Complex-Valued Data Models

Updated 1 October 2025

Complex-valued data models are mathematical frameworks that process complex-domain data by explicitly capturing both magnitude and phase.
They leverage geometric and algebraic insights, including complex convolutions and manifold-based operations, to extend traditional machine learning methods.
Applications span improved MRI reconstruction, fMRI mapping, radar signal processing, and speech analysis, offering robust and interpretable results.

A complex-valued data model is a mathematical or algorithmic framework that processes, analyzes, or predicts data whose domain is the complex field $\mathbb{C}$ , thereby explicitly modeling both magnitude and phase (or equivalently, real and imaginary components). Complex-valued data arises organically in domains such as communications, radar/SAR, fMRI, MRI, audio, and spectral analysis, where signal representation, interferometric phenomena, or analytic transformations (e.g., Fourier, Hilbert) inherently require $\mathbb{C}$ -valued models. Recent advances extend classical machine learning and deep learning architectures to support learning, inference, and generative modeling in the complex domain, exploiting symmetry, geometry, and algebraic structure unique to $\mathbb{C}$ .

1. Geometric and Algebraic Foundations

Complex-valued data models differ fundamentally from real-valued analogues due to the underlying algebraic and geometric structure of $\mathbb{C}$ . The polar form, $z = r e^{i\theta}$ , reveals the interaction between amplitude ( $r$ ) and phase ( $\theta$ ), inducing invariances (e.g., to global phase or scaling) and group actions (e.g., the scaling–rotation group $\mathbb{R}^+ \times SO(2)$ ) that are not present in $\mathbb{R}^n$ -based systems.

Recent frameworks have leveraged this by treating the complex plane as a product manifold ( $\mathbb{R}^+\times SO(2)$ ) (Chakraborty et al., 2019, Chakraborty et al., 2019). Operations such as complex-valued convolution and distance transforms are formulated using the weighted Fréchet mean (wFM) on the manifold. For a set ${z_k}$ with weights ${w_k}$ :

$\operatorname{wFM}(\{z_k\}, \{w_k\}) = \arg\min_m \sum_k w_k d^2(z_k, m)$

where $d(z_1, z_2) = \sqrt{[\log(|z_2|/|z_1|)]^2 + \|\logm(R(z_2) R(z_1)^{-1})\|^2}$ is the geodesic distance, and $R(z)$ the rotation matrix induced by $\arg(z)$ . These manifold-aware models achieve built-in equivariance and invariance with respect to complex scaling and rotations, essential for modalities such as SAR, RF, and MRI (Chakraborty et al., 2019, Chakraborty et al., 2019).

2. Machine Learning and Deep Learning on $\mathbb{C}$

Extensions of classical machine learning (SVM, SVR, LVQ) and deep learning (CNNs, RNNs, transformers, GANs) to complex-valued data are nontrivial due to the breakdown of analyticity (holomorphicity) for bounded nonlinearities and the need for complex-valued optimization.

Support Vector Machines/Regression: The extension of SVM/SVR to $\mathbb{C}$ involves widely linear estimation: $g(z) = \langle \Phi_{\mathbb{C}}(z), w \rangle + \langle \Phi_{\mathbb{C}}^*(z), v \rangle + c$ where $\Phi_{\mathbb{C}}$ maps to a complex RKHS and $w,v$ are complex weights (Bouboulis et al., 2013). The dual optimization for complex SVR decomposes into two real SVRs with an induced real kernel, illustrating a deep structural link between real and complex SVM optimization.

Complex Neural Networks: Complex-valued neural networks (CVNNs) have seen advances in architecture and theory including:

Complex-valued convolutions: With the standard rule $W*d = (X*a - Y*b) + i(Y*a + X*b)$ , preserving phase and magnitude relations (Cole et al., 2020, Chatterjee et al., 2023).
Non-parametric and phase-sensitive activations: Functions such as CReLU, zReLU, modReLU, PC-SS, and kernel activation functions (KAFs) generalize nonlinearities to $\mathbb{C}$ , often working in split or polar domains (Scardapane et al., 2018, Vasudeva et al., 2020).
Training and optimization: The Wirtinger calculus and CR-calculus extend gradient-based learning to non-holomorphic functions (Sarroff et al., 2015, Bouboulis et al., 2013).
Multi-view and hybrid architectures: Architectures that explicitly process real and imaginary (or magnitude and phase) channels in parallel, enforcing analytic or orthogonal constraints (e.g., via the Hilbert transform penalty in Steinmetz networks), achieve improved generalization and robustness (Venkatasubramanian et al., 16 Sep 2024).
Hybrid Real–Complex Networks: Modular architectures combining real-valued and complex-valued pathways and domain-conversion functions yield parameter savings and improved performance on signal tasks (Young et al., 4 Apr 2025).

3. Applications in Signal and Medical Imaging

Complex-valued models have led to performance improvements and new capabilities in multiple scientific domains:

MRI and CS-MRI Reconstruction: Direct modeling in $\mathbb{C}$ improves both magnitude and phase fidelity, necessary for advanced imaging (QSM, velocity mapping). Complex-valued CNNs, GANs, and diffusion models (e.g., Co-VeGAN, PhaseGen) outperform magnitude-only or split-channel real models both in compression sensing and in downstream tasks (Vasudeva et al., 2020, Cole et al., 2020, Rempe et al., 10 Apr 2025). Synthetic phase generation via complex diffusion (PhaseGen) allows pretraining on large image-domain datasets, bridging the gap to k-space data (Rempe et al., 10 Apr 2025).
fMRI Activation Mapping: Fully Bayesian models exploiting the complex-valued signal—notably the CV-M&P and CV-sSGLMM—simultaneously estimate activation in both magnitude and phase. These models account for temporal (AR) structure and spatial correlations (GMRF), are scalable via image partitioning, and yield improved sensitivity, as demonstrated in simulated and real finger-tapping experiments (Wang et al., 2023, Wang et al., 12 Jan 2024).
Speech Processing: Complex-valued Restricted Boltzmann Machines (CRBM) model both amplitude and phase of speech spectra directly, improving quality as measured by PESQ, MOS, and PSNR over amplitude-only strategies (Nakashika et al., 2018).

4. Supervised and Unsupervised Learning with Complex Proximities

For domains where data is not vectorial but given via similarity/dissimilarity measures (possibly indefinite or non-metric), complex embeddings via eigendecomposition and Nyström approximation enable faithful vectorization without information loss (Münch et al., 2020). Machine learning algorithms—including complex-valued LVQ, SVM, and deep nets—can then operate directly on these embeddings, exploiting the information preserved in negative or indefinite eigenvalues.

5. Representation, Invariance, and Symmetry

Many complex-valued models are designed to exploit inherent symmetries:

Scaling/Rotation Invariances: Model layers are constructed to be invariant or equivariant under $s \cdot x$ for $s\in\mathbb{C}^*$ . This mitigates data ambiguities from arbitrary gain or phase shift (prevalent in remote sensing and RF) (Singhal et al., 2022, Chakraborty et al., 2019).
Analyticity and Phase Constraints: Enforcement of analytic signal structure (via discrete Hilbert constraints) regularizes multi-view learning, making the model robust to additive noise and yielding tighter generalization error bounds (Venkatasubramanian et al., 16 Sep 2024).

6. Open Challenges and Future Directions

Activation Functions: The lack of bounded holomorphic activations (Liouville’s theorem) necessitates non-holomorphic, split, or kernel-based approaches; improving these remains an active area (Scardapane et al., 2018, Vasudeva et al., 2020).
Training Stability and Initialization: Complex-valued models are sensitive to initializations and learning rates; development of initialization schemes and regularization for $\mathbb{C}$ -domain remains ongoing (Mönning et al., 2018).
Framework Support: Many mainstream libraries (PyTorch, TensorFlow) lack robust, native complex support; recent efforts provide modular extensions and efficient implementations (e.g., Gauss multiplication trick for complex layers) (Smith, 2023).
Scalable Bayesian Inference: Efficient sampling (e.g., image partitioning and parallelization in fMRI models) is key for practical deployment in large-scale neuroimaging (Wang et al., 2023, Wang et al., 12 Jan 2024).
Generalization Theory: Information-theoretic analysis of generalization in complex-valued architectures provides a new theoretical lens and suggests consistency-penalized schemes (Venkatasubramanian et al., 16 Sep 2024).
Cross-domain Applications: Applications extend to remote sensing, communications, radar, medical imaging, audio, and biological sequence analysis.

7. Summary Table: Key Model Classes and Representative Approaches

Model Class	Key Property	Notable Example(s)
Complex SVM/SVR	Widely linear, complex kernels, quaternary split	(Bouboulis et al., 2013)
Complex Neural Networks	CV-convs, specialized activations, CR-calculus	(Cole et al., 2020, Sarroff et al., 2015)
Kernel Activations (KAF)	Non-parametric, adaptivity	(Scardapane et al., 2018)
Manifold-based Layers	Scaling/rotation equivariance/invariance	(Chakraborty et al., 2019, Chakraborty et al., 2019)
Hybrid Real–Complex	Dual-path, domain conversion, efficient NAS	(Young et al., 4 Apr 2025)
Multi-view Fusion	Latent analytic consistency, Hilbert penalty	(Venkatasubramanian et al., 16 Sep 2024)
Complex Diffusion Models	Synthetic phase generation, conditioning	(Rempe et al., 10 Apr 2025)
Bayesian fMRI Mapping	Full $\mathbb{C}$ -domain, spatial priors	(Wang et al., 2023, Wang et al., 12 Jan 2024)
Complex RBM	Amplitude–phase latent modeling	(Nakashika et al., 2018)

Complex-valued data modeling therefore provides a rigorous, geometry- and symmetry-aware theoretical framework combined with specialized algorithmic innovations spanning kernel methods, deep architectures, generative models, and Bayesian inference. The recent literature demonstrates that these approaches lead to improvements in reconstruction fidelity, robustness, data efficiency, and interpretability across a diverse array of complex-valued application domains.