Efficient Channel Prediction for Beyond Diagonal RIS-Assisted MIMO Systems with Channel Aging

Arthur Sousa de Sena; Matti Latva-aho; Nandana Rajatheva; Nipuni Ginige; Nurul Huda Mahmood

REVIEW 4 major objections 5 minor 35 references

Efficient Channel Prediction for Beyond Diagonal RIS-Assisted MIMO Systems with Channel Aging

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A joint Tucker2-BALS estimator and CNN-AR predictor is proposed that reduces pilot overhead by up to 98% in beyond-diagonal RIS-assisted MIMO systems with channel aging, while preserving near-perfect-CSI downlink sum rates.

desk verdict Tucker2-BALS for BD-RIS is a genuine contribution; the CNN-AR prediction is under-specified and the channel model contradicts its own fading statistics, so the headline 98% overhead claim is not established. read the letter →

arxiv 2411.17725 v1 pith:VPNSZWLZ submitted 2024-11-21 eess.SP

Nipuni Ginige , Arthur Sousa de Sena , Nurul Huda Mahmood , Nandana Rajatheva , Matti Latva-aho This is my paper · ORCID

classification eess.SP

keywords beyonddiagonalRISTucker2decompositionbilinearalternatingleastsquaresCNN-ARpredictionchannelagingpilotoverheaddownlinksumrateMIMOestimation

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's target is the channel state information problem created by beyond-diagonal RISs (BD-RISs): surfaces whose reflecting elements are electrically coupled, so the reflection matrix is not diagonal and the cascade channel has many interdependent entries. The authors claim that in a BD-RIS-assisted MIMO system with fast-fading user channels, the received pilot tensor obeys a Tucker2 decomposition, which lets a bilinear alternating least squares (BALS) estimator recover the RIS-BS and user-RIS channels separately with far fewer pilots than a least-squares baseline. They then feed the estimated user channels to a CNN-augmented autoregressive predictor that forecasts future channels through coherence intervals in which no pilots are sent. If correct, the framework delivers downlink sum rates close to the perfect-CSI benchmark, a 10–50% sum-rate gain over DFT-LS estimation, and a pilot-overhead reduction of up to 98% for fully-connected and 88–97% for group-connected BD-RIS architectures.

What carries the argument

The load-bearing object is the Tucker2 decomposition of the received-signal tensor. Tucker2 factorizes a third-order tensor into a core tensor and two factor matrices while leaving the third mode unconstrained; here the core is $\mathcal{W}$ (the sequence of known reflection matrices) and the factors are $\mathbf{H}$ and $\mathbf{E}[l]^T$. The identity $\operatorname{vec}(\mathbf{A}\mathbf{B}\mathbf{C}^T) = (\mathbf{C}\otimes\mathbf{A})\operatorname{vec}(\mathbf{B})$ converts the cascade relation into linear systems, and the BALS algorithm alternates the two least-squares updates (19) and (20) until the normalized residual stops decreasing. The second mechanism is the CNN-AR predictor: a small convolutional network trained on the autocorrelation function of the aging channel maps a short window of estimated channels to AR coefficients, which are then used recursively to predict $P$ future coherence intervals without pilots.

What would settle it

Hold out a normalized Doppler frequency between two training values (e.g., test at $f_n = 0.006$ when training used $0.005$ and $0.01$) and measure the prediction NMSE; if it jumps by an order of magnitude relative to neighboring trained frequencies, the CNN has memorized the training grid rather than learned a transferable aging pattern. A second decisive test is to run the full pipeline on measured channel traces from a moving terminal; if the CNN-AR predictions are no better than a conventional same-order AR predictor, the learned pattern does not transfer to real fading.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the BD-RIS channel estimation problem, which looks high-dimensional because a non-diagonal reflection matrix couples all elements, can be reshaped into a low-rank tensor model: stacking the received pilot signals across antennas, users, and training blocks gives a third-order tensor $\tilde{\mathcal{Y}}$ whose noiseless part is $\mathcal{Y} = \mathcal{W} \times_1 \mathbf{H} \times_2 \mathbf{E}[l]^T$, where $\mathcal{W}$ holds the known BD-RIS reflection matrices. Fitting this Tucker2 model by BALS—alternating two least-squares updates for $\mathbf{H}$ and $\mathbf{E}[l]$—estimates the two channel factors separately, which the paper argues is both more accurate and lower-complexity than estimating the composite cascade, and it is unique up to a scalar that a one-element calibration step fixes. The user-side estimates feed a CNN that learns the autocorrelation (aging) pattern of the fading process and returns AR coefficients, so the AR recursion predicts $\mathbf{E}[l]$ for the next $P$ coherence intervals without any pilot transmission. Using this CSI in a closed-form BD-RIS reflection optimization, the authors report near-perfect-CSI sum rates, order-of-magnitude NMSE gains, and up to 98% pilot-overhead reduction, with the same machinery adapted to group-connected RIS by processing each group independently.

Load-bearing premise

The load-bearing premise is that the CNN, trained only on synthetic correlated-fading channels and evaluated only at velocities seen during training, outputs prediction coefficients accurate enough for the actual channel; real measured channels or speeds outside the training set are never tested.

Editorial extensions

If this is right

The pilot length for fully-connected RIS becomes $(T+1)Q$ over $Q+P$ coherence intervals instead of $M^2(Q+P)$, which yields the reported 94–98% overhead reduction; the group-connected case scales as $(T+\bar{G})Q$ and saves 88–97%.
Because $\mathbf{H}$ and $\mathbf{E}[l]$ are estimated separately, the per-iteration BALS complexity is $O(M^2T(K+N)+2NKTM)$, quadratic in $M$, whereas the DFT-LS baseline costs $O((Q+P)M^4(1+NK))$; the gap widens as the number of RIS elements grows.
The CNN-AR predictor lowers prediction NMSE by orders of magnitude relative to a same-order AR predictor, because it avoids computing AR coefficients from a short, noisy window and instead reads them from the learned autocorrelation pattern.
At low UE velocities the estimated/predicted CSI yields an average downlink sum rate close to the perfect-CSI bound, while at high velocities prediction accuracy degrades as the coherence time shortens (tested from 9 to 360 km/h at a 3 GHz carrier).
The same framework covers both major BD-RIS families: fully-connected RIS is one Tucker2 problem, and group-connected RIS is solved per group, so no new estimation machinery is needed for either architecture.

Reading between the lines

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

If the CNN were retrained on measured fading traces rather than synthetic correlated-fading realizations, the predictor might extend to non-Rayleigh environments; the paper does not test this transfer, so it is an open question rather than a demonstrated property.
The 98% pilot saving assumes the RIS-BS channel is quasi-static; in deployments with a mobile RIS or moving scatterers, the training phase would need to repeat, shrinking the saving by an amount this paper does not quantify.
The complexity comparison counts online CNN inference only; the offline training cost (300 epochs over $7\times10^4$ samples) would need to be amortized over long operation, or the CNN distilled to a lookup table, before the claimed low complexity holds at the edge.
A natural end-to-end test is to close the loop: feed the predicted CSI into the BD-RIS reflection optimizer and measure realized throughput on a testbed, since the sum-rate curves here assume the prediction errors only affect the bounds, not the actual beamforming.

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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

Short version: the tensor decomposition half of this paper is the contribution worth refereeing. The CNN-AR prediction half is where the paper goes soft. Start there.

What's new: They're the first to apply Tucker2-BALS to channel estimation for fully- and group-connected BD-RIS. The identifiability condition M ≤ min(NT, KT) (and the group analog) is new and plausible. The simulations show order-of-magnitude NMSE gains over DFT-LS, and the pilot-overhead math is transparent: you buy a 94–98% reduction by removing pilots in the prediction phase. The complexity analysis is also useful.

Where it goes soft: The system model in Eq. (2) has E[l] as a static vector times a pure phase rotation e^{j2πfdTs l}. That is not a fading channel with a Jakes ACF. The ACF of a pure tone is a complex exponential, not J0(2πfn|l|), yet the AR model in Eq. (25) and the CNN training assume J0. Either the simulation uses the phase-rotation model (then prediction is trivial and the CNN is unnecessary) or it uses Jakes fading (then Eq. (2) is wrong). The paper doesn't say which, and there's no code or data. That's load-bearing, because the headline overhead reduction relies on the CNN-AR keeping prediction NMSE low.

Second, the CNN-AR is under-specified—no layer sizes, no kernel sizes, no report of whether the five Doppler values in Fig. 10 were held out from the ten used in training. The paper says the CNN 'is capable of adding more fn values in training,' which suggests they never tested on unseen velocities or non-Jakes fading. At minimum, that generalization claim needs to be tested.

Third, the scaling ambiguity removal by switching on the first RIS element and first BS antenna is questionable under the unitary constraint on the reflection matrix; for fully-connected BD-RIS you can't simply turn off other elements. This may be fixable with a known reference pilot, but it needs a concrete procedure.

None of these kill the Tucker2-BALS contribution. That part is plausible and the identifiability analysis is sound within its own assumptions. But the paper as written overclaims the prediction phase. A serious referee should ask the authors to fix Eq. (2) or the fading model, evaluate CNN-AR on held-out Doppler values and non-Jakes autocorrelations, and give a feasible scaling-ambiguity protocol. I'd send it to review with major revision; the tensor part deserves a fair hearing.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a joint channel estimation and channel prediction framework for beyond-diagonal RIS (BD-RIS) assisted MIMO systems under channel aging. Estimation is based on a Tucker2 tensor decomposition solved by bilinear alternating least squares (Tucker2-BALS), applied separately to fully-connected and group-connected architectures. Prediction uses a CNN combined with an AR model (CNN-AR) to extrapolate estimated channels during a pilot-free phase. The estimated/predicted CSI is then used to optimize the BD-RIS reflection matrix for downlink sum-rate maximization. The paper claims high estimation/prediction accuracy, a robust sum-rate performance close to perfect CSI, and a pilot overhead reduction of up to 98% relative to a DFT-LS baseline.

Significance. If the claims are fully supported, the contribution is valuable: it extends tensor-based channel estimation to non-diagonal RIS architectures, separates the RIS-BS and RIS-UE channels with reduced training overhead, and combines this with an ML-based predictor to mitigate channel aging. The Tucker2 formulation and the identifiability condition M <= min(NT,KT) are plausible and provide a useful starting point for BD-RIS channel estimation. The main weaknesses are that the prediction component is only validated in distribution, the channel model in Eq. (2) is inconsistent with the Jakes ACF used to train the predictor, and the scaling-ambiguity removal step appears infeasible for the BD-RIS architectures studied. These issues directly affect the credibility of the headline pilot-overhead and sum-rate claims.

major comments (4)

[II, Eq. (2); IV-A, Eq. (25)] The UE-RIS channel in Eq. (2) is written as a static Rician vector multiplied by a complex exponential e^{j2*pi*fd*Ts*l}. The ACF of this process is a pure tone, not the Jakes ACF R[l] = J0(2*pi*fn*|l|) used in Section IV-A to derive the AR coefficients for the CNN-AR predictor. The manuscript never states which channel model generated Figs. 9 and 10. If the simulator followed Eq. (2), the prediction problem is a trivial known rotation and the CNN-AR machinery is unnecessary; if the simulator used Jakes/Rayleigh fading, then Eq. (2) is incorrect. This ambiguity is load-bearing because the 98% pilot-overhead reduction in Section V-C depends on the prediction phase being accurate on the actual simulated channel.
[III-B2a, uniqueness and scaling ambiguity] The proposed scaling-ambiguity removal requires estimating [H]_{1,1} by 'switching on only the first element in RIS and the first antenna in BS.' For a fully-connected BD-RIS, a reflection matrix with only one nonzero entry violates the unitary constraint Theta*Theta^H = I_M stated in Section II-A; the same issue arises within each group for the group-connected architecture in Section II-B. The manuscript therefore does not provide a feasible calibration procedure for fixing the scalar ambiguity, and the sum-rate and NMSE evaluations implicitly assume that the ambiguity is resolved. This needs to be addressed with a feasible measurement or the performance claims must be revised.
[IV-B and Figs. 9-10] The CNN-AR predictor is under-specified and only evaluated in distribution. The architecture description is limited to activation functions, optimizer, and loss; no layer counts, kernel sizes, filter numbers, or input/output dimensions are given, and Eq. (33) does not fully specify the input tensor layout. Figure 10 shows five normalized Doppler values, but the paper never states that these test Doppler values are held out from the ten values used in training. There is no evaluation at intermediate Doppler values or under non-Jakes/non-Rayleigh fading. Since the prediction-phase pilot removal is the mechanism behind the headline pilot-overhead reduction, this missing generalization and reproducibility evidence is load-bearing.
[V-C and Table III] The pilot-overhead reduction percentages in Table III are not reproducible without specifying the values of Q and P used in the calculation. The expression for the proposed average pilot overhead, (T+1)Q/(Q+P) for fully-connected and (T+G)Q/(Q+P) for group-connected, depends on the ratio Q/(Q+P), but no Q or P values are given in Table III or in the surrounding text. Additionally, the '+1' and '+G' terms are not derived; if they represent calibration pilots for the scaling-ambiguity removal introduced in Section III-B2a, that connection should be made explicit.

minor comments (5)

[V-A, Eq. (36)] In the optimization problem P1, the symbol T in the term (e_k Theta^H T)^H u_k is undefined; it should presumably be the RIS-BS channel matrix H or its transpose, depending on the reciprocity convention. Please clarify the notation.
[II, Eq. (2) and IV-A] The system model introduces a correlated Rician channel for E[l], while Section IV-A adopts a Rayleigh fading assumption for the ACF model. The relationship between the Rician components and the Jakes ACF should be stated explicitly.
[IV-B] The sentence 'Even though we use 10 different fn values to train the CNN model, Our proposed CNN model is capable of adding more fn values in training' is unclear and contains a capitalization error; it should be rewritten to describe how the training set is constructed and how additional Doppler values would be incorporated.
[VI-C and Table III] The paper does not report whether the simulation code or data will be released, which makes it difficult to verify the CNN training details and the exact pilot-overhead computation. A reproducibility statement would strengthen the manuscript.
[Throughout] There are several typographical and grammatical issues, e.g., 'archives' for 'achieves' near Table I, 'the lower bound of MSE' usage, and inconsistent spacing in equations. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Tucker2-BALS estimation and CNN-AR prediction claims are evaluated against external baselines, and the only self-citation ([1]) is not load-bearing.

full rationale

The paper's central claims do not reduce to their own inputs. The Tucker2-BALS channel estimator is derived from the received-signal model in Eq. (6) and the Tucker2 factorizations in Eqs. (15)-(20), and its accuracy is compared against independent external baselines (DFT-LS [29] and PARAFAC [15]) in Figs. 7-8. The CNN-AR predictor is specified in Section IV-B with its own architecture, loss function, and training/validation/test datasets; training on synthetic data generated from the same Jakes ACF model is a generalization limitation rather than circularity, because the prediction NMSE in Figs. 9-10 is evaluated on separate test samples against ground-truth channels. The 98% pilot-overhead reduction in Table III is an accounting consequence of the protocol design that removes pilots in the prediction phase (Section V-C), and its practical validity depends on the prediction accuracy that the paper reports; it is not a fitted parameter renamed as a prediction. The only self-citation, reference [1], is used to point to the authors' earlier CNN-AR proposal and to note partial WCNC results, but the present paper re-implements and extends the method to BD-RIS, so the citation is not load-bearing. A model-consistency concern exists between Eq. (2)'s pure-tone time variation and Eq. (25)'s Jakes ACF, but that is a correctness/generalization risk, not a circularity. Overall, no step in the derivation chain is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper relies on standard channel models and a learned CNN mapping; the main free parameters are the pilot length T and the AR regularization epsilon, which are tuned from simulations rather than derived. No new physical entities are introduced.

free parameters (3)

Minimum pilot length T (per architecture) = 20/36/78 (FC, M=16/32/64); 22/40/84 (GC, Mbar=8/16/32)
Chosen numerically from NMSE convergence curves in Fig. 11; the 94-98% overhead reduction claim uses these values, which are not derived analytically.
ACF regularization epsilon = 0.1
Added to R[0] in Eq. 32 to stabilize high-order AR models; follows a heuristic from [28], not derived for this system.
AR order Q = 8, 16, 24 used in Fig. 9; CNN input length V=Q
Prediction accuracy depends on Q, which is selected by trial rather than optimized or derived.

assumptions (4)

domain assumption The RIS-BS channel H is quasi-static and unaffected by channel aging, remaining constant over many coherence intervals.
Section II: 'channel aging does not affect the RIS-BS channels, which is motivated by the fact that the BS and the RIS are usually fixed.' This limits applicability to setups with fixed BS/RIS.
domain assumption The time-varying RIS-UE channels follow a stationary AR process whose coefficients are derived from the Jakes ACF R[l] = J0(2 pi fn |l|).
Eq. 25-31 in Section IV-A; the AR model with Levinson-Durbin and epsilon=0.1 is assumed to represent the true channel aging process.
ad hoc to paper The CNN trained on 1e4 synthetic test samples and 10 Doppler values generalizes to the evaluation scenarios.
Section IV-B: training uses 70k samples and 10 fn values; test uses the same simulation model. No out-of-distribution evaluation is reported.
ad hoc to paper The scalar scaling ambiguity in H and E can be removed by measuring [H]_{1,1} while switching on only the first RIS element and first BS antenna.
Section III-B2a: this measurement procedure may violate the fully-connected BD-RIS constraint Theta Theta^H = I, which does not allow a reflection matrix that isolates a single element. The paper does not address this inconsistency.

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Cite this review

Pith. "Pith review of Efficient Channel Prediction for Beyond Diagonal RIS-Assisted MIMO Systems with Channel Aging." pith.science (2026). https://pith.science/paper/VPNSZWLZ

@misc{pith2026241117725,
  author       = {Pith},
  title        = {Pith review of: Efficient Channel Prediction for Beyond Diagonal RIS-Assisted MIMO Systems with Channel Aging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPNSZWLZ}},
  note         = {Machine review of arXiv:2411.17725}
}

read the original abstract

Novel reconfigurable intelligent surface (RIS) architectures, known as beyond diagonal RISs (BD-RISs), have been proposed to enhance reflection efficiency and expand RIS capabilities. However, their passive nature, non-diagonal reflection matrix, and the large number of coupled reflecting elements complicate the channel state information (CSI) estimation process. The challenge further escalates in scenarios with fast-varying channels. In this paper, we address this challenge by proposing novel joint channel estimation and prediction strategies with low overhead and high accuracy for two different RIS architectures in a BD-RIS-assisted multiple-input multiple-output system under correlated fast-fading environments with channel aging. The channel estimation procedure utilizes the Tucker2 decomposition with bilinear alternative least squares, which is exploited to decompose the cascade channels of the BD-RIS-assisted system into effective channels of reduced dimension. The channel prediction framework is based on a convolutional neural network combined with an autoregressive predictor. The estimated/predicted CSI is then utilized to optimize the RIS phase shifts aiming at the maximization of the downlink sum rate. Insightful simulation results demonstrate that our proposed approach is robust to channel aging, and exhibits a high estimation accuracy. Moreover, our scheme can deliver a high average downlink sum rate, outperforming other state-of-the-art channel estimation methods. The results also reveal a remarkable reduction in pilot overhead of up to 98\% compared to baseline schemes, all imposing low computational complexity.

Figures

Figures reproduced from arXiv: 2411.17725 by the authors.

**Figure 3.** System architecture: Proposed joint channel estima [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Representation of Tucker2 decomposition. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 5.** Figure 5: Architecture of the proposed CNN-AR predictor. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Training and validation loss of the CNN model in the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: Comparison of average downlink sum rate with the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: Comparison of prediction NMSE among the AR predic [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: Comparison of estimation accuracy of proposed chan [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 12.** Figure 12: Comparison of average pilot overhead when SNR [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

**Figure 11.** Figure 11: NMSE vs training blocks for Tucker2-BALS when [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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