Phase-Aware Signal Prior

• Phase-Aware Signal Prior is a framework that exploits phase characteristics to enhance reconstruction from magnitude-only measurements using statistical, convex, and deep learning models.
• It improves inverse problem solutions in phase retrieval, audio inpainting, and speech enhancement by enforcing phase continuity and reducing sample complexity.
• Recent advances integrate data-driven architectures with traditional signal processing to achieve superior reconstruction fidelity and robustness against noise.

A phase-aware signal prior is a structural prior, explicit or implicit, that exploits information about the phase characteristics of signals—particularly in contexts where only magnitude measurements are available (e.g., phase retrieval) or where phase carries critical reconstructive or perceptual information (e.g., speech enhancement, audio inpainting, source separation). These priors are designed to enforce or leverage coherence or structure in the phase domain, thereby improving the identifiability, reconstruction quality, or sample complexity in inverse problems. The concept encompasses classical analytic forms (e.g., circular or Markovian phase distributions), data-driven implicit schemes (e.g., denoising or deep neural priors), convex regularizers built from physical models (e.g., instantaneous-frequency correction), and multi-stage architectures that condition modern networks on coarse phase predictions.

1. Theoretical Foundations and Types of Phase-Aware Priors

Phase-aware priors formalize domain-specific knowledge about phase continuity, distribution, or dynamics. These priors fall into several broad categories:

• Circular Statistical Priors: Models such as the von Mises or wrapped normal distributions are imposed on the phase of STFT coefficients, capturing circularity and periodicity (Dionelis et al., 2017).
• Sinusoidal/Instantaneous Frequency-Based Correction: Under the sinusoidal model, the phase observed in the STFT domain evolves predictably. Correcting for this evolution collapses harmonic signals to temporally constant trajectories, justifying convex or low-rank priors in the phase-corrected spectrogram domain (Masuyama et al., 2019, Masuyama et al., 2019, Balušík et al., 26 Jan 2026).
• Semi-Algebraic and Deep Generative Priors: Modern approaches encode very general phase-aware structural constraints using generative models or more abstract semi-algebraic sets, covering sparse, low-dimensional, and neural-network–generated manifolds (Bendory et al., 2023, Hand et al., 2018).
• Implicit Priors via Denoisers: The denoiser–score connection relates the residual output of an MMSE denoiser to the gradient of the log-prior, providing a powerful, distributionally robust implicit prior, as in the Alternating Phase Langevin Sampler (Agrawal et al., 2022).

2. Formulation in Inverse Problems and Optimization

Phase-aware priors are central to a range of nonlinear inverse problems, most notably phase retrieval from magnitude-only measurements, and are embedded in the regularized objective:

$$\widehat x = \arg\min_x \; \tfrac12\bigl\|\,|\mathcal{F}x| - b\,\bigr\|_2^2 + R(x)$$

Here, $R(x)$ embodies the phase-aware signal prior. Key instantiations include:

• Known Support or Anchor Constraints: Known spatial or spectral regions provide phase information for the unknown part, converting spatial priors into effective phase constraints and accelerating convergence (Osherovich et al., 2012).
• Score-based/Denoiser Priors: The gradient of the implicit prior is accessed through a denoising function, circumventing the need for an explicit density (Agrawal et al., 2022).
• Phase Correction and Total Variation (iPC-TV) Regularization: Enforces temporal smoothness of instantaneous-phase-corrected spectrograms, leading to convex, efficiently-solvable formulations for tasks such as inpainting and HPSS (Masuyama et al., 2019, Balušík et al., 26 Jan 2026).
• Low-Rank Constraints on Phase-Corrected Spectrograms: Promotes harmonically structured signals by ensuring the (complex, phase-corrected) spectrogram exhibits low-rank structure (Masuyama et al., 2019).

Convex and nonconvex algorithms (e.g., ADMM, primal–dual splitting, Langevin sampling, gradient descent in the latent domain) are employed to optimize the corresponding objectives.

3. Phase-Aware Priors in Audio, Speech, and Imaging Applications

Phase-aware priors enable advances across several domains:

Audio Inpainting and HPSS:

• Convex forms of the signal prior based on instantaneous-phase-corrected total variation (iPCTV) or low-rankness have shown to yield superior gap reconstruction in audio spectrograms, outperforming magnitude-only and deep-prior methods (Balušík et al., 26 Jan 2026, Masuyama et al., 2019, Masuyama et al., 2019). These priors ensure horizontal coherence for sinusoidal structures, thereby addressing the limitations of energy loss and incoherence introduced by phase-unaware methods.

Speech Enhancement:

• Bayesian STSA estimators incorporate a circular phase prior (e.g., von Mises), leading to MMSE-optimal estimators when prior phase information or estimates are available (Samui et al., 2022, Dionelis et al., 2017). Circular statistical models embedded into modulation-domain Kalman filters yield improved phase and amplitude estimation, effectively jointly denoising both quantities.
• Deep learning systems such as PHASEN implement architectural phase-awareness with distinct phase and amplitude streams, frequency transformation blocks encoding harmonic structure, and attention mechanisms between streams, achieving notable improvements in SDR, PESQ, and MOS benchmarks over magnitude-only and single-stream baselines (Yin et al., 2019).

Phase Retrieval:

• Incorporation of denoiser-based, generative, and semi-algebraic priors enables reconstruction of signals from magnitude-only measurements, overcoming limitations of traditional convex or sparsity-based methods. The generalization performance, particularly to out-of-distribution images or signals, is substantially improved when the prior leverages phase structure and is embedded in alternating (phase-estimate/signal-sample) frameworks (Agrawal et al., 2022, Hand et al., 2018, Bendory et al., 2023).

4. Neural Architectures and Learning-Based Phase-Aware Priors

Recent neural network architectures accentuate phase priors by explicit conditioning or multi-stage workflows:

• SP-NSPP: Stage-wise and prior-aware neural architectures first generate a coarse phase prior from the magnitude spectrum, which is then refined in a subsequent network, with adversarial and continuity losses ensuring detailed and smooth phase predictions. This dramatically reduces both instantaneous phase distortion and parameter count, and approximately doubles the efficiency relative to single-stage or no-prior networks (Liu et al., 2024).
• PHASEN: Implements dual amplitude/phase streams with inter-stream gating; frequency transformation blocks learn harmonic correlations crucial for phase recovery in speech. Such explicit architectural phase-awareness is critical for preserving detailed T-F structure (Yin et al., 2019).

These data-driven priors can generalize to a wide array of time-frequency phase recovery contexts, including audio, biomedical, and radar applications.

5. Experimental Validation and Impact

Across domains, phase-aware priors consistently lead to marked improvements:

Task	Phase-Aware Approach	Main Benefit over Baselines	Cited Paper
Phase retrieval (imaging)	APLS/denoiser-implicit	Outperforms prDeep on OOD images	(Agrawal et al., 2022)
Audio inpainting (TF domain)	iPCTV prior	Superior SNR/ODG, less perceptual artifact	(Balušík et al., 26 Jan 2026)
Speech enhancement	Circular prior Kalman Filter	Superior perceptual metrics, joint denoise	(Dionelis et al., 2017)
Source separation (audio)	iPCLR/phase-aware TV	Higher SNR/OPS, reduced artefacts	(Masuyama et al., 2019)
Speech phase prediction (NN)	SP-NSPP	Lower phase distortion, higher SNR/MOS	(Liu et al., 2024)
Music HPSS	iPC-TV convex formulation	Outperforms NMF/mask-based HPSS	(Masuyama et al., 2019)

Empirical results indicate that phase-aware priors not only improve reconstruction fidelity (PSNR, SDR, SNR), but lead to tangible perceptual gains (MOS, ODG), robustness to out-of-distribution data, and reduced sample complexity.

6. Connections to Generative, Statistical, and Convex Paradigms

The landscape of phase-aware priors incorporates and generalizes multiple statistical and learning paradigms:

• Statistical/Predictive Frameworks: Circular statistics (von Mises, wrapped normal), AR-models, and Markovian priors regulate phase uncertainty and temporal coherence in estimation (Dionelis et al., 2017).
• Generative Model Priors: Deep generative networks (VAE, DCGAN) create implicit phase/structure constraints, enabling recoverability at optimal sample complexity and improved task-specific performance (Hand et al., 2018).
• Convex Regularisation: Nuclear norm (iPCLR), TV norms, and other convex penalties can be reinterpreted as phase-aware when modified by instantaneous frequency–corrected operators, yielding tractable optimization with explicit phase continuity enforcement (Masuyama et al., 2019, Balušík et al., 26 Jan 2026).
• Implicit/Plug-and-Play Priors: Score-based connections between denoisers and log-density gradients allow the use of very general learned priors without explicit likelihood forms, as in Langevin sampling approaches (Agrawal et al., 2022).

Phase-awareness thus emerges as a unifying theme that brings together analytic, statistical, convex, and deep-learning–based approaches, and is essential in achieving high-quality, robust signal reconstructions in challenging underdetermined or ill-posed settings.