linear-mmse amplitude-phase estimation

This work addresses the instability of maximum a posteriori (MAP) estimation in two-dimensional blind deconvolution—a canonical blind inverse problem—where non-convexity of the objective function and non-identifiability of solutions often lead to unreliable performance. Under fully controlled experimental conditions, the study systematically compares MAP algorithms employing exact priors against a structurally near-optimal Tikhonov-regularized linear minimum mean square error (LMMSE) estimator. The results demonstrate that LMMSE not only significantly outperforms MAP methods, which require meticulous hyperparameter tuning, but also serves as a robust initialization strategy for MAP optimization, effectively reducing its sensitivity to regularization parameters and enhancing both stability and reconstruction quality. These findings establish LMMSE as playing a dual role in blind deconvolution: a high-performance baseline and a reliable starting point for iterative refinement.

Channel estimation in one-bit quantized massive MIMO systems suffers from severe accuracy degradation due to coarse quantization. Method: Breaking the limitations of conventional Bussgang linear approximation, this work establishes the first rigorous minimum mean-square error (MMSE) channel estimation framework grounded in orthant probabilities of multivariate normal distributions. It integrates Bussgang decomposition, statistical channel modeling, and asymptotic analysis to enable efficient MMSE computation. Contribution/Results: Theoretically, we derive necessary and sufficient conditions for the optimality of the Bussgang-linear MMSE (BLMMSE) estimator and obtain a closed-form, analytically tractable MMSE estimator in the high-dimensional regime—revealing the fundamental gap between linear approximation and global optimality. Experiments under typical massive MIMO configurations demonstrate substantial estimation error reduction. Moreover, we explicitly characterize the applicability boundary of BLMMSE and quantify its performance loss as a function of channel correlation structure and pilot design.

To address the sequential determination of model order and insufficient robustness in high-dimensional line spectral estimation (LSE), this paper proposes the Bilinear Generalized LSE (BiG-LSE) method. BiG-LSE iteratively approximates the nonlinear observation model via Taylor expansion into a bilinear form and employs Expectation Propagation (EP) for joint Bayesian inference of frequencies, model order, and noise variance. It is the first LSE framework enabling fully automatic order selection and rigorous uncertainty quantification. To enhance sequential estimation accuracy, it adopts the von Mises distribution to model frequency priors. Furthermore, low-rank compression of posterior log-density messages significantly reduces computational complexity. Simulation and real-world experiments demonstrate that BiG-LSE achieves estimation accuracy comparable to state-of-the-art methods while exhibiting superior robustness and practicality under high-dimensional nonlinear measurements.

This paper addresses the computational burden of hyperparameter optimization in regularized estimation for system identification. To this end, we propose two novel parameter-free estimators: a generalized Bayesian estimator and a closed-form biased estimator. Theoretically, we construct, for the first time, a family of hyperparameter-free estimators whose excess mean squared error (MSE) strictly matches that of the optimal empirical Bayes regularized estimator. Methodologically, leveraging large-sample asymptotics, ridge regression modeling, and bias–variance trade-off analysis, we derive analytical closed-form solutions—bypassing iterative optimization entirely. Numerical experiments demonstrate that the proposed estimators achieve MSE performance comparable to the best empirical Bayes method, while reducing computational time significantly. This makes them particularly suitable for real-time applications and resource-constrained environments.

Parameter estimation under missing data is doubly sensitive to both misspecification of the underlying data model and deviations from standard missingness mechanisms (MCAR/MAR/MNAR) or Huber-type contamination. Method: This paper proposes a robust M-estimation framework grounded in the Maximum Mean Discrepancy (MMD), leveraging kernel embeddings and functional analysis tools. It avoids explicit specification of either the missingness mechanism or the full-data distribution. Contribution/Results: The method achieves joint robustness against both types of misspecification—theoretically guaranteeing strong consistency and asymptotic normality under MCAR. It yields a decomposable, explicit error bound that cleanly separates model misspecification error from missingness-induced bias. Moreover, it maintains controlled estimation error under MNAR and Huber contamination. By circumventing stringent modeling assumptions, the approach significantly enhances robustness and reliability in practical applications.

This work addresses the lack of theoretical guarantees for plug-and-play (PnP) methods in ill-posed linear inverse problems when the denoiser is not consistent with the physical forward model. The authors propose a PnP forward–backward splitting algorithm based on the minimum mean square error (MMSE) estimator, incorporating a linear operator matched to the covariance structure of the observation noise, and extend it to neural network–parameterized denoisers. They establish, for the first time, that denoisers cannot be designed independently of the forward model. Under mild assumptions, they provide recovery guarantees—both pointwise and in Wasserstein distance—for MMSE and neural network denoisers in the presence of Gaussian noise with non-diagonal covariance, and rigorously prove the convergence and reconstruction performance of the proposed algorithm.

This work addresses the implicit regularization structure and convergence properties of plug-and-play (PnP) methods employing minimum mean squared error (MMSE) denoisers. We establish, for the first time, that MMSE-PnP is equivalent to explicit optimization with a 1-weakly convex regularizer given by the upper Moreau envelope of the negative log marginal density. Building on this insight, we derive the first non-asymptotic sublinear convergence rate for PnP gradient descent—removing restrictive asymptotic or strong convexity assumptions required in prior analyses. Our approach integrates weak convexity theory, proximal operator analysis, and implicit regularization modeling. We validate the theoretical convergence behavior and practical efficacy through synthetic 1D experiments and real-world inverse problems, including image deblurring and CT reconstruction. Key contributions include: (i) characterizing the precise implicit regularizer induced by MMSE denoisers; (ii) providing the first non-asymptotic convergence guarantee for PnP gradient descent; and (iii) demonstrating tight alignment between theory and empirical performance.

The selection of the regularization parameter in low-rank minimum mean square error (MMSE) filters typically relies on empirical tuning or cross-validation, lacking rigorous theoretical guidance. Method: This paper establishes, for the first time, an analytical mapping between the regularization parameter and the filter rank, and proposes a Kronecker-structured automatic parameter selection method. Within the Bayesian MMSE framework, it analytically models the regularization term to enable adaptive, closed-form parameter determination—eliminating the need for grid search or retraining. Contribution/Results: Theoretical analysis uncovers the intrinsic coupling between regularization strength and effective rank. Simulations demonstrate that the proposed method significantly improves estimation accuracy across varying signal-to-noise ratios (SNRs), achieving average SNR gains of 1.8–3.2 dB over conventional Tikhonov regularization and cross-validation approaches, while maintaining high computational efficiency and robustness.

This paper addresses the stable recovery problem in phase retrieval under Poisson and heavy-tailed noise. We develop the first signal-energy-adaptive unified analysis framework for both nonconvex and convex least-squares estimators: at high signal-to-noise ratio (high energy), Poisson noise is modeled as sub-exponential; at low SNR (low energy), it is treated as heavy-tailed—thereby bridging the theoretical gap between these two noise regimes. Leveraging multiplier inequalities, empirical process theory, and random matrix analysis jointly, we derive tight risk bounds for the first time: the high-energy regime achieves the minimax-optimal rate $O(sqrt{n/m})$, while the low-energy regime attains $O(|x|^{2-1/4}(n/m)^{1/4})$; both rates remain optimal under heavy-tailed noise. The framework naturally extends to sparse phase retrieval and low-rank matrix reconstruction.

This work addresses the shrinkage bias of ℓ₁ regularization and the algorithmic instability of ℓ₀ approximation in sparse signal recovery from noisy linear measurements. To overcome these limitations, the authors employ log-sum non-convex regularization as a surrogate for the ℓ₀ norm and introduce an adaptive smoothing mechanism to ensure continuity of the proximal operator. They innovatively extend state evolution (SE) theory to this non-convex framework, enabling, for the first time, accurate prediction of phase transition thresholds and mean squared error. Combining AMP and ADMM algorithms, experiments demonstrate that in the noiseless case, ADMM’s success boundary aligns closely with SE predictions, while in noisy settings, AMP tightly tracks the SE trajectory. Moreover, the log-sum penalty significantly outperforms ℓ₁ regularization under low sparsity or high sampling rates.