Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems

Chintha Tellambura; Gongpu Wang; Yirun Wang; Yongqing Wang; Yuyao Shen

REVIEW 4 major objections 5 minor 55 references

Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems

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

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Pith's one-line read This paper establishes that a fifth-order tensor with one Vandermonde factor can be uniquely decomposed after spatial smoothing even when the other four factors have repeated columns, and uses that decomposition to estimate all multipath…

desk verdict A solid VSCPD extension to fifth-order tensors for active RIS channel estimation with a direct link; the uniqueness claim is plausible but stated a bit too crisply about genericity, and the path-identification similarity step lacks a threshold. read the letter →

arxiv 2411.16420 v2 pith:GNP6KWIZ submitted 2024-11-25 cs.IT eess.SPmath.IT

Yirun Wang , Yongqing Wang , Yuyao Shen , Gongpu Wang , Chintha Tellambura This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords activeRISchannelestimationtensordecompositionVandermondestructuredCPspatialsmoothingmultipathparameterOFDMCramér-Raolowerbound

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

This paper tries to show that channel estimation for an active-RIS-assisted broadband link in a fully multipath environment, including a direct transmitter-receiver path, can be solved by tensor decomposition without any two-dimensional search. The authors construct a fifth-order canonical polyadic tensor whose five modes match the channel's five dimensions, then prove a relaxed uniqueness condition: spatial smoothing along the single Vandermonde factor guarantees uniqueness under $\min(K_1-1,K_2)\geq R$ even though the other four factors have repeated columns. This yields a triple-stage algorithm: algebraic coarse estimation followed by two optional refinements, all built on one-dimensional parameter estimation. A closed-form Cramer-Rao lower bound is derived for the active-RIS case, where thermal noise at the RIS makes the noise covariance depend on multipath parameters. If correct, the result replaces exhaustive 2D search in RIS channel estimation with faster algebraic 1D recovery and extends prior LOS-only active-RIS work to general multipath.

What carries the argument

The central object is the spatial-smoothed fifth-order Vandermonde structured CP decomposition (VSCPD): a canonical polyadic decomposition in which one factor matrix is Vandermonde, with columns $e^{j(m-1)\omega_r}$, and the tensor is augmented by smoothing along that mode. This machinery converts a rank-deficient tensor factorization into linear algebra: a compact SVD of the mode-3 matricization, ESPRIT applied to the shift-invariance of the Vandermonde factor to extract delays, and rank-1 SVDs to split the Khatri-Rao products into individual factor columns for the other four modes.

What would settle it

Simulate the multipath scenario with $L=P=Q=2$, then set two of the direct-link delays equal so $A_1$ has a repeated generator. If the VSCPD still uniquely recovers all parameters and the variance and similarity checks still pass, Theorem 1's distinct-generator assumption is not necessary; if it fails, that assumption is confirmed as load-bearing.

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Extended reading notes

Core claim

The central claim is that a fifth-order CP tensor with only one Vandermonde factor can be uniquely decomposed after spatial smoothing, under the condition $\min(K_1-1,K_2)\geq R$, even though the other four factor matrices contain repeated columns and therefore violate Kruskal's classical uniqueness condition. On this basis, the paper constructs a received-signal tensor of order five matching the SIMO-OFDM channel's five dimensions, decomposes it with a VSCPD algorithm using only SVD and ESPRIT, identifies which columns correspond to direct versus cascaded paths by variance and similarity principles, and recovers delays, RIS angle-related parameters, BS arrival angles, and path gains for both links. A closed-form CRLB is derived in which the noise covariance depends on RIS-BS channel parameters, and simulations show that the triple-stage estimator achieves high accuracy with 100% decomposition success in the tested multipath scenarios.

Load-bearing premise

The decomposition is unique only if the delay generators in the Vandermonde factor are distinct and the cascaded-path columns in the other factors cluster into clean, separable groups; the paper asserts this is generally true but does not analyze what happens when two paths nearly share a delay or when noise smears the clusters.

Editorial extensions

If this is right

Channel parameters for both direct and cascaded links are recovered from one fifth-order CP decomposition with no 2D search; each parameter type is estimated by parallel 1D problems.
VSCPD uniqueness holds with $\min(K_1-1,K_2)\geq R$ even though four of the five factor matrices have repeated columns, so Kruskal's condition is not needed.
The algorithm runs with only linear algebra (SVD, ESPRIT, rank-1 approximations) before optional refinements, making Stage I initialization-free and faster than ALS-CPD.
Optional Stage II (correlation-based search) and Stage III (ALS initialized by Stage II) improve accuracy sequentially, and Stage I alone outperforms ALS-CPD baselines in multipath scenarios.
Active RIS amplification lifts the cascaded-link power enough to make decomposition succeed in the simulated setting, improving both success rate and tensor reconstruction error compared to the passive-RIS case.

Reading between the lines

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

Editorial extension: the same fifth-order VSCPD recipe is portable to other active-RIS and integrated-sensing setups where one tensor mode has Vandermonde structure, such as Doppler or another spatial dimension, so the paper's replacement of 2D search by parallel 1D estimation is not tied to the specific SIMO-OFDM model.
Editorial extension: the variance and similarity checks that the paper uses only to abort on failure could be turned into a data-driven rank and path-number selection rule, since they give a direct measure of whether the decomposition's column clustering is self-consistent.
Editorial extension: because the active-RIS noise covariance depends on RIS-BS parameters, reporting a single SNR-based NMSE may hide large per-parameter differences; the derived CRLB suggests comparing estimators parameter-by-parameter, especially at low SNR.
Editorial extension: the Stage II search refinement could likely be replaced by one Newton step from the Stage I algebraic estimate; the CRLB would then show whether the residual gap is search-resolution-limited or inherent to the ESPRIT initialization.

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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

Read the paper. It is a genuine extension of the VSCPD line: prior tensor work on RIS channel estimation either blocked the direct UE-BS link or stayed in LOS, and this one builds a fifth-order CP tensor for fully multipath with the direct link present. The pilot/RIS-profile design that makes direct-path columns constant, and the variance principle that uses that to separate direct from cascaded paths, is the clever part. The algebraic pipeline - spatial smoothing, ESPRIT-based VSCPD, then two optional refinement stages - is coherent, and the simulations support the main trends: perfect VSCPD success rates where ALS-CPD fails (Tables II and III), the expected gain from the active RIS, and sensible complexity numbers. The simulation settings are described in enough detail that the results are reproducible on paper, even without released code. The soft spots are real but not fatal. The headline uniqueness condition (40) is stated as if it depends only on K1 and K2, but the proof and the ESPRIT step inherit a genericity assumption: the Vandermonde generators (delays) must be distinct, and the similarity principle (41) needs a correlation threshold that the paper never specifies. Two paths sharing a delay, or two RIS-BS angles closer than array resolution, are exactly the cases where the variance/similarity checks either fail or need a tuned threshold; the paper's response is the word 'generally' and perfect success rates in the chosen regime. I don't think this sinks the paper - it matches how the VSCPD literature normally works - but the condition should be stated with its separability assumptions made explicit, and a failure-mode study (or at least a threshold sensitivity analysis) would materially strengthen it. The CRLB derivation is a second, smaller soft spot: the noise covariance depends on the RIS-BS channel, and the paper drops the cross-terms in (55) as 'reasonable' with no bound on the approximation error. Minor, but an error-bounded or numerically verified approximation would be easy to add. No code or data is included; for this venue that is typical, but given the empirical claims, releasing the Tensorlab-based simulator would settle reproducibility questions. Net: this paper deserves a serious referee. I would ask the authors to state the distinctness and threshold assumptions explicitly, quantify the CRLB approximation, and ideally release code. As is, it is a solid conditional-accept candidate.

Referee Report

4 major / 5 minor

Summary. This paper addresses channel parameter estimation for active RIS-assisted SIMO-OFDM systems with both a direct UE-BS link and a cascaded UE-RIS-BS link in full multipath. The authors construct a fifth-order CP tensor via customized pilots, RIS profiles, and a combining matrix, then apply spatial smoothing in the frequency mode and develop an algebraic Vandermonde-structured CPD (VSCPD) with the tensor having only one Vandermonde factor. A triple-stage estimator is proposed: Stage I gives path identification and coarse algebraic estimates, while Stages II and III are optional search- and ALS-based refinements. A closed-form CRLB is derived in the active-RIS case where the noise covariance depends on the multipath parameters, and simulations compare the proposed stages with ALS-CPD-based baselines and with passive RIS.

Significance. The problem is well motivated: existing VSCPD-based channel estimation is limited to third-order tensors or to scenarios without a direct link, and replacing a 2D search with parallel 1D estimations is practically valuable. The tensor construction and the pilot/RIS/combining design are thoughtful, and the paper includes useful numerical evidence such as success rates, run times, and active-versus-passive RIS comparisons. If the uniqueness theorem and the path-identification steps are made rigorous, the proposed framework would be a solid contribution to active-RIS multipath channel estimation. However, the central advertised claim, the relaxed uniqueness condition (40), is currently stated as a general condition while relying on separability assumptions that are not quantified.

major comments (4)

[III-C, Theorem 1 and Eqs. (30)-(40)] The proof of Theorem 1 is not internally consistent as written. Since B1 in (30) has K1 rows, the left-hand side J↑B123 in (35) has (K1-1)G1G2 rows, whereas the right-hand side B1⊙B2⊙B3 has K1G1G2 rows; the displayed identities cannot both hold. In addition, the text after (39) asserts that rank(B1⊙B2⊙B3)=min(K1-1,R) follows from the Khatri-Rao lower bound, but with kB2=kB3=1 that property gives only a lower bound, not the stated equality. The derivation of (40) therefore needs to be corrected, and the conclusion that K≥2R should be re-examined: under (40) the constraint is K≥2R+1.
[III-C, Theorem 1 hypothesis and Eq. (24)] The general applicability of (40) is not established because Theorem 1 assumes that A1 has distinct generators. In the physical channel, two paths can share the same delay or have delays separated by less than the resolution of the training subcarriers; then A1 and its smoothed submatrices B1 and B6 become rank-deficient or ill-conditioned, ESPRIT in Algorithm 1 cannot recover R distinct generators, and the uniqueness claim (40) is neither necessary nor sufficient. The manuscript's only support for this assumption is the word "generally" before Eq. (40). The authors should state the precise separability condition, for example a minimum delay separation relative to 1/(KΔf), and provide a failure-mode analysis for equal or closely spaced delays.
[III-D, Eq. (41) and Algorithm 2] The similarity principle used to group the P duplicate columns in B4 and B5 has no defined threshold and no resolution analysis. The correlation criterion in (41) will fail when two RIS-BS subpaths have angular separations smaller than the array resolution or when the estimated factor columns are too noisy, and the algorithm then aborts through the checks in lines 6-9 of Algorithm 2. Since this grouping is load-bearing for the subsequent mapping of delays, angles, and gains in (43)-(46), the paper needs a quantitative separation criterion and either an error analysis for the grouping step or a remedy when grouping fails.
[IV-A, Eq. (55)] The closed-form CRLB depends on neglecting the second term in (55), the q≠q' cross term in GRGH_R. The paper justifies this by saying that the first term is larger due to aligned phases, but the neglected term contains Q(Q-1) unit-modulus phase factors and also depends on the delay and angle parameters; its magnitude relative to the first term is not established. The authors should either retain the cross term in the FIM or validate the approximation numerically, for example by comparing the approximate CRLB with the full FIM or with the empirical error of a near-ML estimator in the high-SNR asymptotic region.

minor comments (5)

[Eq. (25)] The phrase "direst UE-BS paths" appears to be a typo for "direct UE-BS paths."
[Throughout] The notation "V andermonde" with a space appears repeatedly; it should be written as "Vandermonde."
[III-B and III-D] The element-space and transformed-space ESPRIT models use selection/projection matrices J↑, J↓, Q_n, and F_n, but Q_n and F_n are not defined in the text; please add their definitions or point to the exact equations in the cited references.
[IV-B] The complexity expressions in (60) would benefit from a consistency check; for example, the term O(R^2 K G^2) for delay recovery and the Stage III expression should be re-derived or commented on, since several terms appear to be missing factors of K or G.
[Figs. 4 and 6] The zoomed insets in Figures 4 and 6 are small and not explained in the captions; please enlarge them and state what is being magnified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is proven in-paper from external results ([45], [26]); the estimator is derived from the explicit model and validated on simulated ground truth and the CRLB; no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is self-contained and non-circular. The fifth-order tensor Y is constructed directly from the channel model through explicit pilot, RIS-profile, and combining designs (17)-(19), and its factor matrices (23)-(28) carry the channel parameters by construction; the VSCPD then recovers those factors algebraically rather than fitting them to targets. The central uniqueness claim (Theorem 1, condensed to condition (40)) is established in-paper: the proof derives the full-rank condition (31) from the mode-3 representation (33) and ESPRIT shift-invariance (34)-(36), and the generic version (32) is credited to the external result [45, Theorem 3] (Jiang, Sidiropoulos, ten Berge), with Theorem 1 described as an extension of the external [26, Theorem III.3]; no load-bearing uniqueness step is imported from the authors' own prior work. No parameter is fitted to a subset of data and then renamed a prediction: Stage I reads delays, angles, and path gains from the recovered Vandermonde generators through the defining identities (21) and (43)-(46), and Stages II and III are ML-type and ALS refinements evaluated against simulated ground truth and the likelihood-based CRLB of Section IV-A, whose FIM formula is the standard one cited from Kay [48]. The active-RIS advantage reported in Table IV follows from the amplification model (12)-(13) with eta > 1, but the paper does not tune any parameter to force the outcome; it is a simulation consequence of first-principles settings. The variance and similarity principles of Section III-D classify the recovered columns using structures ('minimal L variances', 'P duplicate columns') that the designs (25)-(28) introduce by construction; this is transparent design-based classification, not a hidden restatement of the claimed predictions. The skeptic's concern -- that Theorem 1 assumes distinct Vandermonde generators and that (41) has no quantified separation threshold -- is a genericity and robustness limitation, which the paper itself acknowledges ('generally a full-rank Vandermonde matrix with distinct generators'); a missing failure-mode analysis is a correctness risk, not a circular reduction of the derivation. Accordingly no step reduces to its own inputs by definition, by fitted parameter, or by a self-citation chain, and the paper is self-contained against external mathematical benchmarks.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the discrete multipath model, the distinctness of delay generators, the feasibility of the Kronecker-structured RIS profile, and an approximation in the CRLB derivation. The hand-picked search and smoothing parameters are design choices rather than fitted constants. No new physical entities are introduced.

free parameters (2)

Stage II search hyperparameters {E_n, I_n, zeta_n, Delta_n^1} = E_n=201, I_n=8, zeta_n=0.5, Delta_n^1=U_n/(10C)
Chosen by hand in Section V-A; they control the optional refinement stage and can affect the reported Stage II accuracy.
Spatial smoothing parameter K1 = K1=15, K2=18 with K=32
Section V-A. This is a design choice that must satisfy min(K1-1,K2)>=R; it is not fitted to data but is a free design parameter of the central uniqueness condition.

assumptions (5)

domain assumption Multipath channels are exactly representable as finite sums of L, P, and Q discrete specular paths with delays and angles as in Eqs. (2)-(4).
Section II-A. The entire tensor model and CRLB depend on this finite-path model; real channels may have diffuse components.
ad hoc to paper The Vandermonde delay generator set is distinct: all R direct and cascaded path delays are different.
Section III-C, condition for VSCPD uniqueness and ESPRIT; the paper says 'generally' but provides no collision analysis.
domain assumption The RIS profile can be exactly set to x^{-1}(T2^H ⊗ T3^H) with T2 and T3 Vandermonde, while respecting the active-RIS phase and amplitude constraint eta.
Eq. (18); this requires |x|*eta=1 and perfect phase control, and hardware impairments are not modeled.
ad hoc to paper The cross-term in the RIS noise covariance in (55) is negligible for the CRLB.
Section IV-A; this approximation affects the claimed closed-form bound, and no validation or error quantification is provided.
standard math Standard linear algebra facts about Khatri-Rao product rank and ESPRIT shift invariance.
Sections III-B and III-C use these facts without proof but they are standard in the tensor signal processing literature.

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Cite this review

Pith. "Pith review of Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems." pith.science (2026). https://pith.science/paper/GNP6KWIZ

@misc{pith2026241116420,
  author       = {Pith},
  title        = {Pith review of: Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNP6KWIZ}},
  note         = {Machine review of arXiv:2411.16420}
}

read the original abstract

Channel parameter recovery is critical for the next-generation reconfigurable intelligent surface (RIS)-empowered communications and sensing. Tensor-based mechanisms are particularly effective, inherently capturing the multi-dimensional nature of wireless channels. However, existing studies assume either a line-of-sight (LOS) scenario or a blocked TX-RX channel. This paper solves a novel problem: tensor-based channel parameter estimation for active RIS-aided multiple-antenna broadband connections in fully multipath environments with the TX-RX link. System settings are customized to construct a fifth-order canonical polyadic (CP) signal tensor that matches the five-dimensional channel. Four tensor factors contain redundant columns, rendering the classical Kruskal's condition for decomposition uniqueness unsatisfied. The fifth-order Vandermonde structured CP decomposition (VSCPD) is developed to address this challenge, making the tensor factorization problem solvable using only linear algebra and offering a relaxed general uniqueness condition. With VSCPD as a perfect decoupling scheme, a sequential triple-stage channel estimation algorithm is proposed based on one-dimensional parameter estimation. The first stage enables multipath identification and algebraic coarse estimation. The following two stages offer optional successive refinements at the cost of increased complexity. The closed-form Cramer-Rao lower bound (CRLB) is derived to assess the estimation performance. Herein, the noise covariance matrix depends on multipath parameters in our active-RIS scenario. Numerical results are provided to verify the effectiveness of proposed algorithms under various evaluation metrics. Our results also show that active RIS can significantly improve channel estimation performance compared to passive RIS.

Figures

Figures reproduced from arXiv: 2411.16420 by the authors.

**Figure 2.** A flowchart of the proposed tensor-based CE framework [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Illustration of reshaping Kronecker product of vect [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: RMSEs of τL, τR, ψ2, ψ3, θL, and θR versus SNR in LOS scenario. B. CE in LOS Scenario Recall that the competing algorithm CPD+ESPRIT+LS is tailored for the LOS scenario, and its performance undoubtedly degrades in the multipath scenario. The estimation performance in …

**Figure 5.** Figure 5: Average computation time with varying SNR in LOS scen [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: RMSEs of τL, τR, ψ2, ψ3, θL, and θR versus SNR in multipath scenario. 10 15 20 25 30 10-6 10-5 10-4 10-3 10-2 10-1 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: NMSE of Y versus SNR in multipath scenario. proposed Stage II has an accuracy advantage over these two existing CBS-based methods, especially in high-SNR cases. Significantly, the complexity of our Stage III (0.14 s) is only 2.1% of LS-refined algorithm (6.81 s), treme…

**Figure 8.** Figure 8: NMSE of Y versus K in multipath scenario [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: Average computation time with varying K and N1 = N2 in multipath scenario. scheme.2 2) Estimation Accuracy in Varying-K Scenario: In this scenario, the number K of training subcarrier is adjusted from 16 to 48 in increments of 8, with K1 consistently set to 0.5K. The S…

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