (CVNNs), trained by gradient descent with explicit Wirtinger (CR-calculus) derivatives. Complex models are underexplored yet natural where data is intrinsically complex -- RF/IQ, MRI k-space, radar/SAR, audio spectra -- and phase carries information real networks discard. CNet is physics-native: a network is a cascade of complex (often unitary) operations 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 finite-difference-verified CUDA kernel; the static, batch-cloned computation graph captures cleanly into CUDA Graphs, outperforming eager and compiled torch.complex64 on an H100. We present three studies. (1) A fully complex causal language model, built on a new causal (triangular-masked) Fourier mixer with an O(N log N) Bluestein evaluation of the forward pass and its Wirtinger adjoint, matches a parameter-matched real causal FNet on character-level tiny-shakespeare in under half the steps. We introduce Born-rule attention, a content-based causal mixer whose weights are the quantum-measurement probabilities |<Q_k, K_j>|^2 -- to our knowledge the first classical, complex-valued attention to replace softmax with the squared-magnitude Born score; with rotary embeddings and per-token complex normalization it reaches 1.52 nats/char, matching softmax self-attention, and transfers unchanged to byte-level (1.31 nats/char) and sub-word WikiText-103. (2) On radio-modulation classification (RML2016.10a/RML22) we trace the pooled-power accuracy ceiling to discarded temporal structure and recover it with a learnable cyclostationary conjugate-lag operator. (3) A bottleneck analysis of the Fourier phase problem in coherent-diffraction imaging isolates where complex-valued networks still need new operators.