Weighted MMSE Alternating Optimization

• The paper demonstrates transforming complex nonconvex problems into block-convex subproblems using weighted MMSE surrogates.
• It details cyclic optimization steps that yield closed or semi-closed form updates, ensuring convergence to local KKT solutions.
• The work highlights applications in wireless MIMO, ISAC, and low-rank recovery, alongside rigorous theoretical recovery guarantees.

Weighted Minimum Mean-Square Error-Based Alternating Optimization (AO) algorithms constitute a broad class of block-coordinate descent methods widely used for nonconvex matrix and vector optimization, especially in communications, signal processing, and statistical learning. They exploit the equivalence between information theoretic or matrix approximation objectives and weighted MMSE (WMMSE) surrogates, enabling the splitting of a difficult joint problem into tractable subproblems that can be solved in closed or semi-closed form and alternately iterated to a stationary point. This article presents a structured account of the methodology and theoretical guarantees underpinning WMMSE-based AO, focusing on rigorous published accounts in wireless MIMO, integrated sensing and communication, and weighted low-rank approximation.

1. Fundamental Problem Formulation

Many multiuser wireless and signal processing tasks require optimizing an objective function (such as sum-rate, spectral efficiency, or estimation error) over a collection of matrices, typically under structural (e.g., low-rank, constant-modulus, power, or sparsity) constraints. In weighted low-rank matrix recovery, the canonical problem is

$\min_{X,Y} \|W \odot (M - XY^\top)\|_F^2$

for a target matrix $M$, deterministic noise $N$, and non-negative weight $W$. In MIMO transceiver design, cellular communications, or ISAC, the task is often to maximize the weighted sum-rate subject to power constraints and hardware-imposed constraints on beamformers or filters. Direct optimization is generally nonconvex and computationally intractable; the WMMSE approach provides structured blockwise decompositions (Li et al., 2016, Wang et al., 9 Jun 2025, Gan et al., 20 Nov 2025, Jiang et al., 2021, Du et al., 2019, Gao et al., 23 Oct 2025).

Modeling assumptions commonly include incoherence of ground-truth subspaces, a spectral gap for $W$, and nondegeneracy (RIP-style conditions) to guarantee convergence or error decay properties (Li et al., 2016, Song et al., 2023). In practical communication models, constraints may encode power budgets, unit-modulus (constant envelope) requirements, or per-device hardware limits (Wang et al., 9 Jun 2025, Li, 4 Oct 2025).

2. WMMSE Transformation and Alternating Optimization Principles

The weighted MMSE equivalence is pivotal: many metrics, especially log-determinant rate functions, can be algebraically transformed into a minimization over auxiliary MMSE-related variables. This enables the original nonconvex program to be expressed as a multi-block objective convex in each block, though not jointly. For example, the classical sum-rate maximization in a multiuser MIMO downlink can be recast as (Gao et al., 23 Oct 2025)

$$\max_{\mathbf{V}} \sum_k \alpha_k R_k(\mathbf{V}) \iff \min_{\{\mathbf{U},\mathbf{W},\mathbf{V}\}} \sum_k \alpha_k[\mathrm{Tr}(\mathbf{W}_k \mathbf{E}_k) - \log\det \mathbf{W}_k]$$

subject to power constraints, where $\mathbf{U}_k$ are equalizers, $\mathbf{W}_k$ positive definite weights, and $\mathbf{E}_k$ the MSE matrices.

In integrated sensing and communication (ISAC), for instance, the posterior Cramér–Rao bound (PCRB) optimal beamforming problem with a communication rate requirement is transformed into a constrained maximization of a trace criterion under a WMMSE surrogate constraint. For the hybrid beamforming case, the AO splits into blocks associated with the transmit/receive analog beamformers, digital beamformers, and WMMSE weights, with associated (often closed-form) solutions for each (Wang et al., 9 Jun 2025).

3. Structure and Detailed Steps of the WMMSE-AO Algorithm

The core of the AO algorithm consists of cyclically optimizing each group of variables (block) while holding the others fixed. The critical substeps and their update structures are summarized below:

Block Variable	Subproblem Structure	Solution Type
MSE Equalizer/Decoder ($Q$)	MMSE estimation, given precoders	Wiener filter (closed form)
MSE Weight ($U$)	Inverse error covariance	Closed form
Precoder/Beamformer	QCQP/Water-filling/FPP-SCA	Closed or semi-closed
Analog Beamformer	Constant-modulus QCQP	FPP-SCA, block coordinate

For example, in the PCRB-constrained ISAC problem with hybrid arrays, the $Q$ (decoder) and $U$ (weight) blocks admit closed-form updates:

• $$Q^{(k+1)} = J^{-1} H V_{\mathrm{RF}}^{(k)} V_{\mathrm{BB}}^{(k)}$$
• $U^{(k+1)} = (E)^{-1}$

The digital covariance $R_{\mathrm{BB}}$ update solves a convex program (generalized water-filling), and the analog beamformer $V_{\mathrm{RF}}$ employs feasible-point-pursuit successive convex approximation (FPP–SCA): quadratic objectives and constraints are linearized and solved via second-order cone programming, iteratively refining the solution (Wang et al., 9 Jun 2025, Gao et al., 23 Oct 2025, Du et al., 2019). Constant-modulus constraints are enforced via quadratic equality constraints or Riemannian manifold optimization (Du et al., 2019).

A typical iteration sequence:

1. Solve for $Q$, $U$ given $V_{\mathrm{RF}}$, $R_{\mathrm{BB}}$, $W_{\mathrm{RF}}$.
2. Update $V_{\mathrm{RF}}$ via convexified QCQP (FPP–SCA).
3. Update $R_{\mathrm{BB}}$ (water-filling or semi-closed form).
4. Update $W_{\mathrm{RF}}$ (closed-form or block coordinate, e.g., Gauss–Seidel).
5. Evaluate objective and check convergence.

4. Theoretical Guarantees and Convergence

Provided each subproblem is solved exactly (or to sufficient accuracy) and all block variables are updated in a cyclic fashion, the WMMSE-based AO algorithm produces a non-increasing sequence of objective values. Under mild compactness and continuity conditions, the sequence of iterates converges to a stationary point—a local KKT solution—of the original nonconvex problem (Li et al., 2016, Song et al., 2023, Wang et al., 9 Jun 2025, Gan et al., 20 Nov 2025, Gao et al., 23 Oct 2025, Jiang et al., 2021).

In weighted low-rank approximation with non-binary deterministic weights, provable global recovery guarantees can be obtained under incoherence, spectral gap, and nondegeneracy conditions. The reconstruction error typically decays at a geometric rate proportional to the noise norm projected onto the weight structure, plus an exponentially vanishing additive term in the number of AO rounds. Advanced variants employing row "clipping" or semidefinite whitening steps enable further relaxation of spectral gap requirements and enhanced robustness (Li et al., 2016, Song et al., 2023).

For large-scale (massive MIMO) systems, first-order variants and sketching enable computational acceleration without significant loss in convergence accuracy (Song et al., 2023, Gao et al., 23 Oct 2025).

5. Extensions and Variants

Recent developments in WMMSE-based AO encompass several directions:

• Hybrid analog-digital beamforming: Riemannian conjugate-gradient optimization for analog precoders subject to constant-modulus constraints paired with closed-form WMMSE solvers for digital subblocks (Du et al., 2019, Wang et al., 9 Jun 2025).
• Penalty and element-wise optimization for deployment: In cell-free and pinching antenna systems, nonconvex deployment constraints are handled by penalty-based AO and element-wise searches (Li, 4 Oct 2025).
• Operator-free probability tools: Large-system deterministic equivalents replace explicit sample-based expectations in cell-free multi-RIS optimization, dramatically reducing computation (Pan et al., 2024).
• Accelerated first-order schemes: To reduce the high per-iteration complexity of $(M\times M)$ matrix inversions, projected-gradient (PGD) steps, Nesterov-style extrapolation, and warm start with unweighted MSE minimization are employed for scaling to very large dimensions (Gao et al., 23 Oct 2025).
• Matrix completion and low-rank recovery: The AO–WMMSE framework is one of the few with rigorous globally convergent recovery bounds for arbitrary noise and deterministic nonbinary weights (Li et al., 2016, Song et al., 2023).

6. Representative Applications

WMMSE-based AO algorithms are deployed across several domains:

• Wireless multi-user beamforming: Downlink/uplink sum-rate maximization, hybrid beamforming, and cell-free MIMO network design, with explicit communication-rate constraints and fronthaul/RIS-assisted architectures (Gan et al., 20 Nov 2025, Pan et al., 2024, Wang et al., 9 Jun 2025, Jiang et al., 2021).
• Integrated Sensing and Communication (ISAC): Simultaneous minimization of sensing mean-square error (PCRB) and maximization of communications rate subject to practical hardware constraints (Wang et al., 9 Jun 2025).
• Relay/MIMO switching: Joint transceiver and relay design via weighted sum-MSE alternating optimization, with asymptotic optimality at both high and low SNR (Wang et al., 2012).
• Matrix approximation and completion: Weighted low-rank recovery from incomplete, noisy, or weighted observations (Li et al., 2016, Song et al., 2023).

7. Complexity, Initialization, and Implementation Considerations

Per-iteration complexity is typically dominated by matrix inversions or large-scale QCQP/SOCP solvers in the most challenging block updates (e.g., analog precoder, digital covariance). Advanced implementations utilize approximate solvers, random sketching, and block-parallelization to achieve near-linear or sub-cubic scaling (Song et al., 2023, Gao et al., 23 Oct 2025). Reliable initialization—via SVD, random orthogonalization, or domain-specific closed-form approximations—is emphasized to avoid poor local minima or slow convergence (Li et al., 2016).

Stopping criteria usually involve the relative change of the WMMSE objective or primary metric. Practical convergence is typically attained within 10–30 iterations for most applications (Du et al., 2019, Li, 4 Oct 2025, Wang et al., 9 Jun 2025).

• Li, Liang, Risteski. "Recovery guarantee of weighted low-rank approximation via alternating minimization" (Li et al., 2016)
• Yang et al. "Hybrid Beamforming Optimization for MIMO ISAC Exploiting Prior Information: A PCRB-based Approach" (Wang et al., 9 Jun 2025)
• Wang et al. "Wireless MIMO Switching: Weighted Sum Mean Square Error and Sum Rate Optimization" (Wang et al., 2012)
• Pan et al. "Pinching Antenna Systems (PASS) for Cell-Free Communications" (Li, 4 Oct 2025)
• Liu et al. "Revealing computation-communication trade-off in Segmented Pinching Antenna System (PASS)" (Gan et al., 20 Nov 2025)
• Zhao et al. "An Accelerated Mixed Weighted-Unweighted MMSE Approach for MU-MIMO Beamforming" (Gao et al., 23 Oct 2025)
• Wang et al. "Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation" (Song et al., 2023)
• Zhou et al. "WMMSE-Based Joint Transceiver Design for Multi-RIS Assisted Cell-free Networks Using Hybrid CSI" (Pan et al., 2024)
• Zhang et al. "Joint Transmit Precoding and Reflect Beamforming Design for IRS-Assisted MIMO Cognitive Radio Systems" (Jiang et al., 2021)
• Shi et al. "Weighted Spectral Efficiency Optimization for Hybrid Beamforming in Multiuser Massive MIMO-OFDM Systems" (Du et al., 2019)