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MIMO Over-the-Air Computation for High-Mobility Multi-Modal Sensing

Guangxu Zhu, Kaibin Huang

arXiv:1803.11129v2cs.IT

TL;DR

High-mobility multi-modal sensor networks need low-latency aggregation, but multi-function AirComp beamforming is NP-hard and requires global channel information. The paper derives a Grassmann-manifold approximation with a weighted-centroid solution, establishes an AirComp–multicasting duality, and designs local-CSI-based concurrent feedback for beamformer acquisition.

  • Problem

    Multi-function AirComp for multi-antenna, multi-modal sensors poses NP-hard receive-beamforming optimization and requires global CSI, making conventional acquisition impractical for ultra-fast sensing.

  • Method

    The paper tightens transmission-power constraints, solves the resulting beamforming problem using differential geometry on a Grassmann manifold, and uses concurrent AirComp feedback based on local CSI.

  • Results

    The approximate normalized receive beamformer is the weighted centroid of sensor-channel subspaces, while the feedback design acquires the beamformer through distributed concurrent transmissions.

  • Takeaways & Limitations

    The beamforming solution transfers to multicast beamforming, and the feedback design has a number of rounds independent of the sensor population.

Abstract

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In future Internet-of-Things networks, sensors or even access points can be mounted on ground/aerial vehicles for smart-city surveillance or environment monitoring. To support the high-mobility sensing with low network latency, a technique called over-the-air-computation (AirComp) was recently developed which enables an access-point to receive a desired function of sensing-data from concurrent-transmissions by exploiting the superposition property of a multi-access-channel. This work aims at further developing AirComp for next-generation multi-antenna multi-modal sensor networks. Specifically, we design beamforming and channel-feedback techniques for multi-function AirComp. Given the objective of minimizing sum-mean-squared-error of computed functions, the optimization of receive-beamforming for multi-function AirComp is a NP-hard problem. The approximate problem based on tightening transmission-power constraints, however, is shown to be solvable using differential-geometry. The solution is proved to be the weighted-centroid of points on a Grassmann-manifold, where each point represents the subspace spanned by the channel matrix of a sensor. As a by-product, the beamforming problem is found to have the same form as the classic problem of multicast-beamforming, establishing the AirComp-multicasting-duality. Its significance lies in making the said Grassmannian-centroid solution transferable to the latter problem which otherwise is solved using the computation-intensive semidefinite-relaxation-technique. Last, building on the AirComp-beamforming solution, an efficient channel-feedback technique is designed for an access-point to receive the beamformer from distributed sensor transmissions of designed signals that are functions of local channel-state-information.

I. INTRODUCTION

High-mobility IoT sensing requires low-latency aggregation beyond transmit-then-compute, motivating MIMO AirComp for multi-modal, multi-function sensing. The paper develops Grassmann-manifold beamforming, establishes an AirComp–multicasting duality, and designs efficient channel feedback.

  • Transmit-then-compute is impractical for large, high-mobility sensor networks because massive radio access causes excessive latency and inefficient spectrum use.
  • AirComp computes nomographic functions from concurrent sensor transmissions by exploiting multi-access-channel signal superposition.It integrates computation and communication, and sensing-noise averaging can improve computation accuracy as simultaneous sensors increase.
  • Existing solutions focus on single-function AirComp with single-antenna, uni-modal sensors, whereas emerging sensing devices collect multiple environmental data streams.
  • MIMO AirComp enables simultaneous multi-function computation through spatial multiplexing and suppresses computation errors using spatial-diversity gain.The intended benefit is reduced data-fusion latency for high-mobility sensing.
  • Multi-Function AirComp Beamforming: Multi-function AirComp receive-beamformer optimization is NP-hard, while tightening transmission-power constraints yields an approximate Grassmann-manifold problem solvable by differential geometry.
  • Multi-Function AirComp Beamforming: The normalized receive beamformer is the weighted centroid of channel eigen-subspaces, with weights given by each channel’s smallest eigenvalue.The resulting efficiently computed beamformer is verified by simulation to be close-to-optimal.
  • AirComp-Multicasting Duality: For single-antenna sensors, AirComp receive-beamforming has the same form as multicast transmit-beamforming, allowing the Grassmannian solution to replace computation-intensive semidefinite relaxation as networks scale.
  • AirComp Channel Feedback: AirComp feedback acquires the globally dependent beamformer through concurrent sensor transmissions based only on local CSI, with feedback rounds independent of sensor population.The techniques sequentially recover the normalized beamformer and its norm using functions of locally available channel information.

II. SYSTEM MODEL

The system models AirComp for K multi-modal MIMO sensors, where concurrent transmissions enable computation of normalized target-function vectors at an access point. Beamforming suppresses channel- and noise-induced distortion, while channel feedback avoids directly estimating global CSI.

  • System model: K multi-modal sensors measure L heterogeneous environmental parameters and transmit their preprocessed data using antenna arrays.The AP is equipped with N_r antennas, while each sensor has N_t antennas.
  • AirComp operations: Nomographic functions are computed through sensor-side preprocessing, multi-access summation, and AP-side post-processing.For geometric means, preprocessing uses logarithms and post-processing uses exponentiation by 1/K.
  • Target-function representation: The preprocessed sensor vectors are normalized to unit covariance, and their sum is treated as the target-function vector corresponding one-to-one with the computed functions.This normalization supports transmission-power control and can be inverted at the AP.
  • Signal model and performance: Joint transmit and receive beamforming seeks coherent combining of simultaneous symbol vectors while minimizing MSE relative to the target-function vector.The channel matrices H_k connect sensors to the AP, and additive white Gaussian noise contributes to the received distortion.
  • Channel feedback: Channel-feedback design uses concurrent sensor transmissions to acquire the beamformer without directly estimating all global channel matrices at the AP.The feedback observation can serve as a sufficient surrogate for global CSI when the transmitted signals are properly designed.

III. PROBLEM FORMULATION

The paper formulates MMSE multi-function AirComp beamforming under per-sensor power constraints and develops an approximate geometric solution for its non-convex receive-beamformer optimization. The formulation also motivates a CSI-based feedback architecture for distributed beamformer acquisition.

  • A. AirComp Beamforming Problem: The MMSE beamforming problem constrains each sensor’s average transmit power to P0 and models the receive beamformer as A = √ηF with F^HF = I.The denoising factor η regulates the tradeoff between noise reduction and transmission-power control.
  • B. AirComp Channel Feedback Problem: The AirComp channel-feedback problem designs functions of local CSI so concurrent sensor transmissions can provide the AP with the derived global-CSI-dependent beamformer.The feedback counterparts are denoted by ˜f and ˜g_k, with X_k = ˜g_k(H_k).
  • IV. MULTI-FUNCTION AIRCOMP: BEAMFORMING: The reduced receive-beamforming problem remains non-convex and is NP-hard because of orthogonality constraints and coupling in the original formulation.The paper connects this difficulty to the equivalent NP-hard multicast-beamforming problem.
  • A. AirComp Beamforming Problem: Given a receive beamformer, zero-forcing transmit beamforming minimizes the MSE objective and allows the joint problem to be reduced to receive-beamformer optimization.The transmit precoder’s power constraint is enforced through the norm of the equalizer, equivalently through η.
  • A. AirComp Beamforming Problem: The number of simultaneously computable functions is limited by L ≤ min{N_t, N_r}, reflecting the maximum number of MIMO spatial streams.This bound follows from requiring the relevant matrix to be invertible.
  • IV. MULTI-FUNCTION AIRCOMP: BEAMFORMING: Tightening the transmission-power constraints produces a smaller feasible problem whose solution is feasible for the original reduced problem but may be suboptimal.This approximation enables a differential-geometry solution based on subspace distances.
  • IV. MULTI-FUNCTION AIRCOMP: BEAMFORMING: The approximate receive beamformer is characterized as a weighted centroid of channel subspaces on a Grassmann manifold using squared projection 2-norm distance.Weights are determined by the smallest channel eigenvalues, so stronger channel gains receive different alignment emphasis.
  • IV. MULTI-FUNCTION AIRCOMP: BEAMFORMING: Replacing projection 2-norm distance with projection F-norm distance yields a closed-form approximation, exact for N_t = 1 and accurate for small principal angles.The resulting approach avoids an iterative algorithm.

2 PF(Uk, F)

The tractable approximation replaces the Grassmannian projection-distance objective with an effective-CSI eigenvector problem. Its closed-form solution uses the principal eigenvectors of a matrix that summarizes the global CSI.

  • 2 PF(U_k, F): The alternative distance formulation remains a weighted-centroid problem over channel subspaces, but uses a different subspace distance metric.
  • 2 PF(U_k, F): The approximated problem remains non-convex because it maximizes a convex objective under orthogonality constraints on F.An equivalent unconstrained formulation enables stationary-point analysis.
  • 2 PF(U_k, F): The effective CSI matrix G is sufficient for computing the normalized AirComp receive beamformer F*.Thus F* depends on global CSI through G rather than requiring the full channel collection directly.
  • 2 PF(U_k, F): F* is formed from the first L principal eigenvectors of G obtained through its singular-value decomposition.
  • 2 PF(U_k, F): The complete MMSE design combines the receive beamformer, denoising factor, and transmit beamformers derived in the preceding lemmas.The denoising factor is obtained from the power constraints.

V. AIRCOMP-MULTICASTING DUALITY

For single-antenna uni-modal sensors, AirComp receive-beamforming is equivalent to downlink multicast transmit-beamforming. This duality transfers the proposed low-complexity AirComp design to the NP-hard multicast problem.

  • V. AIRCOMP-MULTICASTING DUALITY: The single-antenna uni-modal AirComp receive-beamforming problem has the same form as downlink multicast transmit-beamforming.The equivalence establishes the AirComp-multicasting duality and enables transfer of the preceding beamforming design.

A. Review of the Multicast Beamforming Problem

For single-antenna sensors, AirComp receive-beamforming has the same mathematical form as downlink multicast beamforming, enabling the Grassmannian weighted-centroid solution to transfer between them.

  • Multicast Beamforming: Multicast beamforming minimizes AP transmission power subject to users’ SNR-based quality-of-service constraints and is NP-hard.SDR relaxes the rank-one constraint and retrieves an approximate rank-one solution using Gaussian randomization or the principal eigenvector.
  • AirComp Beamforming: For single-antenna sensors, AirComp receive-beamforming reduces to a vector design problem derived from the MMSE formulation.The receive beamformer is an Nr × 1 vector f.
  • AirComp-Multicasting Duality: The AirComp and multicast beamforming problems share the same mathematical form because both align beamformers with multiple vector channels for different objectives.This establishes the AirComp-multicasting duality.
  • Implication: The weighted-centroid AirComp beamforming technique therefore transfers to multicast beamforming as a low-complexity alternative to SDR.The duality makes the Grassmann-manifold solution applicable to the multicast problem.
  • CSI Feedback: The proposed AirComp approach uses effective CSI for one-shot feedback, while SDR requires global CSI feedback from all users.Consequently, SDR channel-training overhead becomes excessive as the user population grows.
  • Complexity: The weighted-centroid method requires one-shot computation with complexity O(N_r^2), whereas SDR has substantially higher iterative complexity.The SDR complexity depends on Nr, K, and solution accuracy, while the centroid method’s stated complexity arises from principal-eigenvector calculation.

VI. MULTI-FUNCTION AIRCOMP: CHANNEL FEEDBACK

The proposed feedback architecture acquires the normalized beamformer and denoising factor through distributed sensor transmissions, using AirComp to compute a matrix centroid and a scalar maximum.

  • Feedback Architecture: The feedback design sequentially acquires the normalized beamformer and denoising factor using functions of sensors’ local channel-state information.The two feedback components correspond to a weighted centroid of matrices and the maximum of scalars.
  • Normalized Beamformer Feedback: The normalized beamformer is obtained by enforcing Y = G through designed sensor feedback signals and extracting the dominant left eigenvectors of the received signal.The construction relies on the effective CSI matrix and compact channel decompositions.
  • Normalized Beamformer Feedback: Scaling each sensor’s feedback signal to satisfy a transmission-power constraint does not change the received normalized beamformer.This invariance follows because only the normalized beamformer is required.
  • Scope: The feedback design targets multi-function AirComp because the maximum function is not directly AirComputable as a nomographic function.The proposed procedure nevertheless acquires the denoising factor over a fixed number of rounds.
  • Denoising-Factor Feedback: For denoising-factor feedback, zero-forcing creates parallel multiple-access channels so simultaneously transmitted signal vectors sum at the AP.The denoising factors are assumed to lie within a fixed finite range.
  • Denoising-Factor Feedback: The AP estimates the maximum denoising factor by quantizing sensor values, detecting the largest nonzero codebook index, and refining the range over feedback rounds.Each sensor transmits a one-hot vector corresponding to its quantized value.
  • Accuracy: The feedback error decreases exponentially with rounds, and each additional round improves resolution by log2 N_t bits.The stated error bound follows from Proposition 1.
  • Accuracy: With η_max = 100, η_min = 0, ε = 10^-4, and N_t = 8, the required number of feedback rounds is M = 7.This is smaller than the number of sensors that determines conventional channel-training rounds in dense networks.

C. Comparison with Conventional Channel Training

Conventional channel training requires orthogonal pilot transmissions from sensors, whereas the proposed AirComp feedback is designed to avoid feedback overhead scaling with the sensor population.

  • Conventional Training: Conventional training requires at least T = K × N_t symbol slots because each sensor transmits pilots separately to estimate its N_r × N_t channel.The proposed normalized-beamformer feedback is contrasted with this sensor-by-sensor process.

VII. SIMULATION RESULTS

Simulations compare the proposed multi-function AirComp beamformer with antenna-selection, eigenmode, and SDR-based baselines, showing accuracy advantages and large computational savings.

  • Simulation Setup: The multi-function AirComp simulations vary the number of functions L, receive antennas N_r, and sensors K.The baseline schemes use zero-forcing transmit beamforming with different receive-beamformer designs.
  • Multi-Function AirComp: Across parameter settings, the proposed beamformer outperforms antenna-selection and eigenmode baselines, with larger gains for large L and K.Performance differences converge as N_r grows because receive diversity improves SNRs.
  • Comparison with SDR: The weighted-centroid solution achieves comparable MSE to SDR while reducing computation time by 100x to 1000x across the considered N_r and K ranges.SDR is described as near-optimal with high probability for the NP-hard multicast beamforming problem.
  • Comparison with SDR: The proposed beamforming complexity is independent of network size K and relatively insensitive to N_r, unlike the SDR complexity.This supports its use in large-scale sensor or multicast networks and with large AP arrays.
  • Conclusion: The paper concludes that multi-function MIMO AirComp combines differential-geometry beamforming, efficient channel feedback, and transferable low-complexity multicast beamforming.The conclusion identifies further directions including sensor scheduling, broadband AirComp, cooperative multi-AP sensing, and distributed learning or inference.

APPENDIX

This appendix introduces Stiefel and Grassmann manifolds, geodesic and principal-angle distances, and proof steps relating matrix inequalities to well-conditioned channels.

  • Manifold preliminaries: The Stiefel manifold V_N,M contains N-by-M tall orthonormal matrices, while the Grassmann manifold G_N,M contains M-dimensional subspaces of C^N.Grassmann points correspond to equivalence classes of Stiefel matrices spanning the same column subspace.
  • Distance metrics: Geodesic distance on a Grassmann manifold is the shortest curve length between two subspaces and is computed from their principal angles.Principal angles measure minimal angles between orthonormal bases spanning the two subspaces.
  • Distance metrics: SVD of X^H Y provides singular values equal to the cosines of the principal angles, enabling several subspace-distance metrics.The appendix highlights projection 2-norm and projection F-norm metrics.
  • MSE bounds: The MSE proof uses positive semidefiniteness to derive an inequality for any equalizer, with zero-forcing matrices achieving equality.The equality follows by setting the matrices B_k to the zero-forcing structure.
  • Channel conditioning: The upper bound becomes exact for well-conditioned channels because unitary invariance preserves the relevant eigen-spectrum.A scaled-identity eigenvalue matrix is also identified as implying a well-conditioned channel.

D. Proof of Lemma 3

The proof transforms the constrained beamforming problem into an equivalent Grassmannian metric problem, then identifies its optimizer through stationary-point analysis.

  • Problem reformulation: The power constraints are consolidated into one active constraint and moved into the objective, yielding an equivalent min-max formulation.The reformulation then maximizes the inverse objective after removing the constant term L P_0.
  • Problem reformulation: The transformed objective is related to the projection 2-norm Grassmannian metric, producing the desired problem P5.The equivalence follows by substituting the metric identity into the reformulated objective.
  • Stationary-point analysis: Relaxing the orthogonality constraint yields an unconstrained problem whose solution is obtained by evaluating stationary points of the smooth objective J(F).The relaxation is shown to preserve a solution satisfying the original orthogonality constraint.
  • Stationary-point analysis: Problem P7 reduces to an objective involving the effective CSI matrix G after simplifying the trace expression.The proof defines an alternative objective J(F) and constructs an equivalent problem P′′.
  • Global minimizer: The nonzero stationary points have the form F = V_L Q, where V_L contains distinct eigenvectors of G and Q is an arbitrary unitary matrix.The zero stationary point is also identified, but it is not globally optimal because the objective is negative at the nonzero eigenvector solutions.
  • Global minimizer: The global minimizer selects the L principal eigenvectors of G and still satisfies the orthogonality constraint, solving the original equivalent problem.At F = V_L Q, the objective equals −tr(Σ_L), which is negative for positive-definite G.

F. Proof of Lemma 6

This proof rewrites the single-antenna projection-metric problem and rescales its optimization variable to obtain the target formulation.

  • Metric reduction: When N_t = 1, the projection 2-norm and projection F-norm coincide, allowing the problem to be rewritten using the projection 2-norm.The proof applies the definition of the projection 2-norm to problem (17).
  • Variable scaling: The resulting optimization is equivalently reformulated before changing variables from f to f̃ = 1/√ρ f.This change removes the scaling factor from the problem formulation.
  • Variable scaling: The solution to the rescaled problem is obtained by scaling the solution of problem (18) to satisfy the unit-norm constraint.This establishes the desired form in (18).
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