Hybrid RIS-aided Digital AirComp

• The paper introduces HRD-AirComp as a task-oriented wireless aggregator that leverages vector quantization, adaptive bit allocation, and hybrid RIS control to optimize edge AI inference.
• It implements a blockwise digital aggregation pipeline where agents convert high-dimensional features into discrete codewords for efficient over-the-air feature summation.
• Experimental findings show 12%-25% accuracy gains and reduced inference uncertainty compared to baselines, underscoring its practical relevance for 6G edge systems.

Searching arXiv for the cited HRD-AirComp and related RIS-AirComp papers to ground the article in current literature. Hybrid RIS-aided Digital AirComp (HRD-AirComp) is a task-oriented wireless feature aggregation scheme for edge AI inference in which distributed agents quantize high-dimensional features into digital codewords, transmit the resulting symbols simultaneously, and rely on controlled waveform superposition—assisted by a hybrid reconfigurable intelligent surface (RIS)—so that an edge node (EN) can reconstruct aggregated features for downstream inference (Fu et al., 25 Sep 2025). In the formulation reported for edge inference, HRD-AirComp combines vector quantization, digital modulation, active-passive RIS reflection, and joint transceiver design under an inference-accuracy objective rather than a communication-only distortion criterion. The reported experimental results show that it outperforms baselines in both inference accuracy and uncertainty, while remaining compatible with digital physical layers such as QAM and BPSK (Fu et al., 25 Sep 2025).

1. Position within AirComp and RIS-enabled edge inference

The motivating scenario is 6G edge inference, where multiple distributed agents observe an environment from different viewpoints and send extracted features to an EN for collaborative AI inference. A critical operation is the aggregation of multi-view sensory features, since combining observations can improve environmental perception accuracy (Fu et al., 25 Sep 2025). AirComp is relevant because it exploits the waveform superposition property of multiple-access channels for fast aggregation, but classical analog AirComp is described as incompatible with existing digital communication systems and sensitive to channel impairments. In parallel, most digital AirComp research is described as using scalar quantization, ignoring the unequal importance of feature dimensions and disregarding task-orientation for AI inference (Fu et al., 25 Sep 2025).

RIS-assisted AirComp predates HRD-AirComp, but the earlier focus was on distortion reduction for wireless aggregation rather than task-oriented feature inference. Passive RIS or IRS designs were introduced to reconfigure propagation channels and mitigate the worst-channel bottleneck in AirComp, typically by jointly optimizing transmit scalars, receive beamforming, and phase shifts to minimize mean-squared error (MSE) distortion (Jiang et al., 2019, Fang et al., 2021). Subsequent work also considered data-driven stochastic beamforming when accurate channel state information is difficult to obtain, optimizing the probability that the AirComp MSE remains below a threshold based on historical channel realizations (Fang et al., 2020). HRD-AirComp extends this line of research by shifting the objective from aggregation distortion alone to inference accuracy and posterior uncertainty, and by replacing purely passive RIS with a hybrid RIS architecture that contains both active and passive elements (Fu et al., 25 Sep 2025).

A common misconception is to treat digital AirComp as merely a quantized version of analog amplitude summation. In the HRD-AirComp formulation, the digital character is more specific: vector quantization maps feature blocks to discrete codewords, codeword indices are associated with digitally modulated sequences, and the EN recovers aggregate codeword sums via sparse signal recovery rather than by directly estimating analog amplitudes (Fu et al., 25 Sep 2025).

2. System model and hybrid RIS architecture

The reported system model consists of multiple single-antenna agents, an EN with multiple antennas, and a hybrid RIS deployed between agents and EN to control wireless propagation (Fu et al., 25 Sep 2025). The agents hold sensors and lightweight AI modules, extract local features from observed scenes, and transmit feature representations to the EN for collaborative inference. The hybrid RIS is explicitly defined by the coexistence of active elements, which can tune both amplitude and phase and therefore enable signal amplification, and passive elements, which only adjust phase and satisfy unit-modulus reflection (Fu et al., 25 Sep 2025).

The effective channel for agent $k$ is written as

$$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$

where the terms correspond to agent–EN, RIS–EN, and agent–RIS propagation components, and $\mathbf{\Phi}$ denotes the RIS reflection matrix (Fu et al., 25 Sep 2025). The reflection matrix is described as the sum of active and passive components, i.e. $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$, with diagonal entries controlled according to element type. This architecture is significant because active elements introduce amplification capability while passive elements preserve the conventional low-complexity phase-shifting role; the reported experiments state that hybrid RIS outperforms purely passive or purely active RIS under various power budgets and numbers of active elements, while also exposing a tradeoff between beamforming gain and amplification noise (Fu et al., 25 Sep 2025).

This channel-control role is consistent with earlier RIS-AirComp literature, which treated RIS as a mechanism for boosting received signal power and equalizing unfavorable channels in multiuser aggregation (Fang et al., 2021). HRD-AirComp preserves that propagation-control perspective but embeds it into a digital, inference-oriented stack instead of an analog MSE-minimization pipeline.

3. Digital aggregation pipeline and feature reconstruction

HRD-AirComp replaces direct analog feature superposition with a blockwise digital aggregation procedure (Fu et al., 25 Sep 2025). First, each agent’s high-dimensional feature vector is split into blocks. Each block is quantized through a shared Grassmannian codebook optimized for that block. The bit budget is allocated non-uniformly across feature blocks according to task importance, subject to

$\sum_{t=1}^{T} B_t = B.$

Blocks that carry more important features receive more bits and therefore lower quantization error (Fu et al., 25 Sep 2025).

Second, each quantized codeword index is mapped to a unique digitally modulated sequence from a shared modulation codebook. The paper explicitly notes compatibility with digital PHYs such as QAM and BPSK. Third, all agents transmit their modulated sequences simultaneously. The hybrid RIS and the AirComp transceivers are adjusted so that the EN observes controlled superposition across agents, allowing the received signal to represent an over-the-air sum of codewords (Fu et al., 25 Sep 2025).

Fourth, the EN performs sparse signal recovery to estimate the aggregate codeword sum. The reported implementation uses compressive sensing algorithms such as SWOMP, after which the EN reconstructs the global feature via the shared codebooks and feeds it to the inference model (Fu et al., 25 Sep 2025). In contrast to one-bit or per-dimension scalar schemes, the pipeline is explicitly vector-quantized and block-structured. The significance of that design is task selectivity: non-uniform bit allocation makes the communication layer sensitive to downstream inference relevance rather than treating all feature dimensions as equally important.

The paper’s baselines clarify this distinction. MD-AirComp is described as digital AirComp with a common codebook, uniform bit allocation, and no RIS, while OBDA uses one-bit quantization per feature dimension mapped to BPSK (Fu et al., 25 Sep 2025). HRD-AirComp differs from both by combining shared blockwise Grassmannian codebooks, adaptive bit allocation, and hybrid RIS-assisted superposition control.

4. Inference-oriented theory and surrogate objective

The task setting is multi-class inference. Rather than optimizing aggregation distortion directly, the paper defines the performance metric as the entropy of posteriors,

$$H = \mathbb{E}_{\mathbf{\hat f}} \left[ -\sum_{l=1}^{L}\Pr(l \mid \mathbf{\hat f}) \ln \Pr(l \mid \mathbf{\hat f}) \right],$$

where $\mathbf{\hat f}$ is the reconstructed global feature vector (Fu et al., 25 Sep 2025). This metric quantifies prediction uncertainty and anchors the task-oriented design principle.

To connect communication design with inference performance, the analysis assumes that the ground-truth global feature $\mathbf{f}$ follows a Gaussian Mixture Distribution. Under that assumption, Theorem 1 gives an upper bound on the aggregation MSE for block $t$, decomposed into misalignment error, channel noise, and quantization distortion: $$\mathbb{E}\|\mathbf{\hat f}_t-\mathbf{f}_t\|^2 \leq \text{Misalignment error} + \text{Channel noise} + \text{Quantization distortion}.$$ The reported explicit form is

$$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$0

where $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$1 is the signal correlation between agents, $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$2 is the transmission coefficient of agent $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$3, $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$4 is the EN receiver beamformer, and $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$5 are the RIS and EN noise variances (Fu et al., 25 Sep 2025).

Theorem 2 then provides a surrogate lower bound on posterior entropy: $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$6 with

$$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$7

The average class separability

$$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$8

is used as the surrogate objective for maximizing inference accuracy, where $$\mathbf{h}_{k} = \mathbf{h}_{\mathrm{AE},k} + \mathbf{H}_{\mathrm{RE}} \mathbf{\Phi}\, \mathbf{h}_{\mathrm{AR},k},$$9 reflects feature importance and $\mathbf{\Phi}$0 is the $\mathbf{\Phi}$1-th diagonal entry of the error covariance (Fu et al., 25 Sep 2025).

This theoretical structure matters because it translates feature quantization and wireless aggregation errors into a task-relevant criterion. A plausible implication is that the framework does not treat all blocks or all feature coordinates symmetrically: it weights them through class-centroid separation and error covariance, which is exactly the rationale for non-uniform bit allocation. At the same time, the Gaussian Mixture Distribution assumption should not be conflated with a proof that all deep feature spaces satisfy that model; the paper reports that the framework applies to real DNN features as demonstrated with MobileNet-v2, but that empirical observation is distinct from the analytical assumption (Fu et al., 25 Sep 2025).

5. Joint optimization problem and JQAPB algorithm

The joint design objective is to maximize the surrogate $\mathbf{\Phi}$2 by optimizing quantization, transmission, reception, and propagation control simultaneously (Fu et al., 25 Sep 2025). The optimization variables are the bit allocation $\mathbf{\Phi}$3, the agent transmission coefficients $\mathbf{\Phi}$4, the EN receiver beamforming vector $\mathbf{\Phi}$5, and the hybrid RIS beamforming matrix $\mathbf{\Phi}$6. The reported constraints include the total bit budget $\mathbf{\Phi}$7, the per-agent power constraint

$\mathbf{\Phi}$8

and hybrid RIS amplitude, phase, and total amplification power budget $\mathbf{\Phi}$9 constraints (Fu et al., 25 Sep 2025).

The resulting problem, denoted as P1, is non-convex because of fractional sums, integer-valued $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$0, and coupling among variables under RIS constraints. The proposed solver is an alternating optimization procedure named JQAPB (Fu et al., 25 Sep 2025). Its updates are organized as follows.

For fixed $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$1, the bit allocation subproblem is convexified via successive convex approximation using Taylor expansion. For fixed other variables, each transmission coefficient $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$2 is solved as a QCQP, or in closed form when the agent signals are uncorrelated: $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$3 where $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$4 is a Lagrange multiplier set by the RIS power constraint (Fu et al., 25 Sep 2025).

For fixed remaining variables, the EN receiving beamformer admits the closed-form solution

$$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$5

The hybrid RIS beamforming step is handled by alternating minimization: with phases fixed, optimize active amplitudes via KKT conditions; with amplitudes fixed, optimize phases in closed form (Fu et al., 25 Sep 2025). The paper states that this design significantly reduces computational complexity compared to SDR-based approaches.

This algorithmic preference aligns with the earlier RIS-AirComp literature, where alternating minimization combined with SCA and closed-form inner updates was already shown to reduce computation time by one to two orders of magnitude relative to SDR/DC baselines while achieving similar MSE performance (Fang et al., 2021). HRD-AirComp adopts the same broad principle of avoiding SDR-heavy formulations, but it extends the optimization scope to include task-oriented bit allocation and hybrid active-passive RIS control (Fu et al., 25 Sep 2025).

6. Experimental findings, baselines, and technical significance

The reported experiments cover both synthetic linear classification on Gaussian mixture data and realistic multi-view object recognition using ModelNet and MobileNet-v2 features (Fu et al., 25 Sep 2025). The baseline set includes MD-AirComp, OBDA, Full Power Transmission, SDR+Gaussian for RIS reflection, and PFA/Ideal Edge Inference as upper bounds with or without quantization and transmission errors. This comparison isolates the effects of vector quantization, adaptive bit allocation, transmit power control, and hybrid RIS beamforming.

Several findings are reported. First, for RIS beamforming, JQAPB is described as approximately $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$6 faster than SDR-based RIS beamforming and as achieving better or comparable local optima due to closed-form iterative steps (Fu et al., 25 Sep 2025). Second, in object recognition, JQAPB achieves a $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$7–$$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$8 accuracy improvement over MD-AirComp and OBDA, and reaches performance within $$\mathbf{\Phi} = \mathbf{\Phi}_{\mathrm{a}} + \mathbf{\Phi}_{\mathrm{p}}$$9 of the perfect aggregation case with suitable bit and codebook sizes (Fu et al., 25 Sep 2025). Third, it outperforms all baselines in both accuracy and lower inference uncertainty, where uncertainty is measured through the entropy of posteriors.

The parameter studies further report that increasing the number of agents $\sum_{t=1}^{T} B_t = B.$0 improves accuracy and reduces uncertainty because of feature diversity, that hybrid RIS outperforms purely passive or purely active RIS under various power budgets and numbers of active elements, and that adaptive bit allocation protects critical features in both synthetic and DNN feature settings (Fu et al., 25 Sep 2025). The paper also states that the theoretical framework applies to real DNN features, as demonstrated with MobileNet-v2, not only to synthetically generated Gaussian data.

These findings place HRD-AirComp at the intersection of wireless aggregation, quantized representation design, and inference-aware communications. A plausible implication is that its main novelty lies less in any single module than in the coupling of four design layers—feature quantization, agent transmission, EN beamforming, and hybrid RIS reflection—under a unified surrogate for inference accuracy. Another plausible implication is that the scheme reframes AirComp from a function-computation primitive into an edge-AI systems primitive: the aggregation target is no longer merely a sum with small MSE, but a reconstructed feature whose class separability and posterior entropy determine downstream utility (Fu et al., 25 Sep 2025).