AREE-Based Decoupled Design of Hybrid Beamformers in mmWave XL-MIMO Systems

Alessandro Bazzi; Anna Guerra; Francesco Guidi; Heng Dong; Jiazhe Li; Nicol\`o Decarli; Zhuoming Li

REVIEW 3 major objections 3 minor 34 references

AREE-Based Decoupled Design of Hybrid Beamformers in mmWave XL-MIMO Systems

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes an alternating residual error elimination (AREE) algorithm that decomposes the hybrid beamforming problem into two low-dimensional subproblems, each decoupling the analog and digital precoders with negligible error, and…

desk verdict Genuinely new block-partition decoupling trick and solid simulations, but the guaranteed-convergence proof in Sec. III-B is not valid as written. read the letter →

arxiv 2507.02802 v1 pith:PD5J6MXN submitted 2025-07-03 eess.SP

Jiazhe Li , Nicolò Decarli , Francesco Guidi , Heng Dong , Anna Guerra , Alessandro Bazzi , Zhuoming Li This is my paper · ORCID

classification eess.SP

keywords mmWavecommunicationshybridbeamformingXL-MIMOalternatingresidualerroreliminationprecoderdecouplingspectralefficiencysparsechannelSVDconstantmodulusconstraint

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the hard part of hybrid beamforming — the analog and digital precoders appear only as a product, so optimizing one requires knowing the other — can be split into two smaller problems that alternately erase each other's mistakes. Each subproblem covers a block of RF chains, and because the corresponding digital block is row full-rank, the analog block can be pulled out of the product with virtually no decoupling error. Alternating between the two subproblems drives the product toward the optimal fully-digital precoder, and the paper's simulations show higher spectral efficiency than existing iterative and one-shot methods at comparable or lower complexity. If correct, this gives a practical way to approach fully-digital performance in mmWave XL-MIMO systems without a fully-digital array.

What carries the argument

The carrying object is the residual-error alternation: $F_{\mathrm{opt}}$ is split into two overlapping approximations $F_{\mathrm{RF1}}F_{\mathrm{BB1}} + F_{\mathrm{RF2}}F_{\mathrm{BB2}}$, and the two subproblems minimize $\|E_2 - F_{\mathrm{RF1}}F_{\mathrm{BB1}}\|_F^2$ and $\|E_1 - F_{\mathrm{RF2}}F_{\mathrm{BB2}}\|_F^2$ with the residuals fed back after each block converges. Within each subproblem the update is a pseudo-inverse for the digital block, $F_{\mathrm{BB}} = F_{\mathrm{RF}}^\dagger E$, and a phase-extraction step for the analog block, $F_{\mathrm{RF}} = \frac{1}{\sqrt{N_t}} \exp\!\big(j\,\arg\big(E F_{\mathrm{BB}}^H (F_{\mathrm{BB}} F_{\mathrm{BB}}^H)^\dagger\big)\big)$. The paper also contributes GC-SVD, which uses the geometric channel's steering matrices to reduce the high-dimensional channel SVD to SVDs of size $N_{\mathrm{cl}}N_{\mathrm{ray}} \times N_{\mathrm{cl}}N_{\mathrm{ray}}$, and two initializations, PE-OMP and PE-SMD, that seed AREE so that only a few inner iterations are needed.

What would settle it

Run Algorithm 1 on a single channel realization with $N_{\mathrm{RF}} > N_s$ and record the global objective $f(t)=\|F_{\mathrm{opt}} - F_{\mathrm{RF1}}F_{\mathrm{BB1}} - F_{\mathrm{RF2}}F_{\mathrm{BB2}}\|_F^2$ after every inner and outer update; if any step that updates $F_{\mathrm{RF1}}$, $F_{\mathrm{RF2}}$, $F_{\mathrm{BB1}}$, or $F_{\mathrm{BB2}}$ strictly increases $f(t)$, the monotone-convergence chain in (29)-(30) is violated. A simpler check: in subproblem (14), compute $F_{\mathrm{RF1}}$ by (19) and compare $\|E_2 - F_{\mathrm{RF1}}F_{\mathrm{BB1}}\|_F$ with the previous iterate to see whether the phase-extraction step can increase the residual.

Watch

Extended reading notes

Core claim

The central claim is that decomposing $F_{\mathrm{opt}} \approx F_{\mathrm{RF}}F_{\mathrm{BB}}$ into two residual-error subproblems, one per block of RF chains, decouples $F_{\mathrm{RF}}$ and $F_{\mathrm{BB}}$ so effectively that the alternating updates converge to a near-optimal hybrid precoder. In the first subproblem, the residual $E_2 = F_{\mathrm{opt}}$ is approximated by $F_{\mathrm{RF1}}F_{\mathrm{BB1}}$ where $F_{\mathrm{BB1}}$ is an $N_s \times N_s$ square matrix; in the second, the new residual $E_1 = F_{\mathrm{opt}} - F_{\mathrm{RF1}}F_{\mathrm{BB1}}$ is approximated by $F_{\mathrm{RF2}}F_{\mathrm{BB2}}$ with $F_{\mathrm{BB2}}$ row full-rank. Because each digital block has full row rank, its right inverse exists, so each analog block can be extracted from the matrix product by phase extraction, avoiding the row-rank deficiency that forces existing algorithms to choose between decoupling error and high complexity. Alternating the residuals $E_1$ and $E_2$ through the two subproblems monotonically decreases the squared Frobenius objective, and when the number of RF chains approaches $2N_s$ the two digital blocks become near-unitary, making the phase-extraction update essentially optimal.

Load-bearing premise

The convergence guarantee assumes each subproblem is solved exactly at every alternation, but the actual update only solves a relaxed constant-modulus problem by extracting phases; nothing in the proof shows that this phase-extraction step always lowers the original objective.

Editorial extensions

If this is right

If AREE delivers what the simulations show, mmWave XL-MIMO transceivers can run hybrid beamforming at complexity comparable to PE-AltMin while getting spectral efficiency closer to the fully-digital bound, without solving the coupled matrix product as a high-dimensional optimization.
With the number of RF chains near $2N_s$, the two subproblem digital blocks become near-unitary, so the phase-extraction update is nearly optimal and the algorithm approaches fully-digital performance.
The PE-OMP initialization, which keeps the residual beam components instead of discarding them, offers a one-shot design that already outperforms standard OMP, and PE-SMD reduces its complexity further with only slight loss.
The GC-SVD channel decomposition lowers SVD cost from $O(N_t N_r^2)$ to $O((N_t+N_r)(N_{\mathrm{cl}}N_{\mathrm{ray}})^2)$, which matters for large arrays in sparse millimeter-wave channels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same residual-error splitting is not limited to the $N_s / (N_{\mathrm{RF}}-N_s)$ partition; the paper itself varies the partition and observes degradation but does not prove optimality, so one could test whether $n=N_s$ is universally optimal or only optimal for the phase-extraction update.
Editorial inference: because the subproblems operate on blocks of RF chains, the AREE structure may extend to partially connected arrays or to wideband OFDM, where the paper says the decoupled design applies; a concrete test would be to run the same alternation on per-subcarrier channel SVDs.
Editorial inference: the convergence chain in (29)-(30) assumes each subproblem is solved exactly at every step, while the actual update is only an approximate phase-extraction solution; a fixed-point analysis of the phase-extraction map rather than the exact minimizer would place the observed fast convergence on firmer ground.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read arXiv:2507.02802. The AREE algorithm is a credible piece of work, not a field-changer. The block partition of FBB into a square block and a row full-rank block is the real contribution: it sidesteps the row-rank deficiency that makes direct decoupling of FRF hard, and the alternating residual-feedback formulation is clean. GC-SVD is a nice complexity reduction using the geometric channel structure, and PE-OMP/PE-SMD are reasonable initialization tricks. The simulations are extensive (1000 realizations, several baselines, SNR and RF-chain sweeps) and support the claim that AREE offers better spectral efficiency than PE-AltMin and OMP-class methods at comparable complexity.

The soft spot is the convergence guarantee. Equations (29)-(30) claim monotone decrease because "the subproblem has been optimized," but the actual updates (19) and (23) are not exact minimizers of (14) and (21). They solve the relaxed constant-modulus problem (18) by phase extraction. The equivalence between (18) and (14) is only established when FBB1 is unitary (eq. 32), and Fig. 8 only shows NMSE below 0.1 empirically. So the strict inequalities in (29)-(30) are not justified, and "guaranteed convergence" is overstated. The algorithm may well converge in practice - the simulations say it does - but the stated proof does not establish it. The same gap weakens the "negligible decoupling errors" claim if you read it as a theorem rather than an observed property.

Minor: Sec. III-D asserts n=Ns is the optimal partition without proof; the authors admit this and back it with Fig. 9. Fine as a conjecture, but label it as such. The appendix on power normalization is plausible and does not bother me.

Bottom line: worth a serious referee. The novelty is real, the simulations are careful, and the flaws are specific and addressable. I would ask for a revision that either proves a descent property for the phase-extraction step under stated conditions or qualifies the convergence claim to "empirically observed fast convergence." The paper is for people working on mmWave hybrid beamforming and XL-MIMO; they will get a useful algorithm and a clear comparison. As it stands I would not lean on it in my own work, but after a revision that fixes the guarantee I would cite it.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes AREE, an alternating algorithm for hybrid precoding in mmWave XL-MIMO systems. AREE decomposes the coupled product FRF FBB into two low-dimensional subproblems, alternately eliminates the residual errors between them, and then recombines the results under the power constraint. The paper also introduces GC-SVD, a low-complexity SVD method exploiting the sparse geometric channel structure, and two initialization schemes, PE-OMP and PE-SMD. Simulations compare the spectral efficiency and convergence speed of AREE against OMP, PE-AltMin, MO-AltMin, TensDecomp, and fully-digital baselines.

Significance. If the algorithmic claims hold, the contribution is practically valuable: AREE combines near-optimal spectral efficiency as the number of RF chains approaches 2Ns with a per-iteration complexity comparable to PE-AltMin. GC-SVD is a clean use of channel sparsity to reduce SVD complexity, and the initialization study is useful. The numerical evaluation is extensive and the claims are empirically falsifiable. The main weakness is theoretical: the convergence guarantee is not established for the actual updates, and the claimed optimality conditions are supported only by simulation.

major comments (3)

[Sec. III-B, Eqs. (29)-(30)] The convergence proof asserts strict descent because subproblems (14) and (21) have been optimized. However, the updates in Algorithm 1, lines 7-8 and 13-14, consist of an exact least-squares FBB update followed by a phase-extraction FRF update that solves the relaxed problem (18), not the original constant-modulus problem. No inequality is provided showing that this approximate update decreases the original objective ||E2 - FRF1 FBB1||_F^2 or ||E1 - FRF2 FBB2||_F^2. Thus the inequalities in (29) and (30) are not justified, and the statement that AREE is guaranteed to converge is unsupported. The authors should either prove monotonicity for the actual updates or explicitly reframe convergence as an empirical observation.
[Sec. III-C and Fig. 8] The optimality of the phase-extraction update for subproblem (14) is shown in (32) only under the condition that FBB1 is unitary, and the analogous claim for subproblem (21) requires FBB2 to be square as NRF grows. Figure 8 reports NMSE(FBB1) and NMSE(FBB2) below 0.1, which is an empirical indication rather than a proof. If near-optimality is used to support the negligible-decoupling-error claim, the paper should prove or clearly qualify this condition.
[Sec. III-D, Eqs. (34)-(35), Fig. 9] The section is titled "The Optimal Matrix Partition" and states that n = Ns is the optimal partition, but the paper acknowledges that a proof of this optimality is beyond its scope and provides only an intuitive explanation supported by Fig. 9. Since Algorithm 1 fixes this partition, the claim should be either proven or softened to a heuristic choice justified by simulation.

minor comments (3)

[Algorithm 3] The symbol FBB is reused for the intermediate least-squares solution and later for the final digital precoder, which can confuse the reader; a separate variable for the intermediate projection would be clearer.
[Sec. III-A and Fig. 9 caption] There are typographical errors: "SinceFBB" at the start of Sec. III-A and "Intial" in the caption of Fig. 9.
[Eq. (33)] The normalization of NMSE is written in a needlessly convoluted way; since the denominator equals one, the definition can be simplified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: AREE is derived from a stated least-squares objective and validated against external baselines; the near-unitarity condition is an empirical check, not a fitted input.

full rationale

The derivation chain is not circular. The AREE algorithm starts from the standard least-squares approximation problem (7) of the fully-digital precoder F_opt; subproblems (14) and (21) are formed by partitioning FRF and FBB, and the updates (16)/(19) and (22)/(23) are obtained by closed-form derivatives and phase extraction rather than by fitting any parameter to the quantities being predicted. GC-SVD in Sec. IV is an exact factorization of H = Ar Hd At^H using projectors and low-dimensional SVDs; it does not presuppose F_opt. PE-OMP and PE-SMD provide initializations using the same channel steering matrices, not the final spectral-efficiency result. The only self-referential element is Sec. III-C, where the claim that (19) is near-optimal for (14) relies on the empirically observed near-unitarity of FBB1 (NMSE < 0.1, Fig. 8). This condition is checked after the fact in simulation and is not fitted to the reported spectral-efficiency curves, so it does not make the derivation circular. The stricter concern raised in Sec. III-B is a correctness gap: Eqs. (29)-(30) assert strict decrease because each subproblem "has been optimized," but the actual phase-extraction updates are not exact minimizers; this is an unsupported proof step, not a circular reduction. Sec. III-D explicitly leaves the optimality of n = Ns unproved, again a completeness limitation rather than a definitional loop. No load-bearing self-citation or imported uniqueness theorem appears; comparisons against OMP, PE-AltMin, MO-AltMin, and fully-digital bounds are external. The paper is therefore self-contained against external benchmarks and merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The algorithm introduces no fitted constants and no post-hoc parameters. The key unstated premises are the faithfulness of the relaxed subproblem to the original, the full-rankness of the digital blocks, and the exact sparse geometric channel model. The n=Ns partition optimality is an additional unproven claim.

assumptions (5)

ad hoc to paper The phase-extraction update (19) is assumed to be a valid suboptimal solution of (14) that decreases the objective, i.e., the relaxed problem (18) is a faithful proxy for (14).
Used in Sec. III-A-III-B; without it the convergence inequalities (29)-(30) do not follow because (19) does not exactly minimize (14) unless FBB1 is unitary.
domain assumption FBB1 and FBB2 are full row-rank throughout the iterations, so their right inverses exist.
Invoked in Sec. III-A around (16), (19), (22)-(23); with random initialization this holds almost surely, but it is not established after alternating updates under all stopping criteria.
domain assumption The mmWave channel is exactly represented by the sparse geometric model (3) with known Ar, Hd, and At (Assumption 1).
Used in Sec. IV for GC-SVD; the low-complexity SVD and the PE-SMD relationship (45) rely on this exact structure.
standard math Standard properties of SVD and pseudo-inverse are used without proof.
Used throughout Sec. III-IV; acceptable as background.
ad hoc to paper The optimal matrix partition n=Ns in (34)-(35) is assumed optimal for spectral efficiency.
Claimed in Sec. III-D and supported only by simulation in Fig. 9; no proof is provided.

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Cite this review

Pith. "Pith review of AREE-Based Decoupled Design of Hybrid Beamformers in mmWave XL-MIMO Systems." pith.science (2026). https://pith.science/paper/PD5J6MXN

@misc{pith2026250702802,
  author       = {Pith},
  title        = {Pith review of: AREE-Based Decoupled Design of Hybrid Beamformers in mmWave XL-MIMO Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PD5J6MXN}},
  note         = {Machine review of arXiv:2507.02802}
}

read the original abstract

Hybrid beamforming has been widely employed in mmWave communications such as vehicular-to-everything (V2X) scenarios, as a compromise between hardware complexity and spectral efficiency. However, the inherent coupling between analog and digital precoders in hybrid array architecture significantly limits the computational and spectral efficiency of existing algorithms. To address this issue, we propose an alternating residual error elimination (AREE) algorithm, which decomposes the hybrid beamforming problem into two low-dimensional subproblems, each exhibiting a favorable matrix structure that enables effective decoupling of analog and digital precoders from the matrix product formulation. These subproblems iteratively eliminate each other's residual errors, driving the original problem toward the optimal hybrid beamforming performance. The proposed initialization ensures rapid convergence, while a low-complexity geometric channel SVD algorithm is developed by transforming the high-dimensional sparse channel into a low-dimensional equivalent, thereby simplifying the derivation of subproblems. Simulation results demonstrate that the AREE algorithm effectively decouples analog and digital precoders with low complexity, achieves fast convergence, and offers higher spectral efficiency than existing beamforming methods.

Figures

Figures reproduced from arXiv: 2507.02802 by the authors.

**Figure 2.** Diagram of the AREE algorithm. A. Alternating Residual Error Elimination Algorithm As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** (a), (b) and (c) are beam patterns of the equivalent channel obtained by different beamforming algorithms, where (e) and (f) are beam patterns of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Relationships between spectral efficiency and SNR for different [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Relationships between spectral efficiency and the number of RF chains [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 9.** Figure 9: The spectral efficiency of different matrix partition schemes, as defined [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 6, 2026 · model on record in the stance chip above.

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