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Over-the-Air Computation via Intelligent Reflecting Surfaces

Tao Jiang, Yuanming Shi

arXiv:1904.12475v2cs.ITeess.SP

TL;DR

AirComp requires fast aggregation but suffers from unfavorable propagation and a difficult coupled transceiver–IRS design problem. The paper introduces IRS-assisted AirComp and an alternating DC approach based on lifted rank-one optimization. Simulations show improved performance over SDR and baselines, with larger IRSs reducing MSE.

Abstract

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Over-the-air computation (AirComp) becomes a promising approach for fast wireless data aggregation via exploiting the superposition property in a multiple access channel. To further overcome the unfavorable signal propagation conditions for AirComp, in this paper, we propose an intelligent reflecting surface (IRS) aided AirComp system to build controllable wireless environments, thereby boosting the received signal power significantly. This is achieved by smartly tuning the phase shifts for the incoming electromagnetic waves at IRS, resulting in reconfigurable signal propagations. Unfortunately, it turns out that the joint design problem for AirComp transceivers and IRS phase shifts becomes a highly intractable nonconvex bi-quadratic programming problem, for which a novel alternating difference-of-convex (DC) programming algorithm is developed. This is achieved by providing a novel DC function representation for the rank-one constraint in the low-rank matrix optimization problem via matrix lifting. Simulation results demonstrate the algorithmic advantages and admirable performance of the proposed approaches compared with the state-of-art solutions.

I. INTRODUCTION

AirComp enables concurrent wireless data aggregation, but uncontrolled propagation and coupled IRS–transceiver design create performance and optimization challenges. The paper addresses these challenges with IRS-assisted propagation control and a DC-based treatment of lifted rank-one constraints.

Motivation

  • AirComp exploits multiple-access-channel superposition for concurrent wireless data aggregation, avoiding the latency and spectrum inefficiency of interference-avoiding access schemes.Source: §I · p. 1
  • Existing AirComp approaches cannot control wireless environments, so harsh propagation conditions can significantly degrade performance.Source: §I · p. 1

IRS-Assisted AirComp

  • The proposed IRS uses passive elements with tunable phase shifts to control communication environments and boost AirComp performance.Source: §I · p. 1, A. System Model · p. 2

Optimization Challenge

  • Jointly optimizing AirComp transceivers and IRS phase shifts yields a highly intractable nonconvex bi-quadratic problem.Source: §I · p. 1

Proposed Solution

  • Matrix lifting and a novel DC representation of the rank-one constraint address the nonconvex optimization, while simulations report significant gains over SDR and strong IRS benefits.Source: §I · p. 2

B. Problem Formulation

The system minimizes AirComp distortion by jointly designing transmitter scalars, the AP decoding vector, and IRS phase shifts. This produces a nonconvex bi-quadratic formulation with quadratic constraints.

Problem Formulation

  • The objective is to minimize MSE by optimizing transmitter scalars, the AP decoding vector, and the IRS phase-shift matrix.Source: B. Problem Formulation · pp. 2–3
  • Given the decoding vector and IRS phase shifts, the transmitter scalars are designed to satisfy each transmitter's power constraint.Source: B. Problem Formulation · p. 3
  • After optimizing transmitter scalars and the normalizing factor, the MSE is rewritten as a function suitable for joint phase-shift and decoding-vector optimization.Source: B. Problem Formulation · p. 3
  • The joint design is constrained by bounded IRS phase shifts and becomes a nonconvex bi-quadratic programming problem.Source: B. Problem Formulation · p. 3

III. ALTERNATING MINIMIZATION

The alternating minimization method updates the AP decoding vector and IRS phase-shift matrix in turn until convergence.

Alternating Minimization

  • The algorithm alternately optimizes the AP decoding vector and IRS phase-shift matrix until convergence.Source: §III · p. 3

A. Alternating Minimization

With one variable block fixed, the original problem becomes either a nonconvex QCQP or a feasibility problem. Homogenization and alternating solution updates provide the resulting procedure and termination conditions.

Subproblems

  • Fixing the IRS phase-shift matrix reduces the problem to a nonconvex QCQP, while fixing the decoding vector reduces it to feasibility detection.Source: A. Alternating Minimization · p. 3
  • The IRS phase shifts are represented through a vector of complex unit-modulus entries, which is used to formulate the phase-shift subproblem.Source: A. Alternating Minimization · p. 3

Homogenization

Recovery and Termination

B. Matrix Lifting

Matrix lifting reformulates the transceiver and IRS subproblems as low-rank matrix optimization problems while retaining rank-one constraints.

  • Defining M = mmH lifts the decoding vector m into a PSD matrix with rank(M) = 1, reformulating problem (12).Source: B. Matrix Lifting · p. 3
  • The IRS-related quadratic constraints are similarly lifted by defining V = ˜v˜vH and expressing quadratic forms through matrix traces.Source: B. Matrix Lifting · pp. 3–4
  • The lifted IRS problem includes PSD, diagonal-unit, trace constraints and the nonconvex condition rank(V ) = 1.Source: B. Matrix Lifting · p. 4

C. Problem Analysis

The SDR approach convexifies the lifted problems by dropping rank-one constraints, but its ability to recover rank-one solutions decreases in high-dimensional settings.

  • SDR drops the nonconvex rank-one constraints and solves the resulting SDP problems with convex optimization solvers.Rank-one decomposition recovers the original solution when the SDP solution is rank-one; otherwise Gaussian randomization may produce a suboptimal solution.Source: C. Problem Analysis · p. 4
  • As the number of antennas increases, SDR has a lower probability of returning rank-one solutions, yielding significant performance deterioration.Source: C. Problem Analysis · p. 4
  • The paper introduces a DC framework to solve the lifted problems without relying on SDR's rank-one recovery.Source: C. Problem Analysis · p. 4

IV. ALTERNATING DC ALGORITHM

The paper represents the rank-one constraint as a difference-of-convex function and incorporates it as a penalty in a matrix optimization objective.

B. DC Algorithm

The DC algorithm iteratively convexifies the penalized problem by linearizing the spectral-norm term, solves convex subproblems, and converges to critical points.

  • At each iteration, the concave term −ρ∥X∥2 is linearized, producing a convex subproblem solvable by existing optimization solvers.Source: B. DC Algorithm · p. 4
  • The subproblem uses Xt−1 from the previous iteration, while the spectral-norm subgradient can be computed from the leading eigenvector.Source: B. DC Algorithm · p. 4
  • The presented DC algorithm converges to critical points of problem (20) from arbitrary initial points.Source: B. DC Algorithm · p. 4
  • Algorithm 1 specifies iterative subgradient computation, convex-subproblem solution, and termination based on objective decrease.Source: B. DC Algorithm · p. 4

C. Proposed Alternating DC Approach

The proposed alternating DC approach solves the AirComp design through alternating subproblems while preserving rank-one feasibility. It uses trace–spectral norm penalties and feasibility checks to recover solutions or identify infeasibility.

Alternating optimization

Rank-one solution

  • A penalty parameter ρ > 0 is used to enforce the rank-one structure in the matrix optimization problem.When the penalty component becomes zero, Cholesky decomposition recovers the vector solution.Source: C. Proposed Alternating DC Approach · p. 5
  • The method detects rank-one solutions by minimizing the difference between trace norm and spectral norm under positive-semidefinite constraints.A zero objective value indicates an exact rank-one optimal solution.Source: C. Proposed Alternating DC Approach · p. 5

Feasibility handling

  • If the feasibility objective remains nonzero, the corresponding IRS subproblem is declared infeasible.If it reaches zero, Cholesky decomposition yields a feasible vector solution.Source: C. Proposed Alternating DC Approach · p. 5

V. SIMULATIONS

The simulations evaluate the proposed alternating DC algorithm and IRS-assisted AirComp in a three-dimensional AP–IRS–user setting. They use path-loss and Rayleigh-fading channel models with fixed algorithmic and SNR parameters.

Simulation setup

  • The simulation uses a 3D coordinate system with uniform linear AP antennas and a uniform rectangular IRS array.The AP is at (0, 0, 25) meters, the IRS at (50, 50, 40) meters, and users occupy [−50, 50] × [50, 150] meters.Source: A. Simulation Settings · p. 5

Channel model

  • The channel model uses distance-dependent path loss with AP-user, AP-IRS, and IRS-user exponents of 3.5, 2.2, and 2.8.The reference path loss is T0 = 30dB at d0 = 1m, and all channels use Rayleigh fading.Source: A. Simulation Settings · p. 5

Simulation parameters

  • The average transmit SNR is set to 30 dB, with ρ = 5, ϵ = 10^-3, and ϵdc = 10^-8.These parameters are used in the numerical evaluation.Source: A. Simulation Settings · p. 5

Compared methods

  • The proposed alternating DC algorithm is compared with an alternating SDR method that uses CVX and Gaussian randomization when rank-one solutions are unavailable.Both methods alternate between the two subproblems and stop based on MSE change or infeasibility.Source: A. Simulation Settings · p. 5

B. Simulation Results

The proposed alternating DC method outperforms alternating SDR across the tested settings, while increasing AP antennas or IRS elements reduces MSE. IRS-assisted AirComp also performs substantially better than AirComp without IRS.

Algorithm convergence

  • Under K = 16, M = 30, and N = 20, alternating SDR stops at the third iteration, whereas alternating DC finds exact rank-one solutions and maintains small MSE.The SDR method fails to find a feasible solution even with Gaussian randomization.Source: B. Simulation Results · p. 5

AP antennas

  • MSE decreases significantly as the number of AP antennas N increases, and alternating DC significantly outperforms alternating SDR and the baseline.The comparison fixes M = 15 and K = 8, averaging each point over 100 channel realizations.Source: B. Simulation Results · pp. 5–6

IRS elements

  • MSE decreases significantly as the number of IRS elements M increases, while alternating DC significantly outperforms alternating SDR.The comparison fixes N = 10 and K = 8, averaging each point over 100 channel realizations.Source: B. Simulation Results · p. 6

IRS benefit

  • AirComp without IRS has poor MSE performance, whereas deploying an IRS significantly enhances AirComp performance.The comparison fixes N = 8 and uses M = 15 for the IRS-assisted case, with results averaged over 100 channel realizations.Source: B. Simulation Results · p. 6

Conclusion

  • The paper concludes that large intelligent surfaces boost AirComp performance through joint transceiver and phase-shift design solved with alternating DC optimization.The approach reformulates the QCQP subproblems using matrix lifting and a DC treatment of rank-one constraints.Source: §VI · p. 6

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