Enhanced WMMSE (eWMMSE) Methods Overview

• Enhanced WMMSE is a family of algorithms that extend the classical WMMSE reformulation to facilitate multi-channel power allocation with QoS and per-user constraints.
• It employs computational enhancements such as projected gradient descent, inverse-free updates, and quadratic majorization to reduce complexity and improve real-time performance.
• Learning-based variants introduce graph-structured unfolding and trainable parameters to achieve low-latency execution and adapt to dynamic wireless network conditions.

Enhanced WMMSE (eWMMSE) denotes a family of weighted minimum mean-square error methods that retain the classical rate–MSE reformulation but extend or modify it to handle additional constraints, multi-channel structure, or lower-complexity computational realizations. In the narrowest explicit usage, papers use the name eWMMSE for multi-channel joint channel-and-power allocation, including per-user total-power constraints and, in one formulation, minimum-rate QoS constraints (Chen et al., 4 Jun 2025, Chen et al., 8 Sep 2025). In a broader and commonly implied sense, closely related works present “enhanced WMMSE” variants without using the exact acronym, for example by eliminating matrix inversions, replacing exact beamformer subproblem solutions by projected-gradient updates, or unfolding WMMSE into trainable architectures for fixed-latency execution (Pellaco et al., 2020, Pellaco et al., 2022).

1. Classical foundation and the scope of enhancement

Classical WMMSE addresses weighted sum-rate maximization by introducing receiver variables and MSE weights so that a nonconvex rate problem can be rewritten as a blockwise-convex weighted-MSE problem. In the downlink MISO formulation with beamformers $\boldsymbol v_i$, receive coefficients $u_i$, and weights $w_i$, the weighted sum-rate objective is

$$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$

with

$$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$

and the equivalent WMMSE surrogate is

$$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$

The standard iteration updates $u_i$, $w_i$, and then $\boldsymbol v_i$ through a Lagrangian-based beamformer step involving $(\boldsymbol A+\mu \boldsymbol I)^{-1}$, where $u_i$0 is chosen to satisfy the power constraint (Pellaco et al., 2020).

The principal computational burden of classical WMMSE lies in the transmit update. In the Shi-style implementation highlighted in the unfolding literature, one iteration requires a matrix inversion, an eigendecomposition, and a bisection search over the power-control multiplier $u_i$1 (Pellaco et al., 2020). In the MU-MIMO setting, inverse-based updates also appear in the receive filter and weight blocks, and these operations are described as implementation-hostile because they are expensive, serial, hard to parallelize, and inconvenient for deep unfolding and low-latency hardware (Pellaco et al., 2022).

Theoretically, later comparative work places WMMSE inside a broader algorithmic taxonomy. In particular, WMMSE is shown to be a special case of weighted sum-rate maximization via fractional programming, and specific WMMSE and WSR-FP instances can be mapped exactly to a minorization-maximization formulation, WSR-MM (Zhang et al., 2023). This suggests that “enhancement” is not tied to a single canonical modification: it may target the surrogate function, the equivalent transform, the constraint treatment, or the numerical realization of the transmit update.

2. Named eWMMSE in multi-channel resource allocation

The most explicit use of the term eWMMSE appears in multi-channel wireless-network resource allocation. In these formulations, the system has $u_i$2 transceiver pairs and $u_i$3 orthogonal channels, each pair may access multiple channels simultaneously, and the original joint channel-and-power allocation problem is reduced to a continuous power-allocation problem by identifying channel occupancy with whether $u_i$4 (Chen et al., 4 Jun 2025). The received signal on channel $u_i$5 is written as

$u_i$6

with rate

$u_i$7

and the simplified optimization becomes

$u_i$8

Introducing $u_i$9, receive coefficients $w_i$0, and weights $w_i$1, the paper rewrites the problem as

$w_i$2

with updates

$w_i$3

and

$w_i$4

Here $w_i$5 is the multiplier enforcing the total-power budget across channels (Chen et al., 4 Jun 2025).

A second, more constrained named eWMMSE formulation adds explicit per-user minimum-rate QoS constraints. The corresponding weighted sum-rate problem is

$w_i$6

and the paper states the rate–WMMSE identity as

$w_i$7

The transmit update acquires an additional QoS multiplier $w_i$8: $w_i$9 with projected-ascent update

$$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$0

while $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$1 is found by bisection so that the power constraint holds (Chen et al., 8 Sep 2025).

The two named eWMMSE variants therefore define a narrow technical meaning of the acronym: WMMSE extended to scalar multi-channel power allocation, with implicit channel assignment via zero/nonzero power, coupled per-user total-power constraints, and optionally explicit QoS constraints. In the QoS-constrained version, the paper states that if the original problem is feasible, eWMMSE is guaranteed to converge to a locally optimal solution, whereas if the problem is infeasible, it produces a suboptimal allocation that minimizes QoS constraint violations while maximizing the total achievable rate (Chen et al., 8 Sep 2025).

3. Computationally enhanced WMMSE: inverse-free and unfoldable realizations

A large strand of the literature enhances WMMSE by targeting its computational bottlenecks rather than altering the underlying rate–MSE principle. In the MISO downlink, one influential approach replaces the exact beamformer subproblem solution by projected gradient descent. For fixed $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$2, the beamformer subproblem is convex in $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$3, so the update is approximated by

$$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$4

with

$$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$5

and projection onto the transmit-power ball

$$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$6

This removes the matrix inversion, eigendecomposition, and bisection steps from the classical transmit update and yields complexity $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$7 for classical WMMSE versus $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$8 for the unfolded/projected-gradient version (Pellaco et al., 2020).

The same work then unfolds $$\begin{aligned} \max_{\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \log_2\!\bigl(1+\mathrm{SINR}_i\bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P, \end{aligned}$$9 outer WMMSE iterations into a neural architecture with $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$0 projected-gradient substeps per layer and trainable step sizes

$$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$1

Importantly, the trainable parameters are only the PGD step sizes; the paper explicitly states that it does not introduce momentum or Nesterov extrapolation (Pellaco et al., 2020). At $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$2 dB SNR with $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$3, the trainable unfolded WMMSE outperforms classical WMMSE for $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$4 and performs essentially equally for $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$5; at $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$6, both reach about $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$7 of the converged WMMSE sum-rate. At $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$8 dB, increasing the number of PGD steps from $$\mathrm{SINR}_i = \frac{|\boldsymbol h_i^H \boldsymbol v_i|^2} {\sum_{j=1,\, j\neq i}^{N} |\boldsymbol h_i^H \boldsymbol v_j|^2 + \sigma^2},$$9 to $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$0 restores competitiveness over a larger range of depths (Pellaco et al., 2020).

For general MU-MIMO, matrix-valued receive filters and weights make the inverse-removal problem harder. A later matrix-inverse-free WMMSE variant addresses this by combining gradient descent for $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$1 and $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$2 with Newton–Schulz iteration for $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$3: $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$4 The paper describes this as the first MU-MIMO WMMSE variant free from matrix inverses and other non-parallelizable operations, proves convergence to a stationary point of the original WSR problem, and states that “matrix-inverse-free” means that no direct inverse operation is executed, not that inversion is irrelevant mathematically (Pellaco et al., 2022). In fully loaded scenarios, unfolded inverse-free configurations such as $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$5 and $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$6 reach about $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$7 of the WMMSE-at-convergence WSR, while IAIDNN reaches about $$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$8 in the same regime (Pellaco et al., 2022).

These inverse-free and projected-gradient formulations establish a broad sense in which eWMMSE denotes a computational enhancement of WMMSE: the objective and block structure are preserved, but the dominant update is converted from inverse-heavy exact minimization into multiplication-dominated first-order or iterative-inverse primitives that are more compatible with GPUs, ASICs, FPGAs, and deep unfolding.

4. Acceleration, surrogate redesign, and reduced-complexity WSR updates

Another enhancement line starts from the observation that WMMSE, WSR-FP, and WSR-MM differ less than their literatures suggest. Comparative analysis shows that WMMSE is a special case of WSR-FP, and that specific WMMSE and WSR-FP instances coincide exactly with WSR-MM through a BCA-to-MM mapping (Zhang et al., 2023). This enables new transmit-block designs that target the same bottleneck as classical WMMSE but are derived through surrogate-function flexibility.

In that framework, WSR-MM+ and its BCA counterpart WSR-FP+ replace the exact transmit-block solution by an additional quadratic majorization. For the MIMO case, choosing

$$\begin{aligned} \min_{\boldsymbol u,\boldsymbol w,\boldsymbol V}\quad & \sum_{i=1}^{N} \alpha_i \bigl(w_i e_i - \log_2 w_i \bigr) \ \text{s.t.}\quad & \operatorname{Tr}(\boldsymbol V \boldsymbol V^H) \le P. \end{aligned}$$9

leads to the subproblem

$u_i$0

with solution

$u_i$1

For MISO, the analogous update is

$u_i$2

These updates eliminate repeated inverse or pseudo-inverse evaluations during multiplier search. The resulting MISO complexity drops from $u_i$3 for WSR-MM to $u_i$4 for WSR-MM+, while the MIMO complexity drops from $u_i$5 to $u_i$6 (Zhang et al., 2023). Although the paper does not name this eWMMSE, WSR-FP+ is explicitly the BCA counterpart of WSR-MM+, so it is naturally interpreted as an enhanced WMMSE-like alternating method.

A related acceleration perspective comes from fractional programming. The quadratic transform paper identifies WMMSE as a special case of QT for $u_i$7, proves a local objective-gap rate of $u_i$8 for conventional QT and its WMMSE specialization, and then shows that a nonhomogeneous QT exposes the method as projected gradient ascent, enabling Nesterov extrapolation. Under Lipschitz-gradient assumptions, the extrapolated QT satisfies

$u_i$9

yielding an $w_i$0 rate versus $w_i$1 for conventional QT/WMMSE (Shen et al., 2023). This supplies a rate-theoretic interpretation of accelerated eWMMSE-like variants.

A further complementary development is A-MMMSE, which compares directly against conventional WMMSE and its enhanced variant, reduced-WMMSE. A-MMMSE replaces the expensive precoder block by projected gradient descent and introduces a two-stage warm start that first solves the unweighted sum-MSE problem and then switches to weighted WMMSE. The paper states that A-MMMSE matches the WSR performance of both conventional WMMSE and reduced-WMMSE, achieves up to $w_i$2 speedup over WMMSE, up to $w_i$3 speedup over reduced-WMMSE in large-user scenarios, and an additional $w_i$4 GPU acceleration over reduced-WMMSE (Gao et al., 23 Oct 2025). This suggests that “enhancement” can also mean implementation-oriented acceleration around WMMSE-type fixed points rather than a new equivalent transform.

5. Learning-based enhanced WMMSE and graph-structured unfolding

Deep unfolding and graph neural parameterization have produced a distinct class of enhanced WMMSE methods in which a small number of trainable layers approximate or improve the behavior of many classical iterations. In UWMMSE for SISO power allocation, the WMMSE recursion is preserved but the $w_i$5-update is modulated by GNN outputs: $w_i$6

$w_i$7

The architecture is permutation equivariant through the graph construction, and with $w_i$8 layers the reported average sum-rates are $w_i$9 for UWMMSE, $\boldsymbol v_i$0 for WMMSE, and $\boldsymbol v_i$1 for truncated WMMSE, while test-time runtimes are $\boldsymbol v_i$2 ms/sample for UWMMSE and $\boldsymbol v_i$3 ms/sample for WMMSE (Chowdhury et al., 2020).

A related knowledge-driven D2D allocator uses a WMMSE-unrolled GNN with two cascaded message-passing modules per layer, one aligned with $\boldsymbol v_i$4 updates and one with the $\boldsymbol v_i$5-update. In the reported normalized performance table, UWGNN attains $\boldsymbol v_i$6, $\boldsymbol v_i$7, and $\boldsymbol v_i$8 relative performance for user counts $\boldsymbol v_i$9, $(\boldsymbol A+\mu \boldsymbol I)^{-1}$0, and $(\boldsymbol A+\mu \boldsymbol I)^{-1}$1, respectively, while a prior UWMMSE baseline degrades to $(\boldsymbol A+\mu \boldsymbol I)^{-1}$2 and $(\boldsymbol A+\mu \boldsymbol I)^{-1}$3 at $(\boldsymbol A+\mu \boldsymbol I)^{-1}$4 and $(\boldsymbol A+\mu \boldsymbol I)^{-1}$5 users (Yang et al., 2023). This indicates that graph-structured enhancement is particularly valuable for size-varying and topology-varying interference networks.

Stability analysis of unfolded WMMSE adds an important counterweight to pure performance claims. For UWMMSE under bounded additive perturbations $(\boldsymbol A+\mu \boldsymbol I)^{-1}$6, the paper derives a recursive bound on $(\boldsymbol A+\mu \boldsymbol I)^{-1}$7 in terms of numerator and denominator perturbation terms, previous-layer errors, and bounded GNN-output changes (Chowdhury et al., 2021). Empirically, over $(\boldsymbol A+\mu \boldsymbol I)^{-1}$8k test nodes, about $(\boldsymbol A+\mu \boldsymbol I)^{-1}$9 have an upper-bound margin close to $u_i$00, about $u_i$01 have margin greater than $u_i$02, and about $u_i$03 violate the bound, which the paper attributes to higher-order terms (Chowdhury et al., 2021). A common misconception is therefore that learning-based eWMMSE variants are merely faster black-box surrogates; this literature instead presents structured, often interpretable WMMSE modifications with explicit perturbation analyses.

Taken together, the unfolding literature suggests a specific meaning of enhanced WMMSE in finite-latency settings: preserve the $u_i$04–$u_i$05–$u_i$06 semantics of WMMSE, but train a small number of channel-conditioned parameters so that truncated execution is no longer a crude approximation of the asymptotic algorithm.

6. Application-specific extensions, limitations, and terminology

The eWMMSE idea has also expanded into application-specific formulations. In integrated sensing and communications, a mutual-information-based WMMSE framework maximizes a weighted sum of communication MI and sensing MI under clutter by introducing receive beamformers $u_i$07, MSE matrices $u_i$08, and weight matrices $u_i$09, so that the sensing term enters the same log-det/MMSE machinery as the communication terms (Peng et al., 2023). In movable-antenna networks, standard WMMSE transceiver updates are embedded inside a block-coordinate framework with majorization-minimization updates for antenna positions, and the planar movement mode reduces computation time by approximately $u_i$10 at a little performance expense (Feng et al., 2024). In RIS-assisted MU-MISO, WMMSE is made compression-aware by recomputing the effective channel from the actual decompressed RIS phase shifts before updating the AP beamformer, and the paper states that accounting for phase-shift compression errors during beamforming significantly improves the sum-rate performance even when the number of control bits is lower than the number of RIS elements (Fernandes et al., 6 Oct 2025).

Robust incomplete-CSI design in FDD provides another eWMMSE-like extension. There the classical rate objective is replaced by a training-aware lower bound based on LMMSE channel estimation, yielding the augmented weighted average MSE

$u_i$11

and a fully closed-form precoder update

$u_i$12

The paper reports runtime approximately $u_i$13 times smaller than SIWMMSE at $u_i$14 dB while retaining strong robustness under few-pilot incomplete-CSI scenarios (Amor et al., 2023). By contrast, in holographic MIMO RS-NOMA ISAC, a paper uses the exact term “Enhanced WMMSE (E-WMMSE)” but only as a partially specified baseline, explicitly stating that it extends conventional WMMSE to handle RS-NOMA with sensing constraints while not providing a full rigorous derivation, convergence proof, or explicit sensing-augmented WMSE formulation (Majhi, 29 Nov 2025).

Three limitations recur across the literature. First, enhanced WMMSE methods remain local or stationary-point methods for nonconvex problems; none of the surveyed variants claim global optimality. Second, learning-based variants depend on training distributions, fixed layer counts, and often fixed system dimensions, although graph-based designs mitigate this to some extent (Pellaco et al., 2020, Chowdhury et al., 2020). Third, the term itself is not standardized: some papers reserve eWMMSE for multi-channel JCPA with or without QoS constraints, while others describe projected-gradient, inverse-free, accelerated, or application-specific designs as enhanced WMMSE in substance but not in name (Chen et al., 8 Sep 2025, Pellaco et al., 2022).

In contemporary usage, eWMMSE is therefore best understood as a technical category rather than a single algorithm. At its core lies the classical WMMSE equivalence between weighted sum-rate and weighted MSE. What changes across the literature is the surrounding architecture: multi-channel power coupling, QoS pricing, matrix-inverse-free realizations, projected-gradient beamformer steps, extrapolated surrogate updates, graph-conditioned unfoldings, or sensing- and geometry-aware extensions. The unifying principle is not the acronym but the preservation of WMMSE structure while improving feasibility, latency, hardware-friendliness, or task specificity.