{
  "abstract": "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 ||^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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08592",
  "author_total": 1,
  "authors": [
    {
      "name": "Marcel Crasmaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3q04p29n3n5md.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08592",
  "fulltext_url": null,
  "id": "3q04p29n3n5md",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08592",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.MS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7220909725",
  "pdf_url": "https://arxiv.org/pdf/2610.08592",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3q04p29n3n5md.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.08592",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08592"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution",
  "title_text": "CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3q04p29n3n5md",
  "venue": "arXiv",
  "versions": []
}