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An Efficient SIM Beamforming Scheme Based on Physical-Empowered Projected Gradient

Yuzhi Yang, Hao Liu, Tianhao Guo, Jiancheng An, Lu Gan, Chau Yuen, Mérouane Debbah

arXiv:2610.39219v1eess.SP

TL;DR

SIM-assisted MIMO makes channel acquisition difficult because SIM elements cannot directly process signals, while beamforming optimization is non-convex. The paper uses TDD reciprocity for projected over-the-air gradient optimization with ZF variants; simulations report BS-P2GD close to its full-CSI reference within ten over-the-air iterations, while the theoretical result is conditional rather than a global guarantee.

  • Problem

    SIM channel acquisition and precoding are challenging because the SIM–UE channel is indirectly observed and SIM beamforming has a highly non-convex structure.

    Source: A. Motivation · p. 1, 1) Metasurfaces and Their Applications… · p. 2

  • Method

    The proposed framework uses over-the-air gradient feedback and channel reciprocity, with two P2GD variants incorporating ZF constraints.

    Source: C. Contributions · p. 4, §VII · p. 15

  • Results

    BS-P2GD remains close to its full-CSI reference within ten over-the-air iterations.

    Source: §VII · p. 15

  • Takeaways & Limitations

    The framework provides a gradient-based alternative to explicit channel estimation and offers flexible ZF-oriented optimization variants for SIM-assisted systems.

    Source: C. Contributions · p. 4

    The theoretical result provides sufficient conditions for structured projected updates, not a global convergence guarantee for the original non-convex beamforming problem.

    Source: B. Diagonal Limitation of SIM Phase… · p. 10

Abstract

from arXiv · show

Stacked Intelligent Metasurfaces (SIMs) have emerged as a promising technology for next-generation wireless systems. Especially in multi-input multi-output (MIMO) systems, it has the potential to realize complicated wave-domain transformations with low hardware costs and energy consumption. However, SIM-assisted MIMO systems face two major challenges in channel estimation and beamforming. As the SIM elements do not have the signal processing capability, channel estimation can only be conducted in an indirect manner, which makes it challenging. Moreover, the precoding optimization after channel estimation is non-convex and also difficult. Targeting these issues, we exploit the time-division duplex (TDD) channel reciprocity to calculate the gradient backpropagation in an over-the-air manner. Based on this structure, we propose a gradient-based optimization framework that optimizes beamforming without explicitly estimating the SIM--UE channel. The proposed method, namely the physical projected gradient descent (P2GD) method, optimizes the achievable sum rate while incorporating zero-forcing (ZF) constraints. Based on the method of incorporating the ZF restriction, we propose two variations of P2GD, namely user equipment (UE)-P2GD and base station (BS)-P2GD. We also provide a theoretical convergence analysis and projected gradient techniques to ensure physical feasibility. Simulation results show that the proposed approach can achieve suboptimal beamforming in the multi-user system within a small number of over-the-air iterations, using a per-direction signaling budget below the pilot cost of explicit SIM--UE channel estimation. The proposed method provides a low-overhead beamforming scheme for SIM systems without explicit SIM--UE channel estimation.

I. INTRODUCTION

SIM-assisted MIMO can provide low-cost wave-domain processing, but channel acquisition and precoding remain difficult because the SIM–UE channel is indirectly observed and the design is highly non-convex. This paper develops a reciprocity-based gradient framework, reviews related work, and presents its algorithmic and analytical contributions.

A. Motivation

  • SIMs enable complex wave-domain transformations with lower hardware complexity, but indirect SIM–UE channel estimation and non-convex precoding make SIM-assisted MIMO design challenging.The BS–SIM channel is assumed known through physical modeling and calibration, whereas external SIM–UE channels are unknown.Source: A. Motivation · p. 1
  • The paper uses TDD reciprocity for over-the-air gradient backpropagation and optimizes system-level metrics without separately estimating the channel; orthogonal pilots support a ZF penalty by estimating inter-user interference power.Because SIM-layer signals are not directly observable, the method uses projected gradient descent; its physical gradient relation assumes sufficiently accurate reciprocity and an unchanged SIM–UE channel during each exchange.Source: A. Motivation · pp. 1–2

1) Metasurfaces and Their Applications in Beamforming:

  • Prior SIM beamforming work uses stacked layers for richer wave-domain processing, yet coupled layer controls and transmit covariance make precoding non-convex while high-dimensional channels and few RF chains make CSI acquisition pilot-hungry.The section also surveys SIM applications and existing estimation approaches, including model-based, structured, hybrid, decomposition, and learning-based methods.Source: 1) Metasurfaces and Their Applications… · p. 2, 2) Channel Estimation in SIM-assisted… · pp. 2–3

2) Channel Estimation in SIM-assisted Systems:

  • Prior SIM channel-estimation methods address underdetermined acquisition through pilot observations, structured recovery, hybrid processing, tensor decomposition, and learning, but near-field effects and channel dimensionality still burden pilots and computation.The reviewed approaches include LS/MMSE estimation, compressed sensing, hybrid wave-digital estimators, nested decomposition, and neural networks.Source: 2) Channel Estimation in SIM-assisted… · pp. 2–3

C. Contributions

  • The paper proposes a gradient-based TDD SIM-MIMO framework, two ZF-oriented variants, convergence analysis, and simulations comparing the algorithms with conventional schemes.Its stated contributions emphasize reducing conventional channel-estimation overhead and approaching traditional-method performance; the framework's over-the-air gradient transmissions may expose information, a concern left for future work.Source: C. Contributions · pp. 3–4

D. Notations

This section defines the paper's matrix and vector notation and specifies the SIM geometry, propagation model, and channel spatial structure. The model treats the BS–SIM propagation as calibrated and known, while noting that within-layer mutual coupling is not explicitly represented.

D. Notations

  • The notation uses bold uppercase matrices, bold italic lowercase vectors, and nonbold scalars; x_i denotes column i, diag forms a diagonal matrix, and ⊤, H, *, and † denote transpose, Hermitian transpose, conjugate, and pseudoinverse.Source: D. Notations · p. 4

A. SIM Model

  • The SIM has L parallel layers with M reconfigurable meta-atoms per layer (M>N), each using a complex transmission coefficient constrained by |ϕ_l,m| ≤ 1.The metasurface layers use square-lattice UPA arrangements, and the SIM thickness D is divided uniformly between layers.Source: A. SIM Model · p. 4
  • Adjacent-layer propagation is modeled with Rayleigh–Sommerfeld diffraction, and the full cascaded response assumes negligible coupling between nonadjacent layers.The BS–SIM propagation coefficients are assumed calibrated and known at the BS.Source: A. SIM Model · pp. 4–5

1) Array Response and Spatial Structure:

  • Meta-atom positions are measured relative to the last-layer center with spacing d_e=λ_c/2; path angles are independently drawn, and electromagnetic mutual coupling within a layer is not explicitly modeled.Source: 1) Array Response and Spatial Structure · p. 5

2) Large-Scale Fading:

The section specifies user-dependent large-scale fading, formulates the downlink objective with ZF structure, and describes UE-side P2GD pilot and feedback operations without explicit SIM–UE channel estimation.

2) Large-Scale Fading:

  • The UE–SIM horizontal distance is uniformly distributed over the stated range, and its large-scale fading coefficient depends on reference path loss and the path-loss exponent.The model attributes user-specific attenuation to geometric spreading and propagation loss.Source: 2) Large-Scale Fading · p. 5

C. Transmission Model

  • The downlink model uses known BS–SIM propagation matrices and controllable SIM layers, while the SIM–UE channel remains unknown to the BS.Under exact ZF, inter-user interference vanishes and the objective becomes SNR-based; the ZF condition is used as a tractable interference-suppression structure, not claimed to be sum-rate optimal.Source: C. Transmission Model · p. 5, D. Problem Formulation · p. 5

D. Problem Formulation

  • The formulation uses the ZF requirement to separate interference suppression from useful-signal enhancement, while noting that its final constraint cannot be calculated directly because G is unknown.A penalty term incorporates the structured ZF objective into the gradient formulation.Source: D. Problem Formulation · p. 5

III. PROPOSED METHOD BASED ON UE-SIDE ZERO-FORCING

  • UE-P2GD optimizes through trials rather than pre-obtained channel estimates, with UEs computing local loss gradients and the BS reconstructing projected gradients from its known forward model.The BS knows the SIM structure, BS–SIM matrices, and pilots, while the SIM–UE channel and intermediate SIM signals are not directly observed.Source: §III · p. 6

A. Downlink Pilot Transmission

  • Orthogonal DFT pilots let each UE observe pilot energy; under perfect ZF, other streams vanish, and the observed energy corresponds in expectation to the rate objective.The BS predicts intermediate SIM signals that cannot be measured directly, while practical loss computation uses the instantaneous received pilot energy.Source: A. Downlink Pilot Transmission · p. 6

B. UE-Side Zero Forcing

UE-side ZF is encouraged through a regularized penalty, while reciprocal uplink feedback and a pseudoinverse mapping let the BS reconstruct an approximate projected gradient.

B. UE-Side Zero Forcing

  • The UE-side ZF penalty uses a logarithmic interference term and regularization, encouraging interference suppression without enforcing an exactly interference-free solution.The regularization constant prevents numerical singularity as residual interference approaches zero.Source: B. UE-Side Zero Forcing · p. 6

C. Uplink Transmission

  • Under the stated ideal TDD reciprocity assumption, uplink SIM coefficients match downlink controls, and the BS observes only the signal received at its side.The algorithm omits background noise in gradient calculations because noisy gradients are reported not to significantly deteriorate performance.Source: C. Uplink Transmission · p. 7

D. Gradient Backpropagation

  • The BS uses the pseudoinverse of the known forward mapping to reconstruct an approximation of the projected gradient from received uplink feedback.This mapping is needed because the metasurfaces cannot directly provide raw gradient results.Source: D. Gradient Backpropagation · p. 7

E. General Algorithm

The general algorithm applies projected gradient updates with physical bounds, while BS-P2GD uses a distinct BS-side ZF formulation based on desired-signal power.

E. General Algorithm

  • The general update projects SIM coefficients back to their magnitude bounds, updates transmit powers with separate learning rates, and limits uplink power when received pilot energy is too small.Only diagonal elements need to be calculated because the optimization variables are diagonal.Source: E. General Algorithm · p. 7

IV. PROPOSED METHOD BASED ON BS-SIDE ZERO-FORCING

  • BS-P2GD formulates ZF on the BS side by directly considering desired-signal power rather than transforming the ZF requirement into a penalty term.Its optimization problem retains the stated ZF and power constraints.Source: §IV · p. 7
  • The BS-side procedure uses reciprocity and a pseudoinverse to obtain projected gradients from signals unavailable for direct observation, then applies the resulting projected-gradient ascent algorithm.The method derives the necessary BS-known variables from the forward model and uplink feedback.Source: §IV · pp. 7–8

A. Projected Gradient Method

The projected-gradient analysis establishes convergence to suboptimal points under smooth convex assumptions, but the practical SIM beamforming problems are generally non-convex. The section also identifies a tradeoff between observable gradient dimension and signaling overhead.

A. Projected Gradient Method

  • The projected-gradient method converges to suboptimal points under the stated smooth convex setting, with a convergence-rate bound for the prescribed step size.The proof uses a decreasing objective with a lower bound.Source: A. Projected Gradient Method · p. 9
  • The theorem does not characterize practical SIM beamforming convergence, because cascaded SIM mappings, logarithmic signal and interference terms, and physical constraints make those problems generally non-convex.The result is intended as a theoretical reference for the projected-gradient mechanism under idealized conditions.Source: A. Projected Gradient Method · p. 9
  • Increasing N/M enriches the observable gradient subspace and reduces potential projection loss, but also increases pilot and signaling overhead.The method is particularly suitable when the observable dimension N is not excessively small relative to the number of SIM variables M.Source: A. Projected Gradient Method · p. 10

B. Diagonal Limitation of SIM Phase Matrices

The cascaded SIM response is restricted by layer structure and propagation ranks, so the analysis first relaxes its realizability constraint. Its projected-gradient conclusions are conditional on stated assumptions and do not establish global convergence for the original non-convex problem.

B. Diagonal Limitation of SIM Phase Matrices

  • The aggregate SIM response is limited by diagonal layer structure, bounded coefficients, and inter-layer ranks, so its achievable set may not span the full matrix space.The analysis therefore assumes the aggregate response is freely variable, relaxing the cascaded SIM realizability constraint.Source: B. Diagonal Limitation of SIM Phase… · p. 10
  • Under the relaxed realizability assumption and directional conditions, the theorem gives sufficient conditions for the structured projected update, not global convergence of the original non-convex beamforming problem.The practical problem remains non-convex because of the cascaded mapping, logarithmic signal and interference terms, and physical SIM constraints.Source: B. Diagonal Limitation of SIM Phase… · p. 10
  • When the stated conditions hold throughout the iteration, the projected-gradient residuals tend to zero, establishing projected stationarity.The proof bounds the sum of squared residuals and uses a telescoping argument.Source: B. Diagonal Limitation of SIM Phase… · pp. 10–11
  • As the number of SIM layers increases, the achievable cascaded-response set generally becomes richer, making the relaxation less restrictive.If the achievable set occupies only a subspace, the analysis can be interpreted within that reachable subspace; the theorem remains conditional on alignment and magnitude-control properties along the trajectory.Source: B. Diagonal Limitation of SIM Phase… · p. 11

C. Influence of Noise

The analysis omits backward-transmission noise, although noise is unavoidable in practice. It discusses both a convergence-rate cost in convex problems and a possible saddle-escape benefit in non-convex problems, without guaranteeing a global optimum.

C. Influence of Noise

  • Backward-transmission noise is omitted in the proposed method, although it is unavoidable in wireless systems.In convex problems, adding noise after each step does not affect convergence but reduces the convergence rate by a constant multiple determined by the noise level.Source: C. Influence of Noise · p. 11
  • Small random perturbations can help iterations escape saddle regions, but convergence to a global optimum is not guaranteed.The SIM-coefficient objectives are non-convex, and the projection can create saddle-like stationary regions where the projected gradient vanishes while the true gradient does not.Source: C. Influence of Noise · p. 11

D. Transmission and Computational Complexity

P2GD exchanges downlink pilots and uplink gradient feedback each iteration, while explicit CSI acquisition must identify the SIM–UE channel. Its signaling and computational costs are most attractive when a satisfactory solution is reached in few iterations.

D. Transmission and Computational Complexity

  • Each P2GD iteration requires N downlink pilot symbols and N uplink feedback symbols, while explicit SIM–UE channel estimation has MN unknown coefficients and an ideal lower bound of M pilot symbols.The downlink and uplink costs are reported separately because they may use different radio resources; practical estimation may require more than the ideal lower bound.Source: D. Transmission and Computational… · p. 11
  • Episode latency includes downlink and uplink symbols plus SIM reconfiguration, control, synchronization, and UE-gradient processing delays.The latency expression scales these per-iteration costs by the iteration count T.Source: D. Transmission and Computational… · p. 11
  • P2GD is particularly attractive when few iterations yield a satisfactory solution, keeping accumulated signaling below or comparable to explicit CSI acquisition.This is increasingly relevant for large M, as explicit channel-estimation cost grows with the unknown spatial dimensions while per-iteration P2GD signaling is determined by N.Source: D. Transmission and Computational… · p. 11
  • O(TLNM^2) total computational complexity lets the proposed framework avoid cubic dependence on SIM size and remain tractable for large-aperture metasurfaces.The dominant per-iteration costs include effective SIM-response updates and matrix multiplications; diagonal-gradient projection is comparatively negligible.Source: D. Transmission and Computational… · pp. 11–12

E. Conclusion Remark

The paper recommends a limited P2GD iteration budget in practical operation because each iteration adds signaling overhead. In the reported default setting, ten iterations require 30 uplink and 30 downlink symbols, compared with 66 uplink training symbols for the estimation-based baselines.

E. Conclusion Remark

  • P2GD uses a limited iteration budget Tmax in practice because excessive iterations add signaling overhead and can diminish the benefit of avoiding explicit channel estimation.A more aggressive step size may be used to obtain a satisfactory solution within the available coherence and signaling budget.Source: E. Conclusion Remark · p. 12

A. Simulation Scenario

  • The simulation averages results over 10,000 realizations, and Fig. 3 error bars show ±0.1 times the empirical standard deviation, not standard errors or confidence intervals.The larger sample reduces fluctuations in average curves but does not imply smooth behavior in every realization.Source: A. Simulation Scenario · p. 12, Figure 3 · p. 13

B. Comparison Algorithms

  • The comparison includes UE-P2GD and BS-P2GD, their full-CSI gradient references, and two channel-estimation-based Yao baselines; the full-CSI references are not guaranteed upper bounds for finite-iteration non-convex optimization.The plotted iterations exclude the Yao baselines' preceding channel-estimation cost, so the figure compares optimization trajectories rather than equal total signaling or computational budgets.Source: B. Comparison Algorithms · pp. 12–13

C. Signaling Overhead Comparison

  • 30 uplink and 30 downlink MIMO symbols are required for ten P2GD iterations, compared with 66 uplink training symbols for the 22-block Yao baselines.The comparison is for N = 3, M = 64, and T = 10; equal iteration counts do not imply equal computational time.Source: C. Signaling Overhead Comparison · p. 13, Table II · p. 13

D. Comparison Among Different Methods

BS-P2GD stays close to its full-CSI reference within ten over-the-air iterations, while explicit-CSI methods achieve higher rates at greater pilot cost. The results also show that finite-iteration performance depends on tuning and system conditions, with BS-P2GD generally less sensitive to the adaptive penalty target than UE-P2GD.

E. Effect of Learning Rate and Iteration Steps

  • Larger learning rates generally improve finite-budget rates, but UE-P2GD can decline slightly late in the run at the largest tested value.The tested learning rates are η ∈ {5, 50, 500, 5000}; step-size choice should reflect the available signaling budget.Source: E. Effect of Learning Rate and… · pp. 13–14, Figure 4 · p. 14

F. Effect of Background Noise Power

G. Effect of SIM Elements per Layer and UE Numbers

  • For the tested SIM arrays and UE counts, rates generally improve with larger arrays, peak at three UEs, then decrease as more users are served.The decline is more pronounced for UE-P2GD, while BS-P2GD performs better at four and five UEs; these are achieved rates, not optimal system rates.Source: G. Effect of SIM Elements per Layer… · p. 14, Figure 6 · p. 14

H. Sensitivity to the Adaptive λ Ratio Target

D. Comparison Among Different Methods

  • Within ten over-the-air iterations, BS-P2GD remains close to its full-CSI reference, while explicit-CSI methods achieve higher average sum rates at larger pilot cost.The comparison averages 10,000 realizations; error bars describe variability across realizations, not stability guarantees for individual runs.Source: D. Comparison Among Different Methods · p. 13, §VII · p. 15

VII. CONCLUSION AND FUTURE DIRECTIONS

  • The authors identify surrogate-loss functions and finite-iteration parameter choices as factors that influence achieved sum rate, especially for UE-P2GD.Improving alignment between the surrogate losses and sum-rate objective remains future work.Source: §VII · p. 15

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