Magnitude–Phase Activations

• Magnitude–Phase Activations are mechanisms that separately manipulate the modulus and phase of complex signals, supporting enhanced modeling in neural and physical systems.
• They underpin dual-stream architectures which decouple magnitude and phase processing, yielding improved accuracy in tasks like speech enhancement and dereverberation.
• Specialized nonlinearities and loss functions in these setups promote robust gradient flow and higher capacity in applications such as complex Hopfield networks.

Magnitude–phase activations are functional, architectural, and analytic constructs that separately manipulate the modulus (magnitude) and argument (phase) of complex- or hypercomplex-valued signals within learning, signal processing, and physical systems. Their principled separation enables independent or interactive treatment of the two fundamental components of complex representations, which is necessary for tasks where magnitude and phase have distinct physical, perceptual, or mathematical roles. Recent research across neural networks, statistical signal models, and engineered electromagnetic devices has introduced architectures, activations, and optimization objectives directly controlling or exploiting magnitude–phase components.

1. Mathematical Foundations and Formalism

The complex (or quaternion) representation of physical and signal features intrinsically provides a polar decomposition: for any complex value $z = re^{j\phi}$ (or quaternion $q = \|q\| e^{n\varphi}$), magnitude $\|z\|$, and phase/angle $\phi$ (or quaternionic phase $\varphi$) parameterize signal energy and structure. Magnitude–phase activations take functional forms that directly modify one or both components:

• For complex signals: $f(z) = g(|z|) e^{j h(\arg(z))}$, where $g$ and $h$ are real-valued operations;
• For quaternion signals: phase and magnitude are treated as separate axes, with activations such as $\sigma_1(q) = g(\|q\|) (q/\|q\|)$ (magnitude) or $\sigma_2(q) = \|q\| e^{n h(\varphi(q))}$ (phase) (Pöppelbaum et al., 2024).

Magnitude–phase separation is essential whenever the underlying physics or statistical properties of data encode more than mere real-imaginary decomposition. Perceptual, geometric, or invariance properties often depend on the configuration of $q = \|q\| e^{n\varphi}$0 and $q = \|q\| e^{n\varphi}$1 rather than the raw coordinates.

2. Dual-Stream Architectures in Neural Networks

Modern deep learning for complex signals, especially in speech enhancement and separation, increasingly adopts architectures that explicitly separate magnitude and phase processing.

Two-stage schemes such as the dereverberation method in "Magnitude or Phase?" (Schwartz et al., 2022) instantiate this paradigm. The pipeline is as follows:

• Stage 1 (“S2S block”): A U-Net estimates enhanced log-magnitude $q = \|q\| e^{n\varphi}$2 given noisy log-magnitude $q = \|q\| e^{n\varphi}$3, trained with a mean-square-error loss in log-magnitude domain.
• Stage 2 (“RI2RI block”): Given the enhanced magnitude $q = \|q\| e^{n\varphi}$4 and noisy phase $q = \|q\| e^{n\varphi}$5, a second U-Net refines real and imaginary components, reconstructing an improved STFT estimate $q = \|q\| e^{n\varphi}$6, trained with scale-invariant signal-to-distortion ratio (SI-SDR) loss for explicit phase improvement.

Manifold-aware dual-stream approaches go further by enforcing geometric constraints on the phase stream. The "Global Rotation Equivariant Phase Modeling" architecture introduces:

• A magnitude stream (real, Euclidean);
• A phase stream (complex, circular $q = \|q\| e^{n\varphi}$7), with network operations (e.g., bias-free complex convolutions; modulus-based gating) designed to preserve global rotation equivariance (GRE).
• Hybrid modules (MPICM, HADF) permit cross-stream interactions, leveraging the rotation-invariant modulus for information exchange, while phase-specific blocks maintain $q = \|q\| e^{n\varphi}$8 topology (Wang et al., 9 Feb 2026).

Such architectures enable more faithful modeling of the distinctive statistics and topology of phase, leading to strong improvements in both phase-sensitive and perceptual metrics across speech tasks.

3. Magnitude–Phase Activations and Nonlinearities

Specialized nonlinear activations that operate on magnitude or phase—rather than simply on real and imaginary components—address crucial challenges in complex and quaternion neural networks.

Quaternion-valued neural networks (QNNs) benefit from activations that preserve quaternion-space structure:

• Magnitude-modulating activations: $q = \|q\| e^{n\varphi}$9 (e.g., norm normalization, tanh shrinkage);
• Phase-modulating activations: $\|z\|$0 (e.g., $\|z\|$1 or $\|z\|$2).

Key findings include:

• Phase-modulating activations do not saturate or suffer dead zones, since their derivatives (with respect to $\|z\|$3) are nonzero almost everywhere (excluding the real axis), ensuring persistent gradient flow and robust optimization.
• Empirically, phase-based activations outperform split-component analogues (split-ReLU, split-Tanh) on standard classification benchmarks; best observed gains are +3–6% (CIFAR-10) and +1–2% (SVHN) over split-ReLU (Pöppelbaum et al., 2024).

For complex-valued Hopfield networks, magnitude–phase quantization via ceiling-type activations substantially increases per-neuron state counts (Ramamurthy et al., 1 Jul 2025). Rectangular (“CoCeil”) and polar (“CoSign”) activations quantize the magnitude and/or phase:

• CoCeil: Cartesian thresholding; $\|z\|$4 states;
• CoSign: $\|z\|$5 magnitude levels $\|z\|$6 $\|z\|$7 phase quanta, $\|z\|$8 states.

This enables orders of magnitude larger associative memory capacity without increasing network width.

4. Optimization Objectives and Compensation Effects

When optimizing complex-valued signals in neural or variational frameworks, the separation of magnitude and phase in the loss function or activation is critical. Purely complex- or time-domain losses:

• Allow implicit compensation: phase errors may be “explained away” by shrinking estimated magnitudes, since the optimal magnitude under squared error with fixed phase deviation is $\|z\|$9;
• Result in lower SI-SDR (signal-level) but worse perceptual and ASR metrics (PESQ, eSTOI, WER) (Wang et al., 2021).

Adding explicit magnitude loss enforces correct envelope reconstruction and reduces undesirable compensation. Networks trained with a magnitude loss achieve higher PESQ and eSTOI at minor SI-SDR cost, and dramatically lower ASR word error rates.

Analytic and empirical findings suggest future designs should include:

• Joint or separate magnitude–phase streams and objectives;
• Architectures or activations that strictly limit undesired magnitude shrinkage in the presence of phase errors;
• Gated or confidence-weighted coupling between magnitude and phase branches to enforce appropriate trade-offs.

5. Applications and Domain-Specific Implementations

Magnitude–phase activation mechanisms have diverse practical instantiations across multiple disciplines:

• Speech enhancement and dereverberation: Dual-stage models with separate magnitude and phase refinement outperform magnitude-only or real/imaginary-only models on objective and perceptual metrics. GRE-based approaches further improve phase accuracy (Phase Distance reduced by 20–25%) and perceptual metrics by respecting phase topology (Schwartz et al., 2022, Wang et al., 9 Feb 2026).
• Metasurface engineering: Real-time programmable reflectors based on coupled SRR–DRR unit cells achieve fully independent control of the reflection magnitude $\phi$0 (via DRR loss) and phase $\phi$1 (via SRR resonance) at fixed frequency. This allows covering the entire amplitude–phase plane, enabling advanced beam structuring, gain control, and multi-beam synthesis at microwave/RF frequencies (Ashoor et al., 2020).
• Complex-valued Hopfield associative memory: By quantizing both magnitude and phase, memory capacity and pattern separability are substantially increased. Polar “CoSign” activations provide a natural fit for tasks where phase carries information (OFDM, radar, multilevel associative memory) (Ramamurthy et al., 1 Jul 2025).
• Complex-valued fMRI analysis: Fully Bayesian models map independent magnitude and phase brain activation in complex-valued fMRI, capturing activations overlooked by magnitude-only models and achieving superior effect-size estimation (Wang et al., 2024).
• Quaternion neural computation: Magnitude–phase activations yield improved optimization and predictive accuracy for QNNs in vision tasks, providing stable gradient flow and respecting hypercomplex number geometry (Pöppelbaum et al., 2024).

6. Evaluation Metrics and Empirical Findings

Objective evaluation of magnitude–phase activation strategies employs metrics sensitive to both components. For speech tasks:

• Magnitude-sensitive: Cepstral Distance (CD), Log-Likelihood Ratio (LLR), PESQ;
• Phase-sensitive: SI-SDR, (frequency-weighted) SNR, Phase Distance (PD), Weighted Overlap PD (WOPD);
• Intelligibility and perceptual: eSTOI, DNSMOS, UTMOS, word error rate (WER) in ASR.

Empirical studies consistently demonstrate:

• Two-stage magnitude–phase models outperform single-stage baselines in dereverberation (e.g., SI-SDR from –10.4 dB to +1.94 dB, PESQ from 1.48 to 2.38) (Schwartz et al., 2022).
• Manifold-aware GRE architectures yield 20–25% PD reduction and perceptual metric improvements across retrieval, denoising, and dereverberation tasks (Wang et al., 9 Feb 2026).
• Explicit magnitude loss in training improves PESQ by $\phi$20.4, eSTOI by 1–2%, and lowers WER by $\phi$36% in speech separation (Wang et al., 2021).
• Magnitude–phase quantized Hopfield networks exhibit stable energy descent and vastly expanded state spaces (Ramamurthy et al., 1 Jul 2025).
• Bayesian cv-fMRI magnitude–phase mapping achieves high accuracy and recall even for phase-only activation regions, which are undetectable by magnitude-only models (Wang et al., 2024).

7. Theoretical and Practical Benefits, Limitations, and Outlook

Magnitude–phase activations provide principled, geometry-respecting processing for complex and hypercomplex signals. Benefits include:

• Decoupling and flexible control over magnitude and phase processing, enabling improved performance, interpretability, and generalization in signal and neural domains;
• Avoidance of vanishing gradients and dead zones—especially in phase-modulating activation designs;
• Increased capacity and functionality in associative memories and complex-valued physical networks.

Limitations include:

• Higher computational cost due to trigonometric function evaluations;
• Potential numerical instability at degenerate points (e.g., near-zero imaginary part in quaternion phase);
• Need for careful architectural and loss design to prevent unintended compensation or trade-offs.

Applications span from end-to-end audio enhancement, MRI/fMRI analysis, and physical metasurface design to hypercomplex neural computation and high-capacity associative memory. Continuing developments integrate geometric priors, equivariance properties, and domain-specific gating—suggesting a broadening role for magnitude–phase activations in learning, signal, and physical systems (Schwartz et al., 2022, Ashoor et al., 2020, Pöppelbaum et al., 2024, Wang et al., 9 Feb 2026, Ramamurthy et al., 1 Jul 2025, Wang et al., 2024, Wang et al., 2021).