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arxiv: 2604.18852 · v1 · submitted 2026-04-20 · 📡 eess.SP

Multi-Target Estimation via Tensor Decomposition for Beyond Diagonal RIS-Aided Bistatic Sensing

Kenneth Ben\'icio , Andr\'e L. F. de Almeida , Fazal-E Asim , Bruno Sokal , Gabor Fodor , Behrooz Makki , A. Lee Swindlehurst This is my paper

Pith reviewed 2026-05-10 03:24 UTC · model grok-4.3

classification 📡 eess.SP
keywords beyond-diagonal RIStensor decompositionmulti-target sensingbistatic MIMOPARAFACKronecker sum approximationdelay-Doppler estimationangle estimation
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The pith

A two-stage tensor estimator recovers multiple target parameters in BD-RIS bistatic sensing without knowledge of the RIS architecture.

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

The paper develops a TenDAE framework that first approximates the multi-target received signal through a Kronecker sum of rank equal to the number of targets. It then applies a nested tensor factorization estimation stage consisting of two parallel decompositions: a standard PARAFAC for angles and a nested PARAFAC for the coupled delay-Doppler parameters. This structure decouples the three parameter domains and works identically for both beyond-diagonal and classical diagonal RIS by switching only the first-stage approximation. The approach is shown to operate with limited sensing resources while delivering estimation accuracy close to the derived Cramér-Rao bound.

Core claim

The nested tensor factorization estimation framework, applied after an initial Kronecker sum approximation, enables simultaneous estimation of all targets' angles, delays, and Doppler shifts in BD-RIS-aided bistatic MIMO sensing by decoupling the domains through two parallel tensor decompositions solved via alternating least squares or higher-order singular value decomposition followed by ESPRIT, and this framework remains blind to whether the RIS is beyond-diagonal or diagonal.

What carries the argument

The nested tensor factorization estimation (NTFE) that uses a PARAFAC decomposition to extract angles and a nested PARAFAC decomposition to extract delay-Doppler parameters after a Kronecker sum approximation of the multi-target signal.

If this is right

  • The same NTFE pipeline applies to both beyond-diagonal and diagonal RIS architectures by changing only the first-stage matrix approximation.
  • Angles are recovered independently of delay-Doppler parameters, allowing separate high-resolution subspace processing.
  • All targets are estimated simultaneously from a single received tensor using a number of measurements that scales with the tensor rank rather than the individual parameter dimensions.
  • Performance is benchmarked against the Cramér-Rao lower bound derived for the joint estimation problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The domain decoupling may allow real-time implementations to allocate different computational budgets to angle versus delay-Doppler processing.
  • Because the method does not embed knowledge of the RIS phase-shift matrix, it could be deployed in scenarios where the surface configuration is reconfigured on the fly without recalibration.
  • The same nested decomposition structure could be tested on other multidimensional sensing problems that exhibit Kronecker or Khatri-Rao type signal models.

Load-bearing premise

The received signal model possesses a multidimensional structure that permits accurate Kronecker sum approximation in stage one and unique separation via nested PARAFAC decompositions in stage two.

What would settle it

Numerical results in which the root-mean-square error fails to decrease toward the derived Cramér-Rao bound as the number of snapshots or antennas increases while the assumed tensor rank equals the number of targets.

Figures

Figures reproduced from arXiv: 2604.18852 by A. Lee Swindlehurst, Andr\'e L. F. de Almeida, Behrooz Makki, Bruno Sokal, Fazal-E Asim, Gabor Fodor, Kenneth Ben\'icio.

Figure 3
Figure 3. Figure 3: Illustration of the noiseless PARAFAC tensor [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time-domain protocol. We assume the target param [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the noiseless nested PARAFAC tensor [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Block diagram of the proposed TenDAE framework: (i) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Complexity as a function of the parameters in Table I. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Channel estimation performance in terms of NMSE as [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Single-target RMSE comparison between the pro [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: RMSE performance of the proposed solution in Alg. 1 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evaluation of the proposed solution in Alg. 1 and Alg. [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
read the original abstract

We investigate the performance of beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) for bistatic MIMO multi-target sensing using a two-stage tensor Doppler-delay-angle estimation (TenDAE). The first stage solves a Kronecker sum approximation (KSA) with a rank equal to the number of targets. The second stage employs a nested tensor factorization estimation (NTFE) that exploits the inherent multidimensional structure via two tensor decompositions that are solved in parallel. The first employs a PARAFAC decomposition to extract the targets' angles, and the second uses a nested PARAFAC decomposition to find the targets' delay and Doppler parameters. This two-stage approach decouples acquisition of the angles and delays/Dopplers using either alternating least squares or a higher-order singular value decomposition, followed by a high-resolution subspace technique, such as ESPRIT. We further compare the performance of a BD-RIS with a classical diagonal RIS. For the latter, we solve a Khatri-Rao sum approximation problem rather than the KSA due to the specific structure of the received signal. Notably, our NTFE framework remains blind to the underlying RIS architecture while simultaneously estimating all targets with minimal sensing resources. Additionally, we show that employing a nested-PARAFAC decomposition enables the decoupling of the delay-Doppler and angle domains. We also derive the Cram\'er-Rao lower bound to further assess the performance of the TenDAE framework. Finally, we numerically evaluate the solutions presented in this paper and demonstrate their efficiency in terms of RMSE compared with state-of-the-art approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a two-stage TenDAE framework for multi-target estimation in BD-RIS-aided bistatic MIMO sensing. Stage 1 applies a Kronecker sum approximation (KSA) of rank equal to the number of targets to the received signal tensor. Stage 2 uses nested tensor factorization estimation (NTFE) via PARAFAC decomposition for target angles and a nested PARAFAC decomposition for delay-Doppler parameters, solved with ALS or HOSVD followed by ESPRIT. The method is claimed to be blind to RIS architecture (switching to Khatri-Rao sum approximation only for diagonal RIS), derives the CRLB, and reports superior RMSE performance versus SOTA methods with minimal sensing resources.

Significance. If the KSA approximation is accurate and the nested decompositions yield unique factors, the work would enable architecture-agnostic, resource-efficient multi-target sensing in advanced RIS systems by decoupling angle and delay-Doppler domains. The explicit BD-RIS versus diagonal RIS comparison and CRLB benchmark add practical value for evaluating tensor methods in bistatic sensing.

major comments (2)
  1. [Abstract] Abstract: The claim that 'our NTFE framework remains blind to the underlying RIS architecture' is contradicted by the explicit statement that a Khatri-Rao sum approximation is solved 'rather than the KSA due to the specific structure of the received signal' when the RIS is diagonal. This indicates that the first-stage solver choice depends on RIS type, so the framework is not fully architecture-blind.
  2. [Stage 1 (KSA)] Stage-1 KSA description: The received signal tensor is formed as the product of Tx steering vectors, the full (non-diagonal) BD-RIS reflection matrix, target responses, and Rx steering vectors. No error bound or residual analysis is provided for the low-rank Kronecker sum approximation; if the residual is non-negligible, uniqueness of the subsequent PARAFAC and nested-PARAFAC factors is not guaranteed, directly undermining the central performance and blindness claims.
minor comments (3)
  1. [Abstract] Abstract: Simulation details (number of targets, SNR range, RIS element count, specific BD-RIS matrix realizations, and Monte Carlo trials) are omitted, making it impossible to assess the reported RMSE gains.
  2. [Signal model section] Notation and tensor dimensions: The dimensions of the received signal tensor and the exact mapping from KSA factors to angle/delay-Doppler parameters should be stated explicitly before the decomposition steps.
  3. [CRLB section] CRLB derivation: The CRLB expression is referenced but its derivation steps (especially handling of the BD-RIS matrix) are not shown; a short appendix or key steps would improve verifiability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address the major comments point by point below, indicating the changes made to the revised version.

read point-by-point responses

  1. Referee: The claim that 'our NTFE framework remains blind to the underlying RIS architecture' is contradicted by the explicit statement that a Khatri-Rao sum approximation is solved 'rather than the KSA due to the specific structure of the received signal' when the RIS is diagonal. This indicates that the first-stage solver choice depends on RIS type, so the framework is not fully architecture-blind.

    Authors: We agree that the original wording in the abstract could be misinterpreted. The NTFE framework is designed to operate without knowledge of the RIS configuration details or target parameters, hence 'blind' in the estimation sense. The adaptation of the approximation method (KSA vs. Khatri-Rao sum) is a preprocessing step that leverages the known RIS architecture type but does not require specific values. In the revised manuscript, we have updated the abstract to clarify: 'our NTFE framework adapts to the RIS architecture in its initial approximation stage while remaining blind to the reflection coefficients and target parameters.' This revision ensures consistency with the method description. revision: yes

  2. Referee: The received signal tensor is formed as the product of Tx steering vectors, the full (non-diagonal) BD-RIS reflection matrix, target responses, and Rx steering vectors. No error bound or residual analysis is provided for the low-rank Kronecker sum approximation; if the residual is non-negligible, uniqueness of the subsequent PARAFAC and nested-PARAFAC factors is not guaranteed, directly undermining the central performance and blindness claims.

    Authors: The referee raises a valid point regarding the lack of theoretical analysis for the KSA residual. In the current work, the validity of the approximation is supported by the numerical results showing superior performance. For the revision, we have added a paragraph discussing the conditions under which the KSA residual is small, namely when the targets are angularly separated and the BD-RIS provides sufficient degrees of freedom. We also note that the nested PARAFAC decompositions are robust to small perturbations, as per existing tensor perturbation theory. A complete closed-form error bound is beyond the scope of this paper but is identified as future work. This addition addresses the concern without changing the main results. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard tensor decompositions to a received-signal model

full rationale

The paper's two-stage TenDAE framework applies established techniques—Kronecker sum approximation (or Khatri-Rao sum for diagonal RIS), PARAFAC, and nested PARAFAC—to a multidimensional received-signal tensor whose structure follows directly from the bistatic channel model. The NTFE claim of architecture blindness is implemented by switching the first-stage solver according to RIS type, which is an explicit adaptation rather than a self-defining or fitted-input reduction. CRLB derivation and numerical RMSE comparisons are independent of the estimation procedure itself. No load-bearing step reduces by construction to a prior result from the same authors or to a parameter fit that is then renamed as a prediction. The central performance assertions therefore rest on the validity of the tensor model and the convergence properties of the cited decompositions, not on tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from array signal processing and tensor algebra rather than new postulates or fitted parameters.

axioms (2)
  • domain assumption The received signal admits a Kronecker sum approximation whose rank equals the number of targets.
    Invoked explicitly for the first stage of TenDAE.
  • domain assumption PARAFAC and nested PARAFAC decompositions can uniquely extract angle, delay, and Doppler parameters from the tensorized signal.
    Core premise of the second-stage NTFE.

pith-pipeline@v0.9.0 · 5617 in / 1172 out tokens · 74317 ms · 2026-05-10T03:24:08.834354+00:00 · methodology

discussion (0)

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Reference graph

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