Hybrid Real & Complex Neural Networks

Updated 27 September 2025

Hybrid neural networks are architectures that integrate real- and complex-valued computation paths using dual-path modular blocks and explicit domain conversion functions.
They leverage efficient fusion strategies, NAS optimization, and tailored activation schemes to reduce parameter counts while capturing both amplitude and phase features.
Empirical studies show hybrid models excel in applications like speech enhancement, EEG classification, and radar processing, offering improved performance and lower computational costs.

Hybrid real- and complex-valued neural networks (“hybrid NNs” or HNNs, Editor’s term) are architectural frameworks that combine real-valued and complex-valued computation paths within a single model, leveraging the expressive benefits of both domains. Such networks are particularly suited to problems where both real and complex data representations play essential roles, as in phase-aware signal processing, communications, image, and biomedical applications. Hybrid NNs address the inefficiencies and limitations of purely real- or complex-valued models—achieving improved performance, reduced parameter counts, and lower computational cost by selectively exploiting the mathematical structures intrinsic to both real and complex spaces.

1. Architectural Principles and Hybridization Strategies

Hybrid NN architectures are designed by integrating separate real-valued and complex-valued branches, often at the granularity of layers or sub-networks, interconnected via explicit domain conversion functions. The key components are:

Dual-path modular building blocks: Each block possesses real and complex branches (see Figure “hybrid_block” (Young et al., 4 Apr 2025)), accepting and producing both real and complex feature tensors.
Domain conversion functions: These mappings translate between real and complex representations as information is exchanged between branches.
- Real-to-complex (Cartesian): $z = x_1 + i x_2$
- Real-to-complex (Polar): $z = x_1 \exp(i\pi x_2)$
- Other mappings such as “Exp” and “MagExp” expand or contract representational dimensionality.
Branch sizing and parameter scaling: To avoid parameter inflation, the hybrid approach assigns halved sizes to real branches and quartered sizes to complex branches, compensating for doubled storage per complex parameter (Fiorio et al., 25 Sep 2025).
Flexible fusion: Information is concatenated or combined at the input, bottleneck, or output stages, with real and complex outputs being cross-injected as auxiliary features.
NAS Optimization: Architecture search (NAS) steps are undertaken to prune redundant paths, optimize conversion placements, and select the best activation and conversion functions (Young et al., 4 Apr 2025).

Table 1: Hybrid Block Structure

Component	Real Branch	Complex Branch
Convolution	$f_\mathbb{R}$	$f_\mathbb{C}$
Activation	$\sigma_\mathbb{R}$	$\sigma_\mathbb{C}$
Domain Conversion	To/From $\mathbb{C}$	To/From $\mathbb{R}$
Output Fusion	[concat, sum, ...]	[concat, sum, ...]

The design enables the network to harness magnitude-based robustness and computational efficiency from real processing, while preserving phase-specific or multi-scale relations via complex sub-networks.

2. Domain Conversion and Activation Functions

Central to hybrid NNs is the design of inter-domain conversion and complex-valued activation mechanisms:

Domain Conversions: Operations such as “Mag” (extracting magnitude), Cartesian or polar composition (see above), and splitting/concatenation between real and imaginary channels enable seamless translation of feature types.
Complex Activation Functions: To generalize classical nonlinearities, hybrid NNs leverage parameterized polynomial and rational forms, e.g.
- $Z_o = \alpha z + \frac{1}{|z|^q + \varepsilon} \sum_{n=0}^p k_n z^n$ (Equation (1), (Young et al., 4 Apr 2025))
- ReLU-analogues via phase-based switching: $Z_o = z\cdot (1 - |\alpha| + \alpha\cdot \text{sign}(v))$
- These constructions efficiently handle amplitude/phase and support a diverse set of nonlinear effects.
Fully complex non-parametric activations: Kernel expansion based activation functions, as in complex-valued kernel activation functions (KAF), further increase nonlinearity flexibility (Scardapane et al., 2018).
Normalization and regularization: Batchnorm and layernorm procedures are adapted to treat both amplitude and phase to maintain numerical stability.

These elements enable selective, differentiable, and parameter-efficient adaptation of both amplitude and phase-sensitive features while facilitating stable training.

3. Theoretical Foundations and Universal Approximation

Hybrid NNs inherit their representational guarantees from both real and complex-valued network theory:

Universal Approximation Theorem in Hybrid and Hypercomplex Contexts: Networks constructed on non-degenerate algebras, as with complex-valued MLPs or quaternionic extensions, are universal approximators for continuous functions on compact sets, provided suitable split or component-wise activations are used (Valle et al., 4 Jan 2024).
Parameter efficiency for invariant functions: For function classes exhibiting radial or rotational symmetries, complex-valued networks can approximate such mappings with polynomial parameter growth, in contrast to the exponential growth needed by real-valued networks (Zhang et al., 2021).
Critical point set reduction: The set of critical points of complex-reaction networks is a proper subset of that of real-valued networks, indicating potentially smoother loss landscapes and more favorable optimization characteristics (Zhang et al., 2021).

This theoretical underpinning motivates the hybridization strategy: by partitioning processing between domains that are mathematically optimal for separate aspects of the task, hybrid NNs can achieve flexibility, compactness, and convergence properties unattainable in solely real or complex models.

4. Empirical Performance and Complexity

Hybrid NNs consistently achieve or exceed the performance of their pure-domain counterparts—often with significantly reduced parameter and operation counts:

Speech enhancement (AudioMNIST, CDAE/CRN):
- Hybrid models (hCDAE, hCRN) achieve higher STOI/PESQ/SI-SDR at identical or lower parameter and MAC counts than both rCDAE/rCRN and cCDAE/cCRN baselines (Fiorio et al., 25 Sep 2025).
- E.g., at −5 dB SNR, hCDAE achieves STOI 0.743 vs. rCDAE 0.714 and cCDAE 0.734, while requiring only 3.31 G MACs vs. 4.72/4.54 G MACs for the real/complex networks.
Audio and radar signal processing: HNNs effectively capture phase and amplitude simultaneously, improving robustness and denoising performance, critical in low SNR regimes (Young et al., 4 Apr 2025).
EEG classification: Early complex convolutional layers for frequency-domain inputs followed by real-valued layers yield accuracy improvements ( $>99.8\%$ on binary tasks) and a 52% reduction in parameter count over conventional architectures (Du et al., 2022).
Efficiency trade-off: The combination of reduced model complexity, improved generalization (lower validation loss, smaller generalization gaps), and stability across noise conditions renders hybrid NNs particularly attractive for edge and real-time deployment.

Model	MACs (G)	STOI	PESQ	SI-SDR
rCDAE	4.72	0.714	1.575	4.044
cCDAE	4.54	0.734	1.580	3.885
hCDAE	3.31	0.743	1.599	4.404

5. Mathematical and Implementation Insights

Hybrid NNs employ specialized mathematical formulations:

Complex Convolution via Real Operations: Complex convolutions are decomposed into real-valued operations on the real/imaginary parts, e.g.

$[o_r\;o_i]^\top = \begin{bmatrix}w_r & -w_i \ w_i & w_r\end{bmatrix}[x_r\;x_i]^\top$

Domain conversion at layer bottlenecks and gating points: E.g., $Z_r = [a_z, b_z]$ for converting complex $Z = a_z + i b_z$ to real-valued concatenation.
Parameter count normalization: For fair hybridization one complex parameter counts as two real parameters in resource calculations (Mönning et al., 2018, Fiorio et al., 25 Sep 2025).
Feature fusion and masking: Hybrid architectures often fuse outputs via concatenation or optimal mixing, enabling magnitude filtering and fine-grained complex-phase correction (Fiorio et al., 25 Sep 2025).

Implementation frameworks typically extend standard deep learning libraries with routines for complex arithmetic, conversion, normalization, and manifold-aware layers (Smith, 2023). Kernel activations, analytic constraints, and learnable parameterized activations further enhance control and expressiveness (Scardapane et al., 2018, Venkatasubramanian et al., 16 Sep 2024, Ataei et al., 25 Jan 2025).

6. Applications and Domain-Specific Impact

Hybrid NNs have demonstrated substantial value across domains requiring phase-aware or multidimensional signal modeling:

Speech Enhancement and Audio Processing: By leveraging magnitude filtering in the real branch and precise phase correction with the complex branch, hybrid networks improve intelligibility (STOI), perceived quality (PESQ), and denoising (SI-SDR) under severe noise, with reduced MAC complexity (Fiorio et al., 25 Sep 2025).
Biomedical Signal Analysis (EEG): Initial complex stages capture phase details crucial for discrimination, while real-valued layers optimize classification with fewer parameters (Du et al., 2022).
Radar, communications, and remote sensing: Phase-sensitive targeting (SAR imaging, modulation classification) benefits from the ability of hybrid NNs to robustly model polarization, rotational symmetry, and scaling ambiguities (Chakraborty et al., 2019, Singhal et al., 2021).
Color and vision: LAB and sliding encodings enable robust hybrid processing of RGB images, ensuring invariance to hue and channel correlations (Singhal et al., 2021).
General multidimensional modeling: Hybrid NNs, using vector map convolution or hypercomplex extensions, scale weight sharing and internal latent relations beyond the constraints of fixed complex (2D) or quaternion (4D) frameworks (Gaudet et al., 2020, Valle et al., 4 Jan 2024).

7. Open Challenges and Directions

Key challenges and frontiers include:

Activation and optimization: Fully holomorphic, non-singular, and bounded activation functions remain elusive due to Liouville’s theorem. Hybrid NNs circumvent this via split or kernel-based activations, but stability and convergence are still areas of active research (Bassey et al., 2021).
Weight initialization and regularization: Complex-valued weights, especially in deep regimes, require novel initialization schemes to avoid training pathologies and variance collapse (Mönning et al., 2018).
Automated hybrid architecture discovery: As the solution space of possible real/complex block arrangements grows, NAS-based and information-theoretic guides (e.g., analytic constraints for generalization) are crucial (Young et al., 4 Apr 2025, Venkatasubramanian et al., 16 Sep 2024).
Framework integration: Expanding support in mainstream deep learning libraries, with efficient, vectorized complex operations and manifold layers, is ongoing (Smith, 2023).
Interpretability: Hybrid NNs using interpretable activation units (e.g., Lehmer transforms) offer transparent access to the feature selection and decision pipeline, which is critical for deployment in sensitive domains (Ataei et al., 25 Jan 2025).

Hybrid real- and complex-valued neural networks are maturing as a crucial architectural tool when phase, multidimensional correlation, and scaling invariance are essential aspects of the data or task. By leveraging the strengths of both domains—selective, efficient, and interpretable processing—hybrid NNs can achieve state-of-the-art empirical results with improved computational and sample efficiency, laying foundations for further advances across signal processing, vision, communications, and biomedical informatics.