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

Marcel Crasmaru

cs.LGOct 6, 2026

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

Authors: Marcel Crasmaru

Abstract

CNet is a C++/CUDA framework for building deep complex-valued neural networks (CVNNs) and optimizing complex functions by gradient descent with Wirtinger derivatives. Complex models are underexplored yet natural where data is intrinsically complex -- RF/IQ communications, 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. Its library of complex layers makes conv(x,k) = IFFT(FFT(x).FFT(k)) learnable via signal-processing primitives -- spectral-padding local kernels (Pad), the inverse DFT, a magnitude nonlinearity |z|^2 (CModulus2), holomorphic powers z^M (CPower), and mean pooling (MeanPool) -- with GPU kernels for every layer, a GPU true-Adam optimizer, and a reduced-memory inference mode. Three studies. (1) A fully complex FNet-style causal language model, built on a new O(N log N) causal Fourier mixer, matches a param-matched real-valued causal FNet on tiny-shakespeare in under half the steps. We introduce Born-rule attention: a content-based causal mixer whose weights are quantum-measurement probabilities |<Q_k,K_j>|^2 of one token against another -- the first attention mechanism built on the Born rule. With rotary position embeddings and per-token complex normalization, a fully complex Born-rule model reaches 1.52 nats/char on tiny-shakespeare, matching softmax attention and beating the Fourier mixer by 0.23. (2,3) Two bottleneck analyses -- radio-modulation classification (RML2016.10a) and the Fourier phase problem of coherent-diffraction imaging -- isolate where CVNNs need new operators. The complex machinery learns the correct structure; open problems are concrete and operator-level. Code: https://github.com/crasmarum/CNet

Figures & tables

Panel of Figure 1 — Figure 1: Causal token mixing, two ways, shown on the triangular ( $j\leq k$ ) support. Left: masked self-attention—each lower-triangular weight $A_{k,j}$ is learned and input-dependent (content-based routing), at $O(N^{2})$ cost. Right: the causal Fourier mixer ( 6 )—the same triangular sum, but the weights are the fixed roots of unity $\omega^{jk}$ (drawn as unit phasors whose rotation rate grows with $jk$ , fastest at the bottom-right), so the layer is parameter-free. The triangular mask breaks the FFT butterfly, but the Bluestein / chirp- $z$ factorization (bottom) restores an $O(N\log N)$ evaluation—a chirp pre-multiply, a causal linear convolution done with three ordinary FFTs, and a chirp post-multiply—for both the forward pass and its Wirtinger backward.

Model	Training choice	Val (nats/char)
6-block, $1.62$ M	full (warmup $+$ cosine, CGelu )	$\mathbf{1.75}$
	constant low lr (no schedule)	${\approx}\,2.4$ (stalls)
	unit-norm head, no CGelu †	stalls
10-block, ${\sim}10$ M	with block_norm	$2.57$
	without block_norm	diverges ( $\to\!16$ )

Table 1: Effect of the main training choices (character-level tiny-shakespeare , per-position nats/char). Top: the

6

-block /

1.62

M model—the learning-rate schedule is decisive: a constant, overly small rate stalls near

2.4

, while warmup

+

cosine reaches

1.75

. Bottom: the deep

10

-block /

{\sim}10

M model, where per-block spectral normalization ( block_norm ) is required for convergence at all; these deep-net runs predate the tuned schedule, so their absolute losses are not directly comparable to the top block. † The CGelu

\to

global unit-norm change was observed to stall but was not separately tuned to a final loss.

Panel of Figure 2 — Figure 2: Born-rule attention mirrors scaled dot-product attention, with the softmax replaced by a measurement. The two operators share the same block structure (dashed links = identical stages); only two boxes change (teal). The dot-product score becomes the squared magnitude of a complex inner product, $s_{kj}=|\langle Q_{k},K_{j}\rangle|^{2}$ —a Born-rule measurement of how aligned two complex token states are, real and non-negative by construction —and the exponential SoftMax is replaced by a plain causal sum-normalization $A_{kj}=s_{kj}/\!\sum_{j^{\prime}\leq k}s_{kj^{\prime}}$ . Non-negativity and normalization thus come from the measurement rule itself, not from an $\exp$ ; the queries/keys are complex phasors (insets) and the output $O_{k}=\sum_{j\leq k}A_{kj}V_{j}$ is the complex combination of values. It is content-based routing like self-attention ( $O(N^{2})$ ), but physics-native rather than a softmaxed dot product.

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