Complex-Valued Neural Networks

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3 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

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Latest papers 33

Oct 6, 2026cs.LG

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

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

Marcel Crasmaru

Complex-Valued Neural Networks Autoregressive Language Modeling Convolutional Neural Networks

Sep 29, 2026cs.LG

Reinforcement Learning with Complex (valued) Memories

Partially observable environments pose a fundamental challenge in deep reinforcement learning, requiring agents to compress temporal information from observations and maintain a memory to make effective decisions. While there exist many approaches ranging from gated recurrence to attention mechanisms and model-based RL, the search for effective representational techniques that can capture long-term dependencies remains an active area of research. In this work we revisit Unitary recurrent networks (uRNNs) [Arjovsky et al., 2016, Jing et al., 2017], that demonstrated superior gradient flow and associative recall, expressing the recurrence and the hidden state in a complex vector space. Their norm preserving unitary dynamics enable information propagation through long sequences. To this end, we propose three different versions of uRNNs as drop-in replacements for recurrent PPO architectures, and demonstrate that the simple recurrence and the added degree of freedom from the phase of the complex representations enable significant gains over baselines on several memory-improvable tasks, including continuous control. We further explore how to preserve the phase information of the complex hidden state for a phase-aware policy by drawing a parallel to how quantum states are measured. With our methods reaching up to 2-3

\times

the reward in environments like rocksample and Craftax compared to the baselines, this work points towards an exciting new direction of representations for RL and the problem of partial observability. Code is available at: https://github.com/Sathya98/qurl

Sathya Kamesh Bhethanabhotla, Efstratios Gavves, André Biedenkapp

University of Freiburg

Partially Observable RL Recurrent Neural Networks Complex-Valued Neural Networks Representation Learning for RL

Sep 22, 2026cs.LG

AURA: Angular Update Rate Adaptation for training complex-valued neural networks

Complex-valued neural networks (CVNNs) are increasingly adopted for complex-valued data; however, they are often trained with first-order optimizers inherited from the real-valued case. The efficiency of these methods depends largely on the step size, and their step-size rules ignore the angular information available in the complex plane. We address step-size adaptation in the complex domain by introducing AURA (Angular Update Rate Adaptation), a per-parameter step-size adaptation that can be added on top of any first-order optimizer, and removed from it, without altering its update direction. AURA measures the agreement between consecutive updates of each complex parameter, in length, alignment, and sense of rotation, and enlarges the step when they are consistent and reduces it when they are not. It requires no additional gradient evaluations and only inexpensive vector operations per step. We combine AURA with Adam and Muon and compare the resulting methods with well-known first-order optimizers on four test cases of increasing complexity, ranging from the approximation of scalar complex functions to physics-informed training. Fully connected neural networks are used throughout this work. All hyperparameters other than the step size are held fixed across test cases; for one case, we also tune the hyperparameters of each optimizer under the same budget. Our empirical tests show that AURA improves the convergence of its base optimizer in most cases with a small per-step overhead, and we identify the conditions under which it fails to do so.

Enrico Ballini, Allan Peter Engsig-Karup, Tito Andriollo

Aarhus University, Aarhus, Denmark · Technical University of Denmark, Kongens Lyngby, Denmark

Neural Network Optimization Complex-Valued Neural Networks Adaptive Gradient Methods

Sep 14, 2026cs.LG

CRFCAN: A Complex-Valued Cross-Domain Residual Network for Joint Channel and Phase Noise Estimation in Sub-THz OFDM Systems

In sub-terahertz (sub-THz) communications, the coupling of ultra-wide bandwidth and severe phase noise (PN) impairments renders conventional joint channel and PN estimation highly complex and computationally prohibitive. To address this, we propose CRFCAN, a complex-valued residual FFT convolutional attention network designed for joint channel and PN estimation. Unlike existing deep learning schemes that rely on cascaded networks or hybrid frameworks combining neural networks with conventional iterative estimators, CRFCAN performs joint recovery in a truly end-to-end fashion through a physics-inspired cross-domain structure. Specifically, Fast Fourier Transform (FFT) and inverse FFT modules are embedded within residual groups to enable iterative feature interaction across the time and frequency domains, thereby capturing both frequency-selective fading and time-varying phase distortions. In addition, two dedicated residual blocks are introduced for complex feature extraction and multiplicative phase-distortion modeling, respectively. A physics-aware PN output tail with soft normalization is further employed to improve estimation stability while preserving the physical characteristics of the effective PN process. Simulation results demonstrate that CRFCAN significantly outperforms conventional algorithms and state-of-the-art deep learning models in terms of normalized mean square error (NMSE) and bit error rate (BER). Notably, CRFCAN achieves superior performance with single-shot, fixed-complexity inference and generalizes well to unseen PN models without fine-tuning, highlighting its robustness and practicality for sub-THz receivers.

Ruilin Wang, Xiaodai Dong

Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8P 5C2, Canada

Complex-Valued Neural Networks Wireless Communications Channel Estimation

Aug 30, 2026cs.LG

ECA-BLS: An Efficient Complex-Augmented Broad Learning System

Broad Learning System (BLS) is an efficient alternative to deep architectures due to its fast training, analytical learning, and strong generalization under limited data. However, existing BLS variants are confined to real-valued representations, restricting their ability to capture nonlinear interactions and second-order statistical dependencies inherent in real-world data. Notably, no prior BLS model fully exploits the complete second-order statistics that naturally emerge when data are embedded in the complex domain. To address this limitation, this paper introduces the first complex augmented Broad Learning System (CA-BLS), which transforms real-valued inputs into phase-encoded complex representations and adopts widely linear modeling to jointly leverage covariance and pseudo-covariance information via complex conjugate augmentation. This enables effective modeling of latent nonlinearities, coherence structures, and second-order dependencies inaccessible to conventional BLS formulations. To mitigate the additional computational cost of complex augmentation, an Efficient Complex Augmented BLS (ECA-BLS) is further developed, reformulating CA-BLS entirely in the real domain while preserving its exact decision function, achieving up to 75% fewer multiplications and over 60% fewer additions. A rigorous theoretical analysis proves the mathematical equivalence between CA-BLS and ECA-BLS, ensuring zero theoretical loss. Extensive experiments on 26 benchmark datasets from the UCI and KEEL repositories demonstrate that ECA-BLS consistently outperforms classical BLS and recent state-of-the-art randomized neural networks in accuracy, average rank, and statistical significance, establishing augmented second-order modeling as a critical and previously missing dimension of BLS research.

A. Rahaman, A. Quadir, M. Sajid +2

Department of Mathematics Indian Institute of Technology Indore Indore, India

Complex-Valued Neural Networks

Aug 20, 2026cs.LG

Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. Under a Calabi-Yau metric, specifically in a non-compact setting with a global potential so defined geometrically rather than invoking the topological requirements of the Calabi conjecture, a wedged nowhere-vanishing holomorphic form is the top exterior product of the Kähler form up to constants, yielding a constant determinant condition with respect to a background metric and ill-conditioned eigenvalues under nonuniform and almost low-rank assumptions. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist in a geometric analytic modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Andrew Gracyk

Department of Mathematics Purdue University West Lafayette, IN 47907, USA

Riemannian Optimization Gradient Descent Neural Network Optimization Complex-Valued Neural Networks Loss Landscape Geometry

Aug 7, 2026cs.CV

Ghost Features and Spooky Transfer Learning for Hypercomplex-Valued Neural Networks

Hypercomplex numbers extend the concept of complex numbers by introducing additional imaginary components. Besides increasing dimensionality, operations on the imaginary parts provide algebraic and geometrical properties that can be beneficial for solving machine learning problems. In this paper, we show how to create hypercomplex-valued neural network layers where the real part corresponds to the output of a traditional real-valued layer. The additional imaginary parts of these hypercomplex-valued layers produce what we call

ghost features,'' which contain enhanced information that is not present in the output of the real-valued layer. Moreover, ghost features can be effectively integrated into a trained neural network through a process we refer to as

spooky transfer learning.'' This approach allows us to harness the richness of ghost features, leading to more efficient neural networks. The source code and Jupyter Notebook are available at https://github.com/mevalle/v-nets/.

Guilherme Vieira Neto, Marcos Eduardo Valle

Universidade Estadual de Campinas (UNICAMP), Brazil

Complex-Valued Neural Networks Transfer Learning

Jul 24, 2026cs.LG

IQ-JEPA: A Joint-Embedding Predictive Architecture with a Hermitian Vision Transformer for Sound Speed and Attenuation Estimation from Ultrasound IQ Data

The speed of sound in tissue is a prerequisite for well-focused imaging and has diagnostic value, but recovering it from raw pulse-echo channel data is fundamentally a nonlinear inverse problem. Learned solvers are fast yet label hungry. Simulated sound-speed labels are expensive, while abundant real channel data is unlabeled. We propose IQ-JEPA to exploit both data types. An encoder is pretrained without labels to predict the latent representation of masked in-phase and quadrature (IQ) regions from visible context, then fine-tuned on simulated maps. Sound speed appears in the IQ signal as a phase difference, invariant to the constant phase offset. The encoder is a Hermitian vision transformer that operates on the complex signal directly. Its attention is equivariant to that phase and its conjugate-product feed-forward is invariant to it, so the encoder reads a quantity analogous to the one classical coherence methods use. On 79,293 Fullwave 2.5 simulations at 2.5 MHz, pretraining on the 63,435 unlabeled acquisitions reaches 15.60 m/s at 10,000 labels. This is a roughly threefold gain in label efficiency over supervised training, growing to over fourfold at 1,000 labels. It is about 2.2x below an InversionNet baseline, and 8.71 m/s at full labels. The gain still grows with more unlabeled pretraining data. Our comparisons point to self-supervision as the dominant factor. The same encoder transfers. Its frozen features expose sound speed and attenuation, and cross-distribution pretraining between layered and abdominal phantoms costs little accuracy. We see this as a first step toward a foundation model for quantitative ultrasound.

Masashi Sode, Gianmarco Pinton

Joint Department of Biomedical Engineering, University of North Carolina at Chapel Hill and North Carolina State University, Chapel Hill, NC 27599 USA

Self-Supervised Pre-Training Joint-Embedding Predictive Architecture Complex-Valued Neural Networks Ultrasound Imaging Image Inverse Problems

Jul 22, 2026cs.CV

Physics-Aware Complex-Valued State Space Model with Scattering-Prior Feature Modulation for PolSAR Image Classification

Polarimetric synthetic aperture radar (PolSAR) image classification is a representative task for physics-aware GeoAI, where land-cover semantics are closely coupled with electromagnetic scattering mechanisms. Many existing complex-valued networks can preserve amplitude-phase information, but they are often limited in long-range spatial dependency modeling and usually incorporate polarimetric priors only as input-level or shallow auxiliary features. As a result, physical knowledge is insufficiently used to guide deep feature evolution. To address this issue, this paper proposes CV-SSMNet, a physics-aware complex-valued state-space network with scattering-aware feature modulation for PolSAR image classification. The proposed method builds a complex-valued state-space model (CV-SSM) in the original complex domain to capture long-range spatial dependencies while preserving polarimetric amplitude-phase coupling. Meanwhile, seven physically meaningful scattering priors, are encoded as FiLM-style modulation signals to adaptively recalibrate complex-valued representations during feature evolution. CV-SSMNet further integrates multi-scale complex convolutions, branch-wise CV-SSM encoding, prior-guided recalibration, and lightweight global context aggregation, enabling physically guided representation learning from local scattering structures to global spatial context. Experiments on three L-band benchmark datasets and an additional P-band BIOMASS evaluation demonstrate that CV-SSMNet achieves competitive accuracy, improved regional consistency, and better boundary preservation, supporting the effectiveness of embedding polarimetric scattering mechanisms into complex-valued long-range GeoAI representation learning.

Fangyan Zhang, Fan Zhang, Shiqi Zhou +3

The College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, China · The School of Information Science and Engineering, Yunnan University, Kunming, China · The Department of Signal Theory and Communications, Polytechnic University of Catalonia, Barcelona, Spain

Complex-Valued Neural Networks Visual SSMs Physics-Informed ML Polarimetric Imaging Image Classification

Jul 14, 2026cs.LG

PolarBM: Complex-valued Boltzmann Machine for Modeling Audio Signals in Polar and Log-polar Coordinates

Although vast amounts of data, such as audio signal spectra, are naturally represented using complex numbers, conventional machine learning methods often simplify complex-domain problems by employing frameworks designed for real-valued variables. While this simplification offers computational benefits, it discards structural information regarding the inherent relationship between amplitude and phase. In this paper, we propose a novel Boltzmann machine (BM), named PolarBM, capable of naturally handling complex-valued variables in the polar coordinate (i.e., an amplitude-phase representation). PolarBM defines a probability density function for complex variables in which the phase explicitly depends on the amplitude, thereby capturing the physically important relationships of complex-valued signals. Furthermore, to process audio signals in accordance with human auditory perception, we propose LogPolarBM, which models amplitude on a logarithmic scale. This extension yields a flexible conditional probability density function, a power-weighted noncentral complex Gaussian (PW-NCCG) distribution, whose marginal amplitude distribution encompasses the Rice, Nakagami, and noncentral chi distributions as special cases. For practical applications, we also introduce the restricted variants of these proposed models: PolarRBM and LogPolarRBM. Experimental results demonstrate that by explicitly modeling the dependency between amplitude and phase, the proposed RBMs achieve superior modeling accuracy compared to conventional models, including deep neural networks. Although our experiments focus on audio signals, the utility of the proposed BMs is not limited to audio applications; their potential extends widely across various fields of science and engineering that involve complex-valued data, such as wireless communications and quantum mechanics.

Toru Nakashika, Kohei Yatabe

Graduate School of Informatics and Engineering, the University of Electro-Communications, Tokyo 182-8585, Japan · Department of Electrical Engineering and Computer Science, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan

Unsupervised Representation Learning Complex-Valued Neural Networks Restricted Boltzmann Machines

Jul 5, 2026physics.flu-dyn

Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics

Repeated prediction of acoustic fields from spatially distributed boundary excitation is computationally expensive when each source realization requires a new wave simulation. This work introduces a quadrature-aware complex-linear boundary operator (CLBO) that maps complex normal velocity on a vibrating surface to complex pressure at receiver locations. The model couples learned source and receiver basis functions through an explicit complex surface-quadrature contraction, so the boundary excitation enters linearly by construction. This preserves complex superposition, homogeneity, and zero response to zero excitation, while representing the source through coordinates, normals, and quadrature weights rather than a fixed flattened input vector. Reference data were generated using a verified three-dimensional multiple-relaxation-time (MRT) lattice Boltzmann solver and stored in a solver-agnostic boundary-to-field format. CLBO was compared with a fixed-sensor complex DeepONet under matched case splits and optimization settings, with additional tests of structural consistency, receiver-coordinate interpolation, source discretization, source-family holdout, label efficiency, physics-informed ablations, unseen source mixtures, and computational cost. Across five training seeds, CLBO achieved a mean complex relative field error of 0.184 +/- 0.00771, compared with 0.367 +/- 0.00742 for DeepONet. Its measured source-superposition error was 1.31 x 10^-7, and its mean error on newly simulated mixed-source cases was 0.237, compared with 0.415 for DeepONet. Inference was 1.83 x 10^4 faster than the reference calculation for the reported query size. These results show that enforcing the known complex-linear boundary-to-field structure improves physical consistency and generalization under distributed acoustic excitation.

Muhammad Idrees Khan, Hua-Dong Yao

Department of Mechanical Engineering, Chalmers University of Technology Gothenburg 41296, Sweden

Complex-Valued Neural Networks Operator Learning Physics-Based Simulation Neural Operators

Jun 30, 2026cs.LG

Dualformer: Efficient Feature Extractor for Complex-valued Blind Communication Signal Analysis

Designing effective feature extractors is critical for blind signal analysis tasks such as automatic modulation recognition (AMR), signal scheme recognition (SSR), and \color{black} signal structure parsing (SSP). In this work, we propose dual-channel neural network (DualNN) that efficiently exploits complex-valued signals through parameter sharing across IQ channels. Unlike traditional real-valued or complex-valued models, DualNN is a groundbreaking framework which shares the network parameters for processing the real and imaginary parts of the complex-valued signals, and is theoretically shown to reduce generalization error while preserving expressive capacity. Specifically, we propose a novel Transformer-based architecture to implement DualNN, called Dualformer. The Dualformer segments input signals into patch-level tokens and captures multi-granularity features, enabling robust performance across diverse signal analysis tasks. Furthermore, we conduct extensive experiments comparing Dualformer with three Transformer-based baselines and four conventional DL-based approaches. Results demonstrate consistent performance improvements on AMR, SSR, and SSP tasks. Besides, the modular design of DualNN allows it to generalize well to blind signal processing tasks such as blind source separation and low-SNR spectrum sensing. This work paves the way for a broader application of DualNN architectures in unsupervised and weakly supervised complex-valued signal analysis scenarios.

Yurui Zhao, Xiang Wang, Jingreng Lei +3

College of Electronic Science and Technology, National University of Defense Technology, Changsha 410073, China. · Department of Electrical and Computer Engineering, The University of Hong Kong

Transformer Automatic Modulation Classification Complex-Valued Neural Networks

Jun 27, 2026cs.NE

Unified Complex-valued Neural Network: A Magnitude-Phase Computational Model for Event-Driven Neuromorphic Learning

Artificial neural networks (ANN) provide accurate continuous-valued representation, whereas spiking neural networks (SNN) offer event-driven temporal processing, yet both paradigms face limitations when value encoding and timing dynamics must be learned within a single computational structure. This paper introduces a network based on Unified Complex-valued Neuron (UCN), a new neural computational model that integrates continuous activation and phase-driven event generation through an asymmetric complex-valued state. In the UCN, magnitude encodes signal strength while phase governs intrinsic temporal evolution and valued spike emission. A foundational training framework combining backpropagation (BP) and backpropagation through time (BPTT) is first developed to optimize magnitude and phase pathways in a unified way. To reduce computational complexity, an event-driven adaptive phase learning (EAPL) rule is then introduced as a more efficient alternative. The proposed model is evaluated through object tracking and Lorenz attractor learning. Results demonstrate that UCN-based Network (UCNN) provides accurate, stable, and interpretable spatiotemporal learning while preserving sparse event-driven computation for neuromorphic and edge-AI applications.

Reza Ahmadvand, Sarah Safura Sharif, Yaser Mike Banad

Department of Electrical and Computer Engineering, University of Oklahoma, Oklahoma, United States

Complex-Valued Neural Networks Spiking Neural Networks Neuromorphic Computing

Jun 17, 2026cs.SD

QC-GAN: A Parameter-Efficient Quaternion Conformer GAN for High-Fidelity Speech Enhancement

We propose a parameter-efficient speech enhancement framework, Quaternion Conformer GAN (QC-GAN), which combines a Quaternion Conformer generator with MetricGAN-based training. The Hamilton product encodes the magnitude and phase via structured weight sharing, reducing the number of layer parameters while preserving their interdependencies. A metric-learning discriminator was employed to maximize perceptual quality by optimizing the approximate perceptual evaluation scores. On the VoiceBank+DEMAND dataset, QC-GAN achieved a Perceptual Evaluation of Speech Quality (PESQ) score of 3.48 with only 0.89M parameters, delivering a performance comparable to state-of-the-art models at less than half their size. A 35K-parameter variant achieved a PESQ score of 3.23, surpassing conventional methods with significantly fewer parameters. Evaluation on the DNS-Challenge 3 dataset further confirmed generalization to real-world conditions.

Shogo Yamauchi, Hideaki Tamori, Makoto Sakai +2

The Asahi Shimbun Company, Japan · Tokyo Woman’s Christian University, Japan

Transformer Complex-Valued Neural Networks Speech Enhancement Generative Adversarial Networks

Jun 16, 2026cs.LG

Why SWAVE May Not Be All You Need:A Concept-Evolution Retrospective on Complex-Valued Recurrent Language Models

SWave is a complex-valued recurrent language model (169.26M parameters, D=384, L=16, T=2048) trained on FineWeb-Edu using 2xH100 NVL. It was designed around three founding premises: that representing language as complex waves rather than real-valued numbers enables richer information encoding; that a Cayley-parameterised unitary transition provides a mathematical guarantee against state decay or explosion; and that a hidden state which rotates rather than shrinks preserves signal integrity over arbitrarily long contexts. The core of SWave evolved substantially across three development phases. The Resonance Head was found to structurally admit imaginary-channel collapse as a global loss minimum (a failure mode we term cos-domination collapse) and was superseded by an untied head with independent real and imaginary embedding tables from the Phase-Associative Memory (PAM) architecture. This resolved the degenerate minimum and enabled stable 200,000-step training (best-step PPL 22.0 at step 89,861). ComplexNorm and the Wave Propagation Scan proved load-bearing throughout all three phases and were retained to the final architecture. ProtectGatedScan was reframed as a structural prior rather than a learned behaviour. The four multi-scale retention concepts showed no measurable improvement under controlled evaluation and were found non-load-bearing. The ComplexGatedUnit was superseded by a real-valued squared-ReLU channel mixer with fewer parameters. The auxiliary training objectives showed no benefit once structural constraints were resolved. The investigation yields a formal characterisation of cos-domination collapse, a parallel scan with a log-space backward pass for numerical stability, six transferable engineering principles for complex-valued recurrent training, and a plan-to-code traceability methodology for catching structural divergences that conventional test suites miss.

Ramprasath Ganesaraja, Swathika N, Sahil Dilip Panse

EdgeVerve Systems Limited

Recurrent Neural Networks Complex-Valued Neural Networks Autoregressive Language Modeling

Jun 15, 2026cs.LG

On the Entropy Formula for Real, Complex, and Quaternionic Deep Linear Networks

We extend the entropy formula of Menon and Yu for the real Deep Linear Network (DLN) to its complex and quaternionic analogues, obtaining a unified formula for DLNs over

\mathbb{R}

,

\mathbb{C}

, and

\mathbb{H}

.

Luis Contreras, Marco Nahas, Tejas Kotwal

Complex-Valued Neural Networks Deep Linear Networks

Jun 11, 2026cond-mat.dis-nn

Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states

Complex neural quantum states are difficult to optimize when their wavefunction phase carries gauge, chiral, fermionic, or topological structure. We show that the major failure mode is not only ansatz expressivity, but the Monte Carlo estimator used to learn this phase. For separated amplitude-phase states, differentiating the local energy at fixed samples gives a different unbiased estimator of the same variational Monte Carlo phase force, without changing the objective. We further extend the construction to coupled two-head networks by keeping the amplitude-gradient contribution and applying the direct derivative only to the phase path. An adaptive minimum-variance mixture interpolates between standard and direct estimators during training. Across flux ladders, chiral chains, two-dimensional flux cylinders, an interacting fermion ladder, shared-network controls, and a fractional quantum Hall benchmark, the resulting estimators reduce phase-gradient variance, suppress seed failures, and often move multi-percent standard-gradient plateaus to sub-percent accuracy.

Yi-Ran Xue, Rui Wang, Baigeng Wang +1

*National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China. · Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA. · Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China. +3

Neural Quantum States Complex-Valued Neural Networks Quantum Many-Body Physics Variance Reduction

Jun 5, 2026eess.SP

C-MambaPose: A Physics-Informed Complex Mamba Framework for Cross-Environment WiFi Human Pose Estimation

Human pose estimation (HPE) utilizing wireless WiFi signals has emerged as a promising technology owing to its device-free nature, privacy preservation, and robustness against occlusion and poor lighting. However, existing methods often overlook the physical complex phase information of WiFi signals and fail to generalize across diverse environments due to severe domain shifts. In this paper, we present C-MambaPose, a physics-informed complex-valued Mamba-GraFormer hybrid framework for robust cross-environment WiFi-based 3D HPE. Our framework first sanitizes raw WiFi Channel State Information (CSI) phase errors and constructs a phase-preserving complex-valued representation. We then employ a Spatiotemporal Complex Mamba encoder with a dynamic selective receptive field to capture fine-grained phase dynamics. A cross-attention joint-query mapper maps the unstructured sequence tokens to human joints, which are decoded by a Graph Convolutional Network (GCN) to predict anatomically coherent 3D coordinates. Extensive evaluations on the MM-Fi dataset show that C-MambaPose achieves competitive or superior performance to state-of-the-art baselines across all settings, setting a new state-of-the-art specifically on the challenging cross-environment split, requiring only 3.78 M parameters-an 83.1% reduction compared to GraphPose-Fi~\cite{chen2026graph} and an 85.7% reduction compared to MetaFi++~~\cite{zhou2023metafi++}, while maintaining a comparable size to DT-Pose~~\cite{chen2025towards} (which is only 18% smaller) but achieving significantly superior performance without requiring any pretraining. Our code is publicly available at https://github.com/phucngvinuni/cmampose.git.

Phuc Nguyen H

VinUniversity, Hanoi, Vietnam

Domain Generalization Complex-Valued Neural Networks 3D Human Pose Estimation Human Pose Estimation Wi-Fi Sensing

Jun 5, 2026cs.LG

Product units in gated recurrent units improve nuclear-mass prediction

The prediction of masses of atomic nuclei using machine learning can complement theoretical models and advance the exploration of poorly known domains of the nuclear chart. We propose a machine learning technique based on gated recurrent units (GRU), which have demonstrated competitive performance in nuclear-mass prediction by exploiting long-term dependencies. By integrating multiplicative interactions and product-unit transformations within recurrent units, we report significant improvements in nuclear-mass prediction. Computations are performed in the complex domain to jointly capture amplitude and phase dynamics. For interpolation and temporal-extrapolation tasks based on the atomic mass evaluation (AME2016 and AME2020), the complex additive-multiplicative product-unit gated recurrent unit (AM-PU-GRU) model consistently achieves the lowest prediction errors, with an interpolation RMSE of 0.227

\pm

0.004 MeV and an extrapolation RMSE of 0.179

\pm

0.015 MeV. These results surpass other state-of-the-art machine learning models and also outperform the real-valued GRU baseline and product-unit ablation variants, while remaining robust to different theoretical priors, including WS4 and SEMF. Our findings establish complex-valued product-unit recurrent networks as a new benchmark for sequence-based nuclear-mass prediction.

Ziyuan Li, Paulo S. A. Freitas, John W. Clark +1

Department of Mathematics, Informatics and Technology, University of Applied Sciences Koblenz, Joseph-Rovan-Allee 2, 53424 Remagen, Germany · Technical University of Munich, Munich, Germany · Department of Mathematics, University of Madeira, Campus Universitário da Penteada, Funchal, 9020-105, Portugal +1

Recurrent Neural Networks Complex-Valued Neural Networks

Jun 3, 2026cs.CV

Data Efficient Complex Feature Fusion Network For Hyperspectral Image Classification

This work presents a data-efficient variant of the Attention-Based Dual-Branch Complex Feature Fusion Network (CFFN) for hyperspectral image classification. The proposed model, termed DE-CFFN, retains the original two-stream structure: the Real-Valued Neural Network (RVNN) processes standard hyperspectral patches, while the Complex-Valued Neural Network (CVNN) handles their Fourier-transformed counterparts. The main contribution of this work lies in the feature extraction process and architectural enhancement. Factor Analysis is used for dimensionality reduction, offering improved latent feature representation over Principal Component Analysis. Additionally, both the RVNN and CVNN streams are structurally modified by successively halving the number of filters in the 3D convolutional layers to reduce complexity. The outputs of both branches are concatenated and passed through a Squeeze and Excitation (SE) block to enhance joint feature representation. Evaluated on the Pavia University and Salinas datasets, DE-CFFN achieves classification performance comparable to CFFN, while significantly reducing model size, memory consumption, and inference latency, making it suitable for real-time hyperspectral imaging applications.

Maitreya Shelare, Atharva Satam, Poonam Sonar +1

Department of Electronics and Telecommunication, Rajiv Gandhi Institute of Technology, University of Mumbai, India

Efficient Neural Network Inference Complex-Valued Neural Networks Hyperspectral Image Classification Neural Network Compression

Jun 3, 2026cs.LG

Shortcomings and capacities of real-constrained neural networks in complex spaces

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

Andrew Gracyk

Department of Mathematics Purdue University West Lafayette, IN 47907, United States

Complex-Valued Neural Networks Statistical Learning Theory

May 29, 2026cs.CV

PolSAR Image Classification using a Hybrid Complex-Valued Network (HybridCVNet)

Recently, convolutional neural networks (CNNs) have become popular for image classification due to their effectiveness in computer vision tasks. Now, researchers are exploring the potential of vision transformers (ViTs) in remote sensing and Earth observation. However, traditional Real-Valued networks often overlook important phase information in Complex-Valued (CV) data like polarimetric synthetic aperture radar (PolSAR) data. To address this, new CV deep architectures have emerged. HybridCVNet, a novel hybrid network, blends CV-CNN and CV vision transformer (CV-ViT) techniques. It efficiently combines CV 3D and 2D CNNs as feature extractors, enhancing PolSAR image classification by extracting complementary information and effectively leveraging interdependencies within the data. Experimental results from widely-used PolSAR datasets show HybridCVNet outperforms other methods, achieving an overall accuracy of 97.39% on the Flevoland dataset and showing promise even with just a 1% sampling ratio, with a Kappa value of 0.972 on the San Francisco dataset. Source code is accessible through https://github.com/mqalkhatib/HybridCVNet

Mohammed Q. Alkhatib

Hybrid CNN-Transformer Architectures Complex-Valued Neural Networks Remote Sensing Polarimetric Imaging Image Classification

May 26, 2026cs.LG

When do complex-valued neural networks help? A study of representation, geometry, and optimization

Complex-valued Neural Networks (CVNNs) are often motivated by domains where information is naturally encoded in magnitude and phase. Yet complex-valued inputs alone do not determine when complex arithmetic improves learning: the label signal may lie in amplitude, phase, their coupling, or a symmetry that real-valued models can also represent under suitable coordinates. We study this through a representation-first evaluation of CVNNs against Cartesian real, polar, phase-only, magnitude-only, parameter-matched real, and FLOP-matched real baselines. Across synthetic RF tasks, complex representations are useful but not universally superior. PSK-only tasks favor phase-aware and complex-valued models, QAM-only tasks favor magnitude-based models, mixed PSK+QAM gives only a small complex-valued advantage, and unseen carrier-phase rotations break coordinate-dependent models without augmentation. Similar patterns appear beyond RF: in quantum-wavefunction prediction, momentum is invisible to

|ψ|

but recoverable from phase, while EEG analytic-signal experiments show that phase locking, amplitude bursts, and phase-amplitude coupling each favor different coordinate views. We also identify a benchmarking artifact on RadioML 2018.01A. Under matched-shared-trial selection, a CReLU complex model exceeds the best real baseline by 22.94 PP; under independent per-family tuning on the same data and 16-trial search space, the gap collapses to 2.46 PP. Gradient analysis traces the inflated gap to high-learning-rate first-step instability in real baselines, while complex parameter coupling distributes the loss signal more robustly. A learning-rate

\times

activation factorial confirms the failure is primarily hyperparameter-driven. Overall, CVNNs are best viewed as structured inductive biases whose gains depend on representation, symmetry, and optimization, not as universally superior architectures.

Ashutosh Kumar

Owl Autonomous Imaging, Inc.

Neural Network Optimization Complex-Valued Neural Networks Neural Network Training Dynamics

May 26, 2026cs.CV

Model discovery for dynamical systems with complex-valued product units

Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.

Martin Brückmann, Babette Dellen, Uwe Jaekel

Department of Mathematics, Informatics, and Technology, RheinAhrCampus Remagen University of Applied Sciences Koblenz Joseph-Rovan-Allee 2, 53424 Remagen, Germany

Nonlinear System Identification Complex-Valued Neural Networks Chaotic Dynamical Systems

May 21, 2026cs.LG

Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics

Complex dynamical systems governed by holomorphic maps such as

z^2 + c

exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex-analytic geometry, yet Multi-Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex-analytic priors, violate the Cauchy--Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN-ODE, a framework that replaces the MLP with a Kolmogorov-Arnold Network (KAN) whose learnable B-spline activations reside on network edges, and incorporates Cauchy--Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters (

16\times

fewer than the MLP baseline), the network achieves velocity-field

R^2 > 0.95

on all six systems, correctly identifies all six governing symbolic families through automatic spline-to-formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0% agreement. Crucially, the model exhibits only 4% MSE degradation under 10% observation noise versus

15.2\times

for MLPs, and achieves 90.4% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black-box architectures. These results establish KANs as a parameter-efficient, interpretable alternative to MLPs for physics-informed discovery of holomorphic dynamics.

Bhaskar Ranjan Karn, Dinesh Kumar

Nonlinear System Identification Complex-Valued Neural Networks Kolmogorov-Arnold Networks Chaotic Dynamical Systems Neural ODEs ODE Discovery

May 11, 2026cs.LG

Complex-Valued Phase-Coherent Transformer

Complex-valued Transformers have largely inherited softmax attention from real-valued architectures. However, row-normalised token competition is not necessarily aligned with phase-preserving computation. In this paper, we introduce the Phase-Coherent Transformer (PCT), which applies a real-valued, element-independent, smooth gate to L2-normalised complex query-key similarities. PCT replaces token competition with token-non-competing attention and is designed to preserve phase information across layers. Across mid-scale benchmarks spanning long-range memory, hierarchical long-range reasoning, positional retrieval, phase-based memory and superposition, and image classification, PCT shows strong generalisation across task categories. Under parameter-fair comparison, PCT consistently outperforms both the standard softmax Transformer and its direct complex-valued counterpart. Moreover, even on tasks traditionally considered difficult for complex-valued neural networks, such as NIAH and LRA-Text, PCT remains competitive with Multiscreen, the strongest real-valued NN baseline in our comparison. Experiments introducing gates that deliberately violate the PCT conditions show that the design is not incidental: smooth gates that preserve negatively aligned phase components remain strong, whereas gates that delete such components collapse on long-range retrieval, and gates whose outputs become excessively large suffer clear performance degradation. PCT also shows no depth-related accuracy collapse across the tested depth range. These results support introducing multi-layer phase-coherent structure into attention as a promising design principle for achieving generalisation in complex-valued Transformers.

Leona Hioki

Softmax Attention Complex-Valued Neural Networks Transformer Attention Attention Mechanisms Sequence Modeling

May 5, 2026cs.LG

Complex Equation Learner: Rational Symbolic Regression with Gradient Descent in Complex Domain

Symbolic regression aims to discover interpretable equations from data, yet modern gradient-based methods fail for operators that introduce singularities or domain constraints, including division, logarithms, and square roots. As a result, Equation Learner-type models typically avoid these operators or impose restrictions, e.g. constraining denominators to prevent poles, which narrows the hypothesis class. We propose a complex weight extension of the Equation Learner that mitigates real-valued optimization pathologies by allowing optimization trajectories to bypass real-axis degeneracies. The proposed approach converges stably even when the target expression has real-domain poles, and it enables unconstrained use of operations such as logarithm and square root. We Validate the method on symbolic regression benchmarks and show it can recover singular behavior from experimental frequency response data.

Sergei Garmaev, Maurice Gauché, Olga Fink

Intelligent Maintenance and Operations Systems Laboratory, EPFL, Lausanne, Switzerland · Mathematics Section, EPFL, Lausanne, Switzerland

Complex-Valued Neural Networks Symbolic Regression

May 1, 2026cs.LG

Batch Normalization for Neural Networks on Complex Domains

Riemannian neural networks have proven effective in solving a variety of machine learning tasks. The key to their success lies in the development of principled Riemannian analogs of fundamental building blocks in deep neural networks (DNNs). Among those, Riemannian batch normalization (BN) layers have shown to enhance training stability and improve accuracy. In this paper, we propose BN layers for neural networks on complex domains. The proposed layers have close connections with existing Riemannian BN layers. We derive essential components for practical implementations of BN layers on some complex domains which are less studied in previous works, e.g., the Siegel disk domain. We conduct experiments on radar clutter classification, node classification, and action recognition demonstrating the efficacy of our method.

Xuan Son Nguyen, Nistor Grozavu

ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, Cergy, France

Complex-Valued Neural Networks

Apr 21, 2026cs.AR

Algorithm and Hardware Co-Design for Efficient Complex-Valued Uncertainty Estimation

Complex-Valued Neural Networks (CVNNs) have significant advantages in handling tasks that involve complex numbers. However, existing CVNNs are unable to quantify predictive uncertainty. We propose, for the first time, dropout-based Bayesian Complex-Valued Neural Networks (BayesCVNNs) to enable uncertainty quantification for complex-valued applications, exhibiting broad applicability and efficiency for hardware implementation due to modularity. Furthermore, as the dual-part nature of complex values significantly broadens the design space and enables novel configurations based on layer-mixing and part-mixing, we introduce an automated search approach to effectively identify optimal configurations for both real and imaginary components. To facilitate deployment, we present a framework that generates customized FPGA-based accelerators for BayesCVNNs, leveraging a set of optimized building blocks. Experiments demonstrate the best configuration can be effectively found via the automated search, attaining higher performance with lower hardware costs compared with manually crafted models. The optimized accelerators achieve approximately 4.5x and 13x speedups on different models with less than 10% power consumption compared to GPU implementations, and outperform existing work in both algorithm and hardware aspects. Our code is publicly available at: https://github.com/zehuanzhang/BayesCVNN.git.

Zehuan Zhang, Mark Chen, He Li +1

Department of Computing, Imperial College London · 2Southeast University

Bayesian Neural Networks Uncertainty Quantification Complex-Valued Neural Networks Hardware-Aware NAS FPGA-Based Neural Network Acceleration

Apr 20, 2026cs.CV

DDF2Pol: A Dual-Domain Feature Fusion Network for PolSAR Image Classification

This paper presents DDF2Pol, a lightweight dual-domain convolutional neural network for PolSAR image classification. The proposed architecture integrates two parallel feature extraction streams, one real-valued and one complex-valued, designed to capture complementary spatial and polarimetric information from PolSAR data. To further refine the extracted features, a depth-wise convolution layer is employed for spatial enhancement, followed by a coordinate attention mechanism to focus on the most informative regions. Experimental evaluations conducted on two benchmark datasets, Flevoland and San Francisco, demonstrate that DDF2Pol achieves superior classification performance while maintaining low model complexity. Specifically, it attains an Overall Accuracy (OA) of 98.16% on the Flevoland dataset and 96.12% on the San Francisco dataset, outperforming several state-of-the-art real- and complex-valued models. With only 91,371 parameters, DDF2Pol offers a practical and efficient solution for accurate PolSAR image analysis, even when training data is limited. The source code is publicly available at https://github.com/mqalkhatib/DDF2Pol

Mohammed Q. Alkhatib

College of Engineering and IT, University of Dubai, Dubai, 14143, UAE

Remote Sensing Image Understanding Complex-Valued Neural Networks Polarimetric Imaging Image Classification

Mar 18, 2026cs.LG

Variational Phasor Circuits for Phase-Native Brain-Computer Interface Classification

We present the Variational Phasor Circuit (VPC), a deterministic classical learning architecture on the continuous

S^1

unit-circle manifold. Inspired by variational quantum circuits, VPC replaces dense weight matrices with trainable phase shifts, local unitary mixing, and structured interference in the ambient complex space, giving a unified method for binary and multi-class classification of spatially distributed signals. We evaluate VPC on real motor-imagery electroencephalography (EEG) from the PhysioNet Motor Movement/Imagery database (10 subjects, Common Spatial Pattern features, subject-wise cross-validation), where it attains a mean decoding accuracy of

0.60

-- the highest among standard brain--computer-interface baselines (linear discriminant analysis, logistic regression, RBF-SVM, and a multilayer perceptron) -- using an order of magnitude fewer parameters and the lowest cross-subject variance. We also characterize capacity honestly: with phase-only shifts and unitary mixing, VPC realizes a linear decision function in a fixed cosine/sine feature lifting, well matched to the largely separable band-power structure of EEG but unable to represent parity-type functions, a ceiling that depth does not raise. These results position unit-circle phase interference as a parameter-efficient alternative to dense neural computation for signal classification, and motivate VPC both as a standalone classifier and a front-end for hybrid phasor-quantum systems.

Dibakar Sigdel

Mindverse Computing LLC, WA 98087

MI Classification Brain-Computer Interfaces EEG Decoding Complex-Valued Neural Networks

Mar 16, 2026cs.LG

PhasorFlow: A Python Library for Unit Circle Based Computing

We present PhasorFlow, an open-source Python library for computing on the

S^1

unit circle. Inputs are encoded as complex phasors

z=e^{iφ}

on the

N

-torus (

\mathbb{T}^N

); as computation proceeds through unitary wave-interference gates, global norm is preserved while components drift into

\mathbb{C}^N

, letting algorithms leverage continuous geometric gradients. PhasorFlow makes three contributions. First, we formalize the Phasor Circuit model (

N

threads,

M

gates) with a 22-gate library spanning standard-unitary, non-linear, neuromorphic, and encoding operations under full matrix-algebra simulation. Second, we introduce the Variational Phasor Circuit (VPC), a trainable phase-native classifier analogous to variational quantum circuits. Third, we introduce the Phasor Transformer block and Large Phasor Model (LPM), replacing

QK^TV

attention with a parameter-free DFT token-mixing layer. We validate the framework on financial volatility detection, neuromorphic associative memory, neural binding, period finding, and algorithmic logic applications that are unique to the library. This positions unit-circle computing as a deterministic, lightweight paradigm on classical hardware. Available at https://github.com/mindverse-computing/phasorflow.

Dibakar Sigdel, Namuna Panday

Mindverse Computing LLC, Lynnwood, WA 98087

Complex-Valued Neural Networks Quantum-Inspired ML

Feb 6, 2026cs.LG

Perturbing the Phase: Analyzing Adversarial Robustness of Complex-Valued Neural Networks

Complex-valued neural networks (CVNNs) are rising in popularity for all kinds of applications. To safely use CVNNs in practice, analyzing their robustness against outliers is crucial. One well known technique to understand the behavior of deep neural networks is to investigate their behavior under adversarial attacks, which can be seen as worst case minimal perturbations. We design Phase Attacks, a kind of attack specifically targeting the phase information of complex-valued inputs. Additionally, we derive complex-valued versions of commonly used adversarial attacks. We show that in some scenarios CVNNs are more robust than RVNNs and that both are very susceptible to phase changes with the Phase Attacks decreasing the model performance more, than equally strong regular attacks, which can attack both phase and magnitude.

Florian Eilers, Christof Duhme, Xiaoyi Jiang

Department of Computer Science University of Münster Münster, Germany

Adversarial Attacks Neural Network Robustness Adversarial Robustness Complex-Valued Neural Networks