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To Stack or Not to Stack? Jacobian Conditioning and the Benefits of Depth in Metasurface-Based Linear Transformation Synthesis

Andrea Abrardo, Andrea Garulli

arXiv:2610.20377v1eess.SP

TL;DR

The paper asks when stacking improves linear transformation synthesis at a fixed phase-control budget, given that Jacobian rank alone does not capture sensitivity strength and balance. It analyzes layer complementarity in a cascade and assesses the mechanism with a multiport Jacobian. In the considered sensing scenarios, distributing controls across layers improves weak-direction sensitivity and synthesis performance.

  • Problem

    At a fixed phase-control budget, it is unclear when stacking helps, because Jacobian rank alone does not capture the strength and balance of transformation sensitivities.

    Source: §I · p. 1

  • Method

    The paper analyzes layer-resolved sensitivities in the cascade model and tests their implications using a multiport Jacobian with electromagnetic coupling and multiple scattering.

    Source: §VII · p. 11

  • Results

    In the considered near- and far-field sensing scenarios, distributing controls across multiple layers improves weak-direction sensitivity and synthesis performance.

    Source: §VII · p. 11

  • Takeaways & Limitations

    Layer complementarity and Jacobian conditioning are useful criteria for designing SIM architectures under a fixed control budget.

    Source: §VII · p. 11

    For any NT > 1, the analyzed single-layer configuration has at least NT (NT −1) inaccessible real transformation directions, independently of phase settings.

    Source: A. Geometric Limitations of a Single… · p. 6

Abstract

from arXiv · show

This paper investigates when and why stacking programmable metasurfaces into Stacked Intelligent Metasurfaces (SIMs) improves linear transformation synthesis at a fixed phase-control budget. The analysis shows that algebraic Jacobian rank alone is insufficient to characterize practical synthesis, which also depends on the strength and balance of the sensitivity directions generated by the available controls. A layer-resolved analysis of the cascade model explains how inter-layer propagation can generate complementary sensitivity directions. The relevance of this mechanism is then assessed using an S-parameter multiport model that includes electromagnetic coupling and multiple scattering. Numerical results in near- and far-field sensing scenarios show that distributing the same control resources across multiple layers can strengthen weak transformation directions, improve Jacobian conditioning, and substantially enhance synthesis accuracy. These findings explain why increasing depth can offer benefits that are not obtained by simply enlarging a single programmable surface, and identify layer complementarity and Jacobian conditioning as useful criteria for SIM architecture design.

I. INTRODUCTION

The paper asks whether stacking helps linear transformation synthesis when the total phase-control budget is fixed. It argues that practical synthesis depends on the strength and balance of sensitivity directions, not only on Jacobian rank.

I. INTRODUCTION

  • At a fixed control count, stacking is useful when it improves sensitivity balance, since even a full-rank Jacobian can have weak directions that make synthesis difficult.Source: §I · p. 1

A. Related Works and Motivation

B. Main Contributions

  • The contributions include a local inverse-problem account of how singular values and residual alignment affect Gauss–Newton corrections and gradient progress.Source: B. Main Contributions · p. 2
  • A layer-resolved cascade analysis explains how lateral enlargement can saturate geometrically while depth can supply complementary directions and improve conditioning.Source: B. Main Contributions · p. 2
  • Exact multiport sensitivities are evaluated for equal-budget wide and deep SIMs in near-field position and far-field angular sensing; effective rank tracks synthesis accuracy in the experiments.Source: B. Main Contributions · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The S-parameter model represents transmitter, programmable, and receiver ports, with programmable phase controls and electromagnetic coupling among programmable ports.Source: §II · p. 2

A. Relation to the Unilateral Cascade

The unilateral cascade is an idealized, analytically tractable SIM representation. It assumes one-way inter-layer propagation and ideal phase-only cells, unlike the complete model’s retained global electromagnetic interactions.

A. Relation to the Unilateral Cascade

B. Linear Transformation Synthesis Problem

The paper formulates synthesis as matching the SIM output to a prescribed linear operation on selected excitations, up to a global complex scale. It then studies local phase updates through a real Jacobian and its singular-value spectrum.

B. Linear Transformation Synthesis Problem

III. LOCAL INVERSE SYNTHESIS AND OPTIMIZATION

  • At each local update, the optimal complex scaling is evaluated first, held fixed while constructing the phase update, and recomputed after the update.Source: §III · p. 4
  • The local phase update uses a real Jacobian because the controls are real while the transformation is complex; the scaling factor is held fixed during differentiation.Source: §III · p. 4
  • The local inverse problem depends on both Jacobian rank, which determines locally accessible directions, and singular values, which determine their response strength.Source: §III · p. 4

1) Gauss–Newton Interpretation:

Weak Jacobian directions affect both local optimization methods: Gauss–Newton can require large corrections, while gradient descent can progress slowly. Residual alignment with those directions influences the practical impact.

1) Gauss–Newton Interpretation:

  • Error components along weak singular directions require larger Gauss–Newton phase corrections than equally sized components along strong directions.Source: 1) Gauss–Newton Interpretation · p. 4
  • In the nonlinear transformation, large corrections may leave the accurate linearization region, so step reduction limits predicted progress across all directions.Source: 1) Gauss–Newton Interpretation · p. 4

2) Gradient-Descent Interpretation:

  • For fixed step size, the weakest-direction contraction factor approaches one as the inverse condition ratio decreases, indicating slower gradient correction.Source: 2) Gradient-Descent Interpretation · p. 4
  • Gradient descent makes limited progress along weak directions when its step is constrained by the strongest sensitivity, and the effect depends on residual alignment.Source: 2) Gradient-Descent Interpretation · pp. 4–5
  • For nonzero scaling, the scaled and unscaled real Jacobians have the same rank, condition number, and normalized spectrum, so the architectural analysis uses the unscaled Jacobian.Source: 2) Gradient-Descent Interpretation · p. 5

IV. CASCADED INTERPRETATION OF THE JACOBIAN CONDITIONING

The cascade model’s layer-resolved Jacobian shows why control count alone is insufficient: synthesis depends on the strength and balance of sensitivity directions. Propagation around each layer reshapes those directions, allowing layers to contribute complementary sensitivity directions.

IV. CASCADED INTERPRETATION OF THE JACOBIAN CONDITIONING

  • Full row rank does not ensure effective synthesis because correlated control sensitivities can leave the Jacobian poorly conditioned.The cascade decomposition makes each control’s contribution explicit, and the mechanism is later assessed with the complete multiport model.Source: §IV · p. 5
  • Each phase control generates a rank-one matrix perturbation whose vectorization forms a Kronecker-structured sensitivity direction.The Jacobian frame can be decomposed by layer.Source: §IV · p. 5
  • A direction weakly supported by one layer can receive substantial contributions from other layers, improving aggregate sensitivity across the cascade.The ratio of weakest to strongest aggregate sensitivities is small when a transformation direction remains weak after combining layers.Source: §IV · p. 5

A. Geometric Limitations of a Single Layer and the Role of Depth

A single phase screen has structural geometric limits, while independently controlled layers with inter-layer mixing can generate directions outside its symmetric subspace. Stacking removes the universal single-layer restriction but does not guarantee recovery of every direction or good conditioning.

A. Geometric Limitations of a Single Layer and the Role of Depth

  • A poorly conditioned geometry-dependent matrix K imposes a phase-independent upper bound on real-Jacobian conditioning.The signatures are fixed by propagation geometry, so phase optimization cannot overcome a small eigenvalue ratio of K.Source: A. Geometric Limitations of a Single… · p. 6
  • At least NT (NT −1) real transformation directions are inaccessible in the single-layer configuration, independently of phase settings.The single-layer signatures lie in the complex symmetric subspace, forcing the minimum eigenvalue of G(ϕ) to zero for NT > 1.Source: A. Geometric Limitations of a Single… · p. 6
  • With independently controlled layers, a symmetric operating transformation can have Jacobian directions beyond the single-layer symmetric subspace.This occurs when the inter-layer contribution generates a nonzero antisymmetric terminal direction; symmetric terminal placement does not impose the same restriction on a two-layer architecture.Source: A. Geometric Limitations of a Single… · p. 6
  • Stacking does not guarantee recovery of all missing directions or a well-conditioned Jacobian.The mechanism requires inter-layer spatial mixing and suitable terminal coupling; a diagonal inter-layer propagation matrix preserves the single-layer symmetry restriction.Source: A. Geometric Limitations of a Single… · p. 6

B. Numerical Illustration with Square SIM Layers

At approximately equal phase-control budgets, the square-SIM experiments show conditioning improving with depth across terminal spacings. A single layer loses antisymmetric sensitivities at equal Tx/Rx distances, while multilayer propagation restores them; increasing single-layer aperture cannot recover those excluded directions.

B. Numerical Illustration with Square SIM Layers

  • The scalar-cascade comparison uses square grids and approximately equal total phase-control budgets across Q = 1, 2, 3, and 4 architectures.Each layer has half-wavelength element spacing; the transmitter and receiver each have four elements.Source: B. Numerical Illustration with Square… · pp. 6–7
  • Conditioning improves with layer count across the tested terminal spacings, with a significant gain already at Q = 2 over the uniformly poorly conditioned single layer.The heatmaps average 20 random phase profiles and mark each architecture’s maximum conditioning.Source: B. Numerical Illustration with Square… · p. 7, Figure 1 · p. 7
  • At dT = dR, mirror symmetry limits the single-layer Jacobian to at most 20 of 32 real dimensions, giving γ = 0; multilayer panels retain nonzero conditioning there.Inter-layer propagation and independent phase control permit the additional antisymmetric sensitivities.Source: B. Numerical Illustration with Square… · p. 7
  • Increasing single-layer aperture cannot recover the antisymmetric transformation directions excluded by structural sensitivity loss, whereas tested multilayer configurations provide substantially better conditioning.The single-layer curve flattens near 0.02, and the phase-independent Gramian bound closely follows its mean conditioning across the aperture range.Source: B. Numerical Illustration with Square… · pp. 7–8

V. MULTIPORT TRANSFORMATION JACOBIAN

The multiport transformation Jacobian is derived from the complete coupled electromagnetic model. Its exact sensitivities retain a left–local perturbation–right structure while incorporating the coupled network solution.

V. MULTIPORT TRANSFORMATION JACOBIAN

  • The complete multiport model is used to evaluate local transformation sensitivities while accounting for reciprocal coupling, multiple reflections, and load-dependent field redistribution.This extends the cascade-based sensitivity analysis to the coupled electromagnetic network.Source: §V · p. 8
  • Equation (58) preserves a left–local perturbation–right factorization, but its surrounding factors contain the solution of the complete coupled network.A programmable-cell perturbation can therefore affect the transformation through reciprocal coupling and repeated scattering, not only forward cascade paths.Source: §V · p. 8
  • The block-diagonal cell structure makes each cell’s perturbation matrix nonzero only in its associated block, enabling efficient sensitivity computation.The passage relates this structure to low-complexity optimization gradient descent approaches in, [19].Source: §V · p. 8

A. Simulation Setup and Performance Metrics

The simulations compare wide and deep architectures under a fixed overall phase budget using a complete multiport electromagnetic model. Near-field position and far-field angular sensing geometries are assessed through Jacobian effective rank and synthesis error.

A. Simulation Setup and Performance Metrics

  • Architectures are compared using the complete multiport electromagnetic model while preserving the same total number of programmable phase variables.The physical model includes dipole self and mutual impedances, intra-layer coupling, reciprocal inter-layer coupling, and multiple scattering.Source: A. Simulation Setup and Performance… · p. 8
  • Increasing depth reduces programmable cells per layer, so the width–depth tradeoff is tested without changing the total number of controllable real parameters.The listed wide and deep configurations all use P = 6144 phase controls.Source: A. Simulation Setup and Performance… · pp. 8–9
  • The sensing inputs cover near-field angle–range positions NF(8, 2) and NF(8, 4), and far-field spatial DFT modes FF(16) and FF(32).The corresponding output dimension is D = 2MI, with M = I.Source: A. Simulation Setup and Performance… · p. 9
  • Normalized effective rank ρJ(τ) reports the fraction of Jacobian singular values above a prescribed relative threshold, complementing rank and condition number.For example, ρJ(10^-5) = 1 means all D singular values lie within five orders of magnitude of the largest.Source: A. Simulation Setup and Performance… · p. 9, Figure 3 · p. 9

B. Random-Phase Multiport Jacobian

In NF(8, 4), only deep configurations reach the displayed synthesis accuracy floor for reachable targets, while wide configurations make little progress. Random-phase spectra show deeper architectures can better retain weak directions in demanding scenarios, and the layer-resolved test supports complementary sensitivities across layers.

B. Random-Phase Multiport Jacobian

  • At smaller relative thresholds, deep architectures retain a larger fraction of weak output directions in NF and FF(32), especially D[32×16, 12] in the NF cases.The ordering is not uniform at every threshold: some wide architectures retain more strong directions, and W[96×64] has a favorable FF(16) spectrum.Source: B. Random-Phase Multiport Jacobian · p. 9

C. Synthesis of a Reachable Random Target

  • Only deep configurations reach the displayed synthesis NMSE floor in NF(8, 4), while wide configurations show little error reduction despite targets being reachable by construction.Across 20 trials per architecture, deep configurations also begin with smaller dmax and keep Armijo factors near unity; wide configurations retain large dmax and require substantially reduced steps.Source: C. Synthesis of a Reachable Random… · p. 10, Figure 4 · p. 10

D. Multiport Test of Layer Complementarity

  • For all six tested layers, the remaining layers compensate the layer’s weakest direction, with minimum sensitivity gaining proportionally more than maximum sensitivity and conditioning improving.The layer-resolved experiment uses D[64×16, 6] in NF(8, 2) at 100 random configurations without optimization.Source: D. Multiport Test of Layer… · p. 10

E. Identity-Target Synthesis as an Additional Check

Identity-target synthesis provides an additional sensing-style check, while the conclusions report that distributing a fixed control budget across layers improves synthesis in the considered near- and far-field scenarios.

E. Identity-Target Synthesis as an Additional Check

  • The identity-target transformation associates each input hypothesis with one output port while suppressing others, for angular discrimination in far-field and position discrimination in near-field sensing.Unlike the preceding random targets, this transformation is not guaranteed reachable by construction.Source: E. Identity-Target Synthesis as an… · p. 10

VII. CONCLUSIONS

  • The cascade model explains layer complementarity, and multiport Jacobian tests show layers compensating weak sensitivity directions despite electromagnetic coupling and multiple scattering.Source: §VII · p. 11
  • Figures 7–8 track identity-target normalized synthesis error over iterations for gradient descent and Gauss–Newton across four sensing scenarios.Source: Figure 7 · p. 11, Figure 8 · p. 11
  • In the considered near- and far-field sensing scenarios, distributing controls across layers improves weak-direction sensitivity and synthesis performance at a fixed total phase budget.Source: §VII · p. 11

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