Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.08592

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

Marcel Crasmaru
View original source ↗Read source document ↗Export metadata ↓

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 | |^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

Explore related subjects

cs.LGcs.MSmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marcel Crasmaru. 2026-10-07. CNet: A Complex-Valued Deep Learning Framework with Wirtinger Autodifferentiation and FFT--Hadamard Convolution. https://arxiv.org/abs/2610.08592

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Policy Learning with a Language Bottleneck

Modern AI systems such as self-driving cars and game-playing agents can achieve superhuman performance, but often lack human-like generalization, interpretability, and inter-operability with human users. Inspired by the rich interactions between language and decision-making in humans, we introduce Policy Learning with a Language Bottleneck (PLLB), a framework enabling AI agents to generate linguistic rules that capture the high-level strategies underlying rewarding behaviors. PLLB alternates between a *rule generation* step guided by language models, and an *update* step where agents learn new policies guided by rules, even when a rule is insufficient to describe an entire complex policy. Across five diverse tasks, including a two-player signaling game, maze navigation, image reconstruction, and robot grasp planning, we show that PLLB agents are not only able to learn more interpretable and generalizable behaviors, but can also share the learned rules with human users, enabling more effective human-AI coordination. We provide source code for our experiments at https://github.com/meghabyte/bottleneck .

cs.LG↗
arXiv2026

BEAT: Balanced Frequency Adaptive Tuning for Long-Term Time-Series Forecasting

Long-term time-series forecasting supports a wide range of applications, including weather prediction and electricity demand planning. Frequency-domain methods address this task by decomposing observations into components that describe temporal variations at different scales. However, separate representations do not by themselves provide an explicit mechanism for adjusting the training emphasis across components. Under a shared forecasting objective, the frequency-specific networks can retain different levels of coefficient prediction error, motivating an error-dependent adjustment to their gradients. To this end, we propose BEAT (Balanced frEquency Adaptive Tuning), a framework that combines frequency-specific error monitoring with adaptive gradient modulation. We design a Frequency-Specific Monitor that compares predicted and target wavelet coefficients in a common normalized space and expresses each discrepancy relative to a reference error computed from the detail components. We further introduce a Dynamical Gradient Balancer that converts these ratios into positive, bounded coefficients. Components with higher relative errors receive larger gradient weights, whereas those with lower relative errors receive smaller weights. A shared modulation-strength parameter controls the departure from unmodulated training, and the monitoring and balancing operations are used only during training. Experiments on seven real-world datasets show that BEAT achieves competitive performance against state-of-the-art forecasting methods.

cs.LG↗
arXiv2026

C-LoRA: Continual Low-Rank Adaptation for Pre-trained Visual Models

Pre-trained visual models have become fundamental in computer vision, but they face challenges in continual learning scenarios where data and tasks evolve over time. Low-Rank Adaptation (LoRA) offers efficient fine-tuning capabilities but remains limited for such dynamic environments. Standard LoRA cannot distinguish important subspaces, causing critical knowledge to be overwritten in sequential training. Existing approaches address this by dynamically expanding the set of LoRA adapters, either maintaining a growing pool of task-specific modules or merging new adapters into prior ones, at the cost of unbounded parameter growth or increasing inference complexity. We propose Continual Low-Rank Adaptation (C-LoRA), a method that enables a single, shared LoRA adapter to handle sequential tasks without catastrophic forgetting, without requiring any module selection or fusion at inference. The core of C-LoRA is a learnable routing matrix R that explicitly controls how each rank-one subspace contributes to the weight update. This matrix is decomposed into a stability component (R_base), which preserves knowledge from prior tasks, and a plasticity component (R_delta), which drives adaptation to the current task, providing direct control over the stability-plasticity trade-off. We analyze how R governs gradient flow during sequential training, and demonstrate competitive performance across multiple benchmarks.

cs.LG↗