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Closed-Form Global Optimization of Beyond Diagonal Reconfigurable Intelligent Surfaces

Matteo Nerini, Shanpu Shen, Bruno Clerckx

arXiv:2211.06117v2cs.ITeess.SPmath.IT

TL;DR

Group- and fully connected BD-RISs lacked a closed-form globally optimal scattering-matrix design, leaving upper-bound tightness supported only numerically. This paper derives a closed-form solution and extends it to other system settings. It proves exact upper-bound attainment for SISO across channel realizations and for specified multi-antenna settings.

  • Problem

    A closed-form globally optimal scattering matrix for group- and fully connected BD-RISs was unavailable, and upper-bound tightness had been shown only numerically.

    Source: §I · p. 2

  • Method

    The paper derives a closed-form global-optimal design for group- and fully connected RISs and extends the strategy to single-user MIMO and multi-user MISO systems.

    Source: §I · p. 2, §VIII · p. 12

  • Results

    The solution exactly achieves SISO received-power upper bounds for any channel realization, with linear complexity for group-connected and cubic complexity for fully connected RISs.

    Source: §I · p. 2, §VIII · p. 13

  • Takeaways & Limitations

    The SISO upper bounds are mathematically proved tight rather than established only numerically.

    Source: §I · p. 2

Abstract

from arXiv · show

Reconfigurable intelligent surfaces (RISs) allow controlling the propagation environment in wireless networks by tuning multiple reflecting elements. RISs have been traditionally realized through single connected architectures, mathematically characterized by a diagonal scattering matrix. Recently, beyond diagonal RISs (BD-RISs) have been proposed as a novel branch of RISs whose scattering matrix is not limited to be diagonal, which creates new benefits and opportunities for RISs. Efficient BD-RIS architectures have been realized based on group and fully connected reconfigurable impedance networks. However, a closed-form solution for the global optimal scattering matrix of these architectures is not yet available. In this paper, we provide such a closed-form solution proving that the theoretical performance upper bounds can be exactly achieved for any channel realization. We first consider the received signal power maximization in single-user single-input single-output (SISO) systems aided by a BD-RIS working in reflective or transmissive mode. Then, we extend our solution to single-user multiple-input multiple-output (MIMO) and multi-user multiple-input single-output (MISO) systems. We show that our algorithm is less complex than the iterative optimization algorithms employed in the previous literature. The complexity of our algorithm grows linearly (resp. cubically) with the number of RIS elements in the case of group (resp. fully) connected architectures.

I. INTRODUCTION

The introduction motivates beyond-diagonal RISs as architectures whose non-diagonal scattering matrices offer more design flexibility, while noting that globally optimal designs had relied on costly iterative methods. The paper presents closed-form designs and extensions across several system settings, and outlines the system model and single-connected architecture.

I. INTRODUCTION

  • The algorithm is less complex than prior iterative methods, with complexity growing linearly for group-connected and cubically for fully connected RISs.The paper describes this as the first closed-form global-optimal solution for BD-RISs despite the optimization problems’ non-convexity.Source: §I · p. 2
  • A closed-form globally optimal scattering matrix for group- and fully connected BD-RISs exactly attains SISO received-power upper bounds for any channel realization.The guarantee does not assume a particular channel distribution or correlation.Source: §I · p. 2
  • For transmissive BD-RISs enabled by the cell-wise group-connected architecture, the solution also exactly achieves performance upper bounds.The cell-wise group-connected architecture is proposed in.Source: §I · p. 2
  • The strategy extends to single-user MIMO, including MISO; for fully connected RISs with negligible direct links, it exactly achieves a tight received-power upper bound.With non-negligible direct links, the paper proposes a suboptimal solution and notes that a tight bound is unknown.Source: §I · p. 2
  • For multi-user MISO weighted-sum-power maximization, a fully connected RIS with negligible direct links has a tight upper bound and an optimal solution that achieves it.For non-negligible direct links, the paper provides a suboptimal design because a tight bound is unavailable.Source: §I · p. 2

A. Single Connected RIS Architecture

  • In the system model, the conventional single-connected architecture links each network port to ground independently, yielding a diagonal scattering matrix.The SISO model assumes no mutual coupling between RIS antennas.Source: §II · p. 3, A. Single Connected RIS Architecture · p. 3

B. Fully Connected RIS Architecture

Fully connected RISs allow a more flexible scattering matrix than single-connected RISs, while group-connected RISs trade flexibility for lower complexity. For reflective single-user SISO optimization, the paper derives a closed-form route to the received-power upper bound.

B. Fully Connected RIS Architecture

  • Fully connected RISs use a complex symmetric unitary scattering matrix, allowing more flexibility than diagonal single-connected RISs at higher circuit and optimization complexity.Group-connected RISs offer a performance-complexity trade-off, with group size bridging the single- and fully connected architectures.Source: B. Fully Connected RIS Architecture · p. 3

C. Group Connected RIS Architecture

  • Group-connected RISs partition elements into groups connected internally but not across groups, producing a block-diagonal scattering matrix with symmetric unitary blocks.Single- and fully connected RISs are the special cases with group size 1 and the total number of elements, respectively.Source: C. Group Connected RIS Architecture · p. 3

III. OPTIMAL DESIGN FOR BD-RIS-AIDED SINGLE-USER SISO SYSTEMS: REFLECTIVE MODE

  • The reflective-mode SISO design maximizes received signal power; it assumes unit transmit power, normalizes the channels, and first omits the direct link.The direct-link phase is handled afterward, and the analysis excludes discretized hardware constraints.Source: §III · p. 4

A. Closed-Form Solution for Optimal Fully Connected RIS

  • For fully connected RISs, the solution seeks a real orthonormal matrix satisfying an equivalent optimality condition, then constructs the scattering matrix from that matrix and a diagonal phase matrix.The channel-dependent symmetric matrix is diagonalized to convert the condition into a diagonal quadratic form.Source: A. Closed-Form Solution for Optimal… · pp. 4–5
  • The resulting fully connected design is globally optimal because it satisfies the equivalent condition and exactly achieves the objective upper bound.For linearly dependent channels, the paper gives the simpler choice V = I.Source: A. Closed-Form Solution for Optimal… · p. 4
  • Using the special structure of δ, the authors solve (21) in closed form by treating separately the cases N_I = 2, N_I = 3, and N_I ≥ 4.The vector δ has respectively two, three, and four nonzero elements in those cases. For N_I ∈ {2, 3}, Proposition 1 states that A has rank N_I and trace zero.Source: 2) NI = 3 · p. 5

1) NI = 2:

For a fully connected RIS with two elements, the closed-form construction uses the opposite signs of the two eigenvalues to choose orthonormal vectors satisfying the optimality condition.

1) NI = 2:

  • With N_I = 2, the two nonzero eigenvalues are opposites, so the optimality condition sets the squared components of each vector equal.The unit-norm constraint then gives squared components of 1/2, and the chosen vector ensures orthonormality.Source: 1) NI = 2 · p. 5
  • The construction uses a vector with only its first and third entries nonzero, which exists because the first eigenvalue is positive and the third is negative.The components are solved from the optimality condition and unit-norm constraint.Source: 1) NI = 2 · p. 5

2) NI = 3:

The fully connected construction builds orthonormal vectors from the nonzero eigenspace to satisfy the optimality condition, then forms a scattering matrix that attains the upper bound.

2) NI = 3:

  • For N_I ≥ 4, the channel-derived matrix has rank four, zero trace, and four nonzero eigenvalues split evenly between positive and negative values.The other N_I − 4 solutions include coordinate basis vectors spanning the zero-eigenvalue positions.Source: 1) NI = 2 · p. 5
  • The first two vectors are chosen with nonzero entries at opposite-sign eigenvalue positions, yielding orthonormal vectors that satisfy the reduced condition.The remaining vectors are formed from a basis of the null space of the first two vectors.Source: 1) NI = 2 · pp. 5–6
  • The resulting orthonormal matrix T is assembled according to the number of RIS elements and used to construct the optimal scattering matrix as Θ̄ = VDV^T.V is obtained from T and the eigenvectors of the channel-derived matrix, while D is designed from the diagonal phase terms.Source: 1) NI = 2 · p. 6
  • The construction is globally optimal because it solves the reduced condition, which is equivalent to the condition for attaining the objective upper bound.The direct-link phase can then be aligned with the reflected signal term.Source: 1) NI = 2 · p. 6

B. Closed-Form Solution for Optimal Group Connected RIS

The group-connected solution applies the fully connected design separately to each group and co-phases their contributions to maximize received power.

B. Closed-Form Solution for Optimal Group Connected RIS

  • For group-connected RISs, each group’s scattering block is designed using the fully connected algorithm on that group’s truncated channels.The blocks maximize their respective channel products in absolute value and make those products co-phased; the block-diagonal matrix then follows from the group constraints.Source: B. Closed-Form Solution for Optimal… · p. 7

IV. OPTIMAL DESIGN FOR BD-RIS-AIDED SINGLE-USER SISO SYSTEMS: TRANSMISSIVE MODE

In transmissive mode, each RIS cell contains two back-to-back elements, with the transmitter and receiver on opposite sides of the surface. The received-power problem is handled by applying the earlier optimal design to the channels with their mode-specific zero entries.

IV. OPTIMAL DESIGN FOR BD-RIS-AIDED SINGLE-USER SISO SYSTEMS: TRANSMISSIVE MODE

  • Each of the M_I cells pairs two back-to-back RIS elements, dividing the surface into odd- and even-element sectors on opposite sides.The transmitter is assumed in sector 1 and the receiver in sector 2.Source: §IV · p. 7
  • With the transmitter in sector 1 and receiver in sector 2, odd entries of the RIS-to-receiver channel and even entries of the transmitter-to-RIS channel are zero.Source: §IV · p. 7
  • The received signal power maximization problem retains the block-diagonal scattering constraint, and the Section III-B optimal solution applies with the zero channel entries.Source: §IV · p. 7
  • For hybrid transmissive and reflective BD-RIS optimization in multi-user scenarios, the text refers readers to.Source: §IV · p. 7

V. BD-RIS-AIDED SINGLE-USER MIMO SYSTEMS

The single-user MIMO setup uses a BD-RIS in reflective mode and single-stream transmit and receive beamforming. Its received-power objective is optimized jointly over the precoder, combiner, and block-diagonal scattering matrix.

V. BD-RIS-AIDED SINGLE-USER MIMO SYSTEMS

  • The equivalent MIMO channel is H = H_RT + H_RIΘH_IT, with the direct, RIS-to-receiver, and transmitter-to-RIS channels defined in the setup.Source: §V · p. 7
  • For single-stream transmission, unit-norm precoding and combining vectors w and g yield received power P_R^MIMO = P_T|g(H_RT + H_RIΘH_IT)w|^2.Source: §V · p. 7
  • The received-power maximization jointly designs w, g, and the block-diagonal scattering matrix Θ.Source: §V · p. 7

A. Optimizing Fully Connected RIS-Aided Systems Without Direct Link

For a fully connected RIS with negligible direct link, the MIMO received-power upper bound is tight: the required scattering matrix follows from the SISO design applied to dominant singular vectors.

A. Optimizing Fully Connected RIS-Aided Systems Without Direct Link

¯P MIMO

  • Tightness requires alignment of the dominant singular vectors of H_RI and H_IT, up to phase; this is equivalent to maximizing their cosine similarity to 1.Source: ¯P MIMO · p. 7
  • Applying Algorithm 1 to v_RI^H and u_IT gives a scattering matrix that satisfies the upper-bound condition and makes the bound tight.In the MISO setting, the corresponding vectors are h_RI and u_IT.Source: ¯P MIMO · p. 8

B. Optimizing Fully/Group Connected RIS-Aided Systems With Direct Link

With a direct link or group-connected RIS, the single-user MIMO design alternates beamforming and scattering-matrix updates. The multi-user MISO section formulates weighted sum power and identifies a common single-stream precoder.

B. Optimizing Fully/Group Connected RIS-Aided Systems With Direct Link

  • Alternating updates of Θ and the beamforming vectors produce a non-decreasing received-power objective bounded above, guaranteeing convergence.With fixed w and g, Θ is updated by the SISO strategy; with fixed Θ, w and g are dominant singular vectors of the effective channel.Source: B. Optimizing Fully/Group Connected… · p. 8
  • The initialization uses the direct-channel dominant mode when the direct link is strong, or the reflected-link mode when it is weak or many RIS elements are used.Source: B. Optimizing Fully/Group Connected… · p. 8
  • The first iteration’s received power provides a lower bound on the power achieved by the subsequent optimization.Source: B. Optimizing Fully/Group Connected… · p. 8

VI. BD-RIS-AIDED MULTI-USER MISO SYSTEMS

  • The multi-user MISO setup maximizes weighted sum received power across K single-antenna receivers using per-user weights α_k.Source: §VI · pp. 8–9
  • For the weighted sum power objective, the optimal user precoders align with the dominant eigenvector of S, so single-stream precoding incurs no loss of optimality.Source: §VI · p. 9

A. Optimizing Fully Connected RIS-Aided Systems Without Direct Links

For fully connected RISs without direct links, dominant singular vectors guide a closed-form scattering-matrix design that achieves the received-power upper bound. The supplied numerical passages also report results for more general settings and reflective-mode SISO systems.

A. Optimizing Fully Connected RIS-Aided Systems Without Direct Links

  • For fully connected systems without direct links, applying Alg. 1 to dominant singular vectors of the channel matrices yields a scattering matrix that makes the received-power upper bound tight.The equivalent channel is H = G_RI Θ H_IT.Source: A. Optimizing Fully Connected… · p. 9

B. Optimizing Fully/Group Connected RIS-Aided Systems With Direct Links

  • When group-connected RISs are used or direct links are non-negligible, tight upper bounds are unavailable, and the paper applies the suboptimal Alg. 2 strategy.The maximization is formulated using auxiliary unit-norm vectors.Source: B. Optimizing Fully/Group Connected… · p. 9

VII. NUMERICAL RESULTS

  • The numerical setup places the RIS at (50, 2), uses path-loss exponents 3.5, 2.8, and 2 for the RT, RI, and IT links, and models Rayleigh and Rician fading.The Rician factor is 3 dB, and transmit power is 10 W.Source: §VII · p. 9

A. RIS-Aided Single-User SISO Systems: Reflective Mode

  • As N_I→∞ under Rayleigh fading, fully connected RISs provide a power gain of 16/π^2 over single connected RISs, while the closed-form design achieves the stated upper bounds.The SISO comparison also reports an average received signal power of 9.88 nW without an RIS.Source: A. RIS-Aided Single-User SISO Systems:… · p. 10

B. RIS-Aided Single-User SISO Systems: Transmissive Mode

The paper extends its RIS scattering-matrix design to transmissive SISO, single-user MIMO, and multi-user MISO settings, then compares its computational complexity with quasi-Newton optimization. The reported results include exact upper-bound attainment in specified settings and lower complexity than the iterative method.

B. RIS-Aided Single-User SISO Systems: Transmissive Mode

  • In transmissive SISO, the proposed design achieves the received-power upper bounds; with correlated Rayleigh fading, single- and fully connected performance is unchanged while group-connected performance slightly decreases.The correlation comparison uses an exponential correlation model with coefficient ρ = 0.5.Source: B. RIS-Aided Single-User SISO Systems:… · pp. 10–11

C. RIS-Aided Single-User MIMO Systems

  • For single-user MIMO, Alg. 1 exactly achieves the upper bound without a direct link, while Alg. 2 results show fully connected RISs outperform group-connected, which outperform single-connected RISs.Increasing the numbers of transmit and receive antennas raises received power; Rayleigh fading gives slightly higher power than Rician fading in the reported comparison.Source: C. RIS-Aided Single-User MIMO Systems · p. 11

D. RIS-Aided Multi-User MISO Systems

  • In multi-user MISO, Alg. 1 attains the upper bound for fully connected RISs without direct links; with Alg. 2, fully connected RISs lead the architecture comparison and weighted sum power rises with receiver count.The reported setup uses N_T = 4.Source: D. RIS-Aided Multi-User MISO Systems · pp. 11–12

E. Computational Complexity

  • Alg. 1 has O(N_I^3) complexity for fully connected RISs and O(N_G^2N_I) for group-connected RISs, below the cited per-iteration quasi-Newton complexity.For group-connected designs, Alg. 1 is run G = N_I/N_G times; the comparison is also shown in Fig. 8.Source: E. Computational Complexity · p. 12

VIII. CONCLUSION

  • The closed-form strategy is reported to achieve received-signal-power upper bounds in both reflective and transmissive modes, and its complexity grows linearly for group-connected and cubically for fully connected RISs.The conclusion states that the result holds for any channel realization without assumptions on channel distribution.Source: B. RIS-Aided Single-User SISO Systems:… · p. 11, §VIII · pp. 12–13

APPENDIX

The appendix establishes matrix properties used in the paper’s proofs: the constructed matrix has a specified rank and trace, and a related matrix has positive and negative eigenvalues. It treats the four-element and larger cases separately for the latter result.

A. Proof of Proposition 1

  • For the matrix A in Proposition 1, the proof uses full rank of B and trace linearity to establish r(A) = min{4, N_I} and Tr(A) = 0 under the stated independence discussion.The passage notes that the linearly dependent case is addressed separately.Source: A. Proof of Proposition 1 · p. 13

B. Proof of Proposition 2

  • For N_I = 4, the proof establishes det(A) > 0 and uses the zero trace and four nonzero eigenvalues to infer two positive and two negative eigenvalues.The determinant argument uses a lower bound and a permutation of eigenvalue indices.Source: B. Proof of Proposition 2 · p. 13
  • For N_I > 4, the proof applies Weyl inequalities to the rank-two positive-semidefinite matrices A_RI and A_IT to establish two positive and two negative eigenvalues of A.The argument derives zero eigenvalues at selected indices before determining the signs of the remaining eigenvalues.Source: B. Proof of Proposition 2 · p. 13
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