Weighted Sum Mean Square Error (WSMSE)

• Weighted Sum Mean Square Error (WSMSE) is a framework that aggregates weighted mean square errors to optimize trade-offs among fairness, throughput, and energy efficiency.
• It employs methods like alternating optimization, majorization theory, water-filling, and convex relaxations to efficiently handle nonconvex design problems in multi-user systems.
• WSMSE is applied in wireless MIMO, sensor networks, pilot design, and integrated sensing systems, achieving notable SNR and MSE gains as demonstrated through extensive simulations.

Weighted Sum Mean Square Error (WSMSE) is a fundamental design criterion in multi-user, multi-stream signal processing and communications. It serves as a unifying framework for system optimization problems in wireless MIMO, network MIMO, distributed estimation, sensor networks, pilot design, and transceiver optimization. Through a weighted aggregation of the mean square errors experienced by differing users, streams, or sensor measurements, WSMSE enables trade-offs among conflicting objectives such as fairness, throughput, energy efficiency, and robustness. This article reviews its mathematical foundations, key methodologies, prominent algorithmic strategies, and its central role in contemporary communication and estimation systems.

1. Mathematical Definition and Problem Structure

WSMSE quantifies the aggregate performance of vector estimators or multiuser demodulators by assigning explicit weights to the individual MSE terms associated with each stream, symbol, or sensor. For a system where $K$ component errors $e_k$ are tracked (often corresponding to users, streams, or nodes), the general form is:

$$J_{\text{WSMSE}} = \sum_{k=1}^K w_k \, \mathbb{E}[|e_k|^2]$$

where $w_k$ is the weight for component $k$. This approach scalarizes the vector-valued mean square error into an overall objective, enabling convex or nonconvex optimization over design variables such as precoders, receive filters, or pilot sequences.

Extensions of this framework move beyond scalar weights on diagonals, incorporating weighting matrices and matrix-monotone functions over the full error covariance, as introduced in the matrix-field WSMSE model (Xing et al., 2013, Xing et al., 2016). In these extended models,

$$\Psi(G, F) = \sum_{k=1}^K W_k^H \Phi_{\rm MSE}(G, F) W_k + \Xi$$

with $W_k$ denoting (possibly non-diagonal) weighting matrices and $\Xi$ a positive semi-definite offset.

2. Methodologies for WSMSE Optimization

2.1 Alternating Optimization and Block Coordinate Descent

Most practical WSMSE minimization problems are nonconvex and feature coupled design variables. Algorithmic designs exploit alternating optimization, updating one group of variables (e.g., relay precoding matrices, user receive filters) while holding the others fixed, as in (Wang et al., 2012, Kaviani et al., 2013):

• Update precoders by solving quadratic programs or applying Karush–Kuhn–Tucker conditions.
• Update receive filters using closed-form MMSE solutions given fixed precoders.
• Repeat until cost improvement becomes negligible.

2.2 Lagrange Multipliers and Majorization Theory

Analytical approaches use the Lagrange multiplier method to incorporate power or linear constraints, derive necessary optimality conditions, and construct SVD-based precoder structures (Xing et al., 2016). Majorization theory, exploiting Schur-convexity or concavity properties of MSE functions, identifies when fairness or max-min criteria can be achieved by eigenvalue ordering.

• Limitation: Permutation ambiguity and turning-off effects can arise, meaning solution structure depends sensitively on the weight ordering and the eigenchannel assignments.

2.3 Matrix-Monotone Functions and Water-Filling Solutions

Matrix-field WSMSE models allow optimization over more general criteria, such as trace or log-determinant functions of the error covariance, yielding water-filling-type power allocation rules over the eigenmodes (Xing et al., 2013). For instance,

• Trace minimization recovers conventional sum-MSE designs.
• Log-determinant minimization corresponds to capacity-maximizing transceiver structures.

2.4 Geometric Programming and Convex Relaxation

In the context of multiuser downlink and network MIMO, WSMSE problems involving power allocation are transformed into geometric programs, which are convex after logarithmic variable mapping (Bogale et al., 2013). Semi-definite programming (SDP) relaxations are deployed for pilot optimization and distributed sensor placement (Bogale et al., 2014, Vahidian et al., 2019).

3. Asymptotic Analysis and Regime-Dependent Behavior

WSMSE-optimized designs exhibit distinct regime behavior:

• Low SNR: Performance governed by noise, with optimal designs derivable via eigenvalue maximization or power iteration methods (Wang et al., 2012).
• High SNR: Interference dominates, zero-forcing (ZF) precoding or interference alignment techniques become optimal. Initialization sensitivity arises, and algorithms should exploit regime-adapted initial points for efficiency (Wang et al., 2012, Kaviani et al., 2013).

These results enable system designers to anticipate limiting behavior and select algorithms, weights, and initialization strategies best suited to the operating region.

4. Impact of Structured Weights and Generalized Error Models

The introduction of non-scalar, matrix-valued weights greatly expands the WSMSE framework's applicability:

• Designs can balance error covariances across streams, enforce min–max objectives for fairness, or jointly optimize for BER and capacity (Xing et al., 2016).
• The extended model accommodates nonlinear transceiver architectures (Tomlinson-Harashima Precoding, Decision-Feedback Equalization), distributed M-estimation with heterogeneity (Gu et al., 2022), and integrated sensing and communications where mutual information-based weighted error criteria unify multi-objective design (Peng et al., 2023).

5. Applications in Communication and Estimation Systems

5.1 Wireless MIMO Switching and Relay Networks

Unified iterative algorithms achieve joint relay precoder and receiver design for MIMO switching (Wang et al., 2012). By incorporating self-interference cancellation (physical-layer network coding), designs realize marked SNR and throughput improvements over baselines, with simulation showing up to 6 dB MSE gains due to PNC.

5.2 Network MIMO and Distributed Base Stations

Diagonalization and interference alignment algorithms provide tractable solutions to WSMSE minimization in multi-cell scenarios with partial base station cooperation (Kaviani et al., 2013), outperforming suboptimal sum-rate maximization approaches in cases where stream count is limited.

5.3 Distributed Estimation and Sensor Networks

Weighted diffusion LMP algorithms adaptively update sensor weights using steepest descent recursions, facilitating robust estimation even under non-uniform noise and sensor reliability (Zayyani et al., 2016). Tight bounds on global WSMSE in distributed estimation are derived by constraining input distribution within a KL-ball of a Gaussian reference, yielding minimax robust estimators and new engineering guidelines for sensor fusion (Fauß et al., 2019).

5.4 Pilot Design in Massive MIMO

Joint WSMSE formulations for pilot optimization, employing MMSE and generalized Rayleigh quotient tools, significantly mitigate pilot contamination, especially when pilot resources are insufficient (N < K) (Bogale et al., 2014).

5.5 Coreset Construction and Efficient Sampling

WSMSE provides rigorous error bounds for efficient mean estimation using limited, weighted samples in graph and sensor selection—enabling fast-convergent, cost-aware greedy algorithms that outperform clustering and random selection (Vahidian et al., 2019).

5.6 Integrated Sensing and Communication

Weighted MMSE transceiver design for ISAC maximizes a joint mutual-information criterion for communication and sensing, allowing dynamic trade-off via adjustable weighting parameters. Closed-form beamforming solutions and numerical experiments highlight the adaptability and superiority of WMMSE-based ISAC over conventional approaches (Peng et al., 2023).

5.7 Movable Antenna Optimization

Weighted sum-rate maximization in movable antenna systems is tractably reformulated via WMMSE, solved by block coordinate descent and majorization-minimization, with further acceleration through planar movement mode constraints, yielding practical computation–performance trade-offs (Feng et al., 15 Apr 2024).

6. Numerical Performance and Practical Considerations

Extensive simulations in numerous works demonstrate the effectiveness of WSMSE-based optimizations. For example:

• In MIMO switching, WSMSE minimization with PNC achieves SNR improvements of ~6dB and monotonic decrease of MSE with increasing SNR (Wang et al., 2012).
• DMMSE designs in network MIMO deliver highest per-cell sum rates when stream counts are limited (Kaviani et al., 2013).
• Adaptive weighting in sensor networks provides lower mean square deviation and faster convergence compared to uniform-weight diffusion LMP, especially in environments with highly non-uniform noise (Zayyani et al., 2016).
• SCGIGA coreset algorithms consistently outperform random and clustering baselines in convergence rate and cost efficiency (Vahidian et al., 2019).
• In movable antenna systems, imposing planar constraints reduces computation time by 30% with minor performance loss (Feng et al., 15 Apr 2024).

7. Summary Table of Key Concepts and Algorithms

Concept	Key Equation / Principle	Typical Application
Scalar WSMSE	$J = \sum_k w_k \, \mathbb{E}[|e_k|^2]$	MIMO, pilot, sensor net
Matrix-field WSMSE	$\Psi = \sum_k W_k^H \Phi W_k + \Xi$	Nonlinear, fairness
Alternating Optimization	Iteratively update precoder/filters	Relay, MIMO switching
Majorization/Water-Filling	SVD structure, eigenordering	Capacity, min–max MSE
Adaptive Sensor Weighting	$\alpha_k(n+1) = \alpha_k(n) - \mu \cdot \nabla$	Distributed estimation
Planar Movement Mode	$t_m^{n+1} = \Pi_{\mathcal{C}_m^t}( ... )$	Movable antennas

8. Conclusion

Weighted Sum Mean Square Error (WSMSE) is a versatile optimization criterion intrinsic to advanced signal processing, communication theory, and distributed estimation. It reconciles disparate objectives—efficiency, fairness, robustness, sparsity, resource allocation—across a spectrum of physical and computational resource constraints. The evolution of WSMSE methodology, from scalar stream weights to full matrix-field and multi-objective formulations, reflects the complexity and diversity of modern integrated systems. Algorithmic advances, such as alternating optimization, majorization, matrix-monotone functionals, and geometric programming, continue to expand the scope and effectiveness of WSMSE approaches, driving innovation in wireless, sensing, and learning systems.