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XL MIMO Extreme Learning Machines: Wave-Domain Inference with Stacked Metasurfaces

Kyriakos Stylianopoulos, Mattia Fabiani, Davide Dardari, George C. Alexandropoulos

arXiv:2610.35469v1eess.SPcs.ETcs.LG

TL;DR

Conventional acquisition, transmission, and processing can entail unnecessary data transfer and redundant computation, while many MS-based physical architectures remain predominantly linear. This paper models XL-MIMO inference as a complex-domain ELM, using a heterogeneous RX SIM and closed-form wave-domain combining. Its evaluations report degradation at low measurement SNR and convergence toward ideal noise-free-SNR results, while performance also declines as the channel becomes LoS-dominant.

  • Problem

    Conventional wireless processing can involve unnecessary data transmission, increased processing complexity, and redundant computations at both ends of the link.

    Source: §I · p. 1

  • Method

    The XL-MIMO-ELM uses channel coefficients as random hidden-layer weights and a heterogeneous nonlinear/linear RX SIM for activation, bias, and trainable weights, with closed-form configuration of the final linear layer.

    Source: C. Wave-Domain Combining with Linear SIM · p. 7, §VI · p. 10

  • Results

    Performance degrades at low measurement SNR but converges to ideal noise-free-SNR results, and classification accuracy degrades as the channel becomes LoS-dominant.

    Source: §V · p. 9

  • Takeaways & Limitations

    The proposed system performs machine-learning inference on transmitted data entirely over the air using XL-MIMO and receiver-side metasurfaces.

    Source: §VI · p. 9

    The analysis assumes rich scattering for its universal-approximation baseline; as the channel becomes LoS-dominant, the columns of G become linearly dependent and accuracy degrades.

    Source: §V · p. 9

Abstract

from arXiv · show

The recently envisioned goal-oriented communications paradigm requires machine learning inference to be performed directly on wirelessly transferred data, by splitting neural network computations between the transmitter, the Receiver (RX), and the wireless channel. Such approaches aim to offload large portions of the computation onto the channel itself, which is often controlled by reconfigurable MetaSurfaces (MSs). Typical MS-parametrized systems, however, enable only limited computation offloading in the wave domain, usually corresponding to a single neural network layer, due to adverse channel fading effects and the almost linear operation of the channel and conventional MS models. In this paper, we demonstrate that an eXtremely Large (XL) Multiple-Input Multiple-Output (MIMO) system equipped with Stacked Intelligent MSs (SIM) at the RX side, together with a single antenna and its accompanying radio-frequency chain, may act as a two-layer neural network operating exclusively over the air. The first (uncontrollable) network layer corresponds to the transformations offered by wireless fading, while the second (controllable) layer is implemented via SIM operations. In particular, only the first, front-end MS layer needs to exhibit nonlinear response to serve as an activation function. Leveraging the Extreme Learning Machine (ELM) framework, the proposed XL MIMO system architecture is shown to be a universal approximator, and the optimal linear-SIM-based combining is derived in closed form, allowing for fast training even and partial signal observability. Numerical evaluations across diverse datasets showcase that the proposed XL-MIMO-ELM framework achieves classification accuracy comparable to idealized digital models, thereby, proving the viability of low complexity, wave-domain neural computations.

I. INTRODUCTION

The introduction motivates task-oriented inference and wave-domain computing as ways to reduce data transmission and digital processing, while noting that existing MS architectures are often predominantly linear. The paper proposes an XL-MIMO ELM using a heterogeneous RX SIM and reports binary-classification performance approaching fully digital models in the XL-MIMO regime.

I. INTRODUCTION

  • Edge inference jointly trains transmitter and receiver processing for a target task, letting the receiver infer target values without reconstructing the original input.The introduction presents this as reducing conveyed information and downstream computation –.Source: §I · p. 1
  • Existing MS architectures remain predominantly linear, so their physical-domain savings may not offset channel-acquisition, configuration, and control overhead.Earlier physical neural architectures introduced depth through repeated transmissions, multi-stage topologies, or specialized nonlinear hardware –[23].Source: §I · pp. 1–2
  • The proposed RX places a fixed nonlinear front-end MS before linear SIM layers that implement trainable wave-domain combining for a complex-domain ELM.The architecture uses a single RX antenna and associated radio-frequency chain, [31].Source: §I · p. 2
  • The authors describe gradient-free training, a reduced-overhead observation procedure for a single-RFC receiver, and closed-form wave-domain combining through the last linear MS layer.The paper also extends its conference version with physically grounded SIM propagation, a sketching-based measurement procedure, and noise analysis.Source: §I · p. 2
  • The paper formulates the XL-MIMO system as an ELM operating entirely over the air and presents a theoretical proof of universal approximation.The framework uses the complex-domain ELM [27].Source: §I · p. 2
  • The framework’s numerical results approach fully digital models on binary-classification examples in the XL-MIMO regime.The authors characterize it as a fully analog network with an analog multi-antenna transmitter, Ricean MIMO channel, and wave-domain MS activation and combining.Source: §I · p. 2

II. XL MIMO SYSTEM MODEL

The system is trained end-to-end to approximate an input-to-target mapping through an XL-MIMO link, with inference performed over the air. Its channel model is Ricean, and the transmitter uses amplitude-modulated data symbols.

II. XL MIMO SYSTEM MODEL

  • The XL-MIMO system is trained as an over-the-air function approximator mapping transmitter-observed inputs to receiver-inferred target features.The considered receiver has a SIM, a single antenna, and an accompanying radio-frequency chain.Source: §II · p. 3

A. Signal and Channel Models

  • The input dimension equals the number of transmit antennas, targets are real-valued within [−1, +1], and transmitted symbols use amplitude modulation.The inputs are specified to lie in (0, 1)^Nt.Source: A. Signal and Channel Models · p. 3
  • The channel is modeled as quasi-static during training and Ricean, combining a rank-one line-of-sight component with a complex Gaussian non-line-of-sight component.The Ricean factor K controls the dominance of the two components; the transmitter and receiver are modeled with uniform planar arrays.Source: A. Signal and Channel Models · p. 3

B. The Proposed RX Architecture

The receiver combines a nonlinear front-end metasurface with a stack of linear metasurfaces, followed by one antenna and radio-frequency chain. The nonlinear layer supplies an activation-like transformation, while the remaining layers process its output through tunable wave-domain responses.

B. The Proposed RX Architecture

  • The RX SIM has a nonlinear front-end MS followed by L linear MS layers, with a single antenna and radio-frequency chain for digital conversion.The paper excludes additional transmitter feature extraction and digital pre- or post-processing from this framework.Source: B. The Proposed RX Architecture · p. 3

1) Front-End Nonlinear Metasurface:

  • Each front-end element applies a memoryless nonlinear transformation and zonal filtering; the equivalent low-pass mapping preserves input phase while transforming its envelope through the AM/AM characteristic.The paper considers element-wise thresholding with positive bias, and approximates its response by a softplus activation after replacing the transition with a continuous quasi-linear slope.Source: 1) Front-End Nonlinear Metasurface · p. 4

2) Stacked Linear Metasurfaces:

  • The following linear MS layers process the nonlinear output, with inter-layer propagation represented by matrices and each element offering controllable amplitude and phase responses.The linear SIM response is formed by composing propagation and diagonal response matrices.Source: 2) Stacked Linear Metasurfaces · p. 4

3) Single-Antenna Reception:

The receiver uses a single antenna and RF chain, with the linear SIM implementing wave-domain combining. In the ELM formulation, channel transformations act as random hidden features and the linear-SIM weights are trained by least squares; under stated high-SNR conditions, the system has a universal-approximation guarantee.

3) Single-Antenna Reception:

  • With a single antenna and RF chain, the linear SIM implements the receiver’s wave-domain combining operation.The combining response is controlled to solve the inference task.Source: 3) Single-Antenna Reception · p. 4

III. THE PROPOSED XL-MIMO-ELM FRAMEWORK

  • The XL-MIMO system maps to an ELM: channel transformations provide random hidden-layer coefficients, while the combining weights serve as trainable output-layer weights.The ELM framework supports analytical least-squares training and theoretical guarantees.Source: 1) The Idealistic Noise-Free Case · p. 5, §III · p. 5

1) The Idealistic Noise-Free Case:

  • With high SNR, Ricean fading, continuously drawn positive biases, and Nr = D, the least-squares output achieves zero residual error with probability 1.The proposition also states a universal-approximation property for arbitrary datasets under its stated conditions.Source: 1) The Idealistic Noise-Free Case · p. 5

2) Effect of the Reception Noise:

The analysis models receiver noise as independent complex Gaussian noise aggregated across the dataset and quantifies prediction error relative to the noise-free ELM. The remaining analytical presentation assumes noise-free operation, while the numerical section evaluates noise effects.

Effect of the Reception Noise

  • The analysis models receiver noise as independent AWGN and compares noise-imputed ELM predictions with the ideal noise-free predictions.The noise is aggregated across the dataset into a complex Gaussian noise vector.Source: 2) Effect of the Reception Noise · p. 5
  • The analytical description assumes noise-free operation for simplicity after quantifying expected MSE as a function of reception noise.The effects of noise are quantified numerically later in the paper.Source: 2) Effect of the Reception Noise · p. 5
  • For classification, accuracy is described as less susceptible to noise than MSE because noise must change a class decision threshold to affect accuracy.This qualification applies to the classification problems considered in the paper.Source: 2) Effect of the Reception Noise · p. 6

B. Formulation of the Activation Matrix G

The paper estimates the least-squares combining weights using measurements from selected reception combiners, avoiding dedicated CSI estimation. A reduced-dimensional sketch recovers the original weights at full measurement dimension, while nested sketch sizes yield non-increasing residual error.

B. Formulation of the Activation Matrix G

  • The proposed low-overhead training procedure measures each data point using M ≪ Nr selected reception combiners, avoiding a dedicated CSI-estimation process.The received measurements are used to derive a least-squares solution.Source: B. Formulation of the Activation… · p. 6
  • The sketch uses a reduced M-dimensional observation space to estimate intermediate weights, then projects them back to synthesize the physical Nr-dimensional combining vector.The measured outputs are collected across the data points using the selected combiners.Source: B. Formulation of the Activation… · p. 6
  • At M = Nr, a full-rank sketch recovers the original least-squares weights and produces exactly the same predictions as the unsketched system.The weight-recovery result and prediction equivalence are stated separately.Source: B. Formulation of the Activation… · p. 6
  • For nested sketch sizes, the sketching residual prediction error is non-increasing as measurement dimension grows.The sketching matrix must be full rank; nested DFT or randomized unit-amplitude phase-shifting matrices are identified as suitable choices.Source: B. Formulation of the Activation… · p. 6

C. Wave-Domain Combining with Linear SIM

The paper analytically configures the final linear metasurface layer to realize desired combining weights over the air. Under full-rank propagation and nonzero final propagation coefficients, this configuration exactly matches the noise-free received signal to a scaled target functional.

C. Wave-Domain Combining with Linear SIM

  • The analytical SIM design replaces iterative response optimization that may not achieve desired responses with sufficiently small error, and realizes the inference output entirely over the air.The training procedure configures the final linear layer using the computed sketching weights.Source: C. Wave-Domain Combining with Linear SIM · p. 7
  • A closed-form configuration of the final linear MS layer exactly realizes an arbitrary combining vector in the noise-free physical received signal, subject to full-rank intermediate propagation and nonzero final propagation coefficients.The configuration targets a scaled functional of the signal after the nonlinear front-end MS.Source: C. Wave-Domain Combining with Linear SIM · p. 7
  • With randomly sampled responses and nondegenerate element placements, the linear-SIM propagation matrix is full rank almost surely, enabling the closed-form configuration.The paper states that one intermediate linear MS layer suffices for full rank, so L = 2 suffices for the considered inference operations.Source: C. Wave-Domain Combining with Linear SIM · p. 7

IV. XL-MIMO-ELM TRAINING AND DEPLOYMENT

Training uses repeated wave-domain observations to derive and configure a linear-SIM combiner; inference then requires one transmission and no digital computation. Both phases take place within the same channel coherence block.

IV. XL-MIMO-ELM TRAINING AND DEPLOYMENT

  • Training transmits each data point once and observes it M times with changing final-layer configurations, then computes and implements the least-squares combiner.The procedure uses O(D) transmissions and M + 1 reconfigurations of the final linear MS layer.Source: §IV · p. 7
  • The training observations can be collected without explicit channel-state information, using a single RX radio-frequency chain.The first L − 1 linear MS layers may remain static; their front-end propagation quantities are assumed measured during manufacture.Source: §IV · p. 7
  • For a feasibility example, a 5 ms coherence interval accommodates approximately 893 transmissions at a 5.6 µs symbol duration.The passage notes that this can suffice for reasonably small datasets when the M measurements are collected within each symbol interval.Source: §IV · p. 8
  • A single inference transmission uses the trained linear SIM to implement the combiner, requiring neither sketching nor digital computation.The resulting inference complexity is described as pseudo-constant time.Source: §IV · p. 8
  • Training and inference occur within one channel coherence block; the authors propose batch inference to avoid frequent retraining across blocks.Handling time-evolving channels through iterative training is left for future work.Source: §IV · p. 8

V. NUMERICAL RESULTS AND DISCUSSION

Across the reported classification experiments, XL-MIMO-ELM generally approaches or outperforms digital baselines, with performance depending on measurement SNR and channel scattering. The conclusion presents its OTA ELM implementation and identifies digital training calculations that remain.

V. NUMERICAL RESULTS AND DISCUSSION

  • Experiments use Parkinson’s, Wisconsin Breast Cancer, and even/odd MNIST classification, with an 80:20 train-test split and mean accuracy over 200 random initializations.The compared baselines are a parameter-matched Digital ELM and an RX MLP trained with idealized CSI and front-end signal knowledge.Source: §V · p. 8
  • XL-MIMO-ELM generally approaches or outperforms the digital baselines at matched parameter counts, though limited overfitting occurs as the model grows.Higher sketching dimension M generally improves results, while differences across M diminish as Nr increases and the model becomes over-parameterized.Source: §V · p. 8, Figure 3 · p. 9
  • Performance degrades at low measurement SNR but converges to the ideal noise-free SNR results; larger Nr and M improve performance.The RX MLP is less affected by noise because its channel equalization uses zero forcing.Source: §V · p. 9, Figure 4 · p. 9
  • Performance remains relatively stable across examined Ricean factors under sufficiently diverse channels, but degrades as the channel becomes LoS-dominant.The stated explanation is that columns of G become linearly dependent, so the conditions for universal approximation no longer hold; overfitting is more pronounced for Parkinson’s with Nr = 512.Source: §V · p. 9, Figure 5 · p. 9

VI. CONCLUSIONS AND FUTURE WORK

  • The conclusion describes channel coefficients as random hidden-layer weights and heterogeneous nonlinear/linear SIM layers as implementing activation, bias, and trainable weights.The analysis establishes universal approximation; during training, digital processing still computes the least-squares solution and linear-MS configuration weights.Source: §VI · pp. 9–10

APPENDIX A USED DEFINITIONS, PROPOSITIONS, AND LEMMAS

The appendices establish almost-sure invertibility by applying analytic zero-set arguments to the channel- and bias-dependent measurement matrix. They provide the definitions and domain conditions needed for that proof.

APPENDIX A USED DEFINITIONS, PROPOSITIONS, AND LEMMAS

  • A nonzero real-analytic function on an open connected domain has a zero set of Lebesgue measure zero, the criterion used to establish generic invertibility.The proof applies this criterion to det(G) as a function of channel and bias parameters.Source: Appendix A · p. 10, Appendix B · p. 11
  • Analyticity is retained under composition, supporting the proof that the determinant function is real analytic on its domain.The appendix also identifies polynomials, exponentials, affine transformations, and softplus as real analytic.Source: Appendix A · p. 10, Appendix B · p. 11

APPENDIX B PROOF OF PROPOSITION 1

  • The proof's domain excludes zero-modulus channel responses and nonpositive biases; removing the excluded complex-linear constraints preserves connectedness.Those constraints have real codimension two, while positive biases form a convex region.Source: Appendix A · p. 10, Appendix B · p. 11
  • The proof concludes that G is invertible almost everywhere, completing the argument for Proposition 1.It shows det(G) is nonzero almost everywhere on the specified parameter domain.Source: Appendix B · p. 11

APPENDIX C PROOF OF PROPOSITION 4

The proof compares predictions from nested sketch dimensions by viewing them as projections onto nested subspaces. Enlarging the sketch cannot increase the minimum distance to the unsketched prediction.

APPENDIX C PROOF OF PROPOSITION 4

  • Appending measurement directions makes the sketched activation-matrix ranges nested, with the smaller range contained in the larger one.The proof constructs the larger sketch by adding directions to the smaller sketch.Source: Appendix C · p. 11
  • The sketched predictions are orthogonal projections of the target onto their respective activation-matrix ranges, and each minimizes distance within that range.The unsketched prediction is likewise the projection onto the full range of G.Source: Appendix C · p. 11
  • Because the larger sketch spans a superset, its minimum distance from the unsketched prediction cannot increase.This follows from minimizing over the larger of the two nested subspaces.Source: Appendix C · p. 11

APPENDIX D PROOF OF PROPOSITION 5

The proof derives a final linear-SIM configuration that realizes a desired linear functional of the nonlinear metasurface output at the single receive antenna. Passive-element constraints permit that realization up to an overall scale.

APPENDIX D PROOF OF PROPOSITION 5

  • The final linear SIM is configured to produce an arbitrary linear functional of the front-end nonlinear metasurface output at the single receive antenna.The transfer function Q(ϕL) is matched to the target combining vector up to a positive real scaling factor.Source: Appendix D · pp. 11–12
  • Since Ω is invertible, the nonlinear output g(y) and its transformed coordinate s have a one-to-one correspondence for deriving the target weights.The transformed target weight vector is defined in the s-coordinate domain.Source: Appendix D · p. 12
  • With no perfect nulls in the channel to the antenna, the elementwise equations for the final metasurface responses can be solved independently.The proof uses this condition to obtain the final-layer configuration.Source: Appendix D · p. 12
  • For passive elements, the scaling factor must satisfy γ ≤ 1/ζ, and the resulting configuration computes the desired ELM functional scaled by γ.The constraint enforces element responses of magnitude at most one.Source: Appendix D · p. 12

APPENDIX E PROOF OF PROPOSITION 6

The proof establishes that the SIM propagation matrix is nonsingular almost surely by showing its determinant is nonzero except on a measure-zero set. It treats the per-layer configuration and propagation matrices separately.

APPENDIX E PROOF OF PROPOSITION 6

  • The determinant factorization makes Ω nonsingular when the configuration and propagation matrices at each intermediate layer are nonsingular.The proof applies multiplicativity of the determinant to the layer factors.Source: Appendix E · p. 12
  • Absolutely continuous sampling of nonzero element amplitudes makes each configuration matrix Φ_l nonsingular almost surely.Its determinant is the product of the element responses.Source: Appendix E · p. 12
  • The propagation matrix Ψ_l has an analytic determinant that is not identically zero, so its singular configurations have measure zero.The proof uses the large-pitch limit to show the determinant is not identically zero.Source: Appendix E · p. 12
  • Under the stated random-amplitude and nondegenerate-placement conditions, rank(Ω) = Nr almost surely.The proof obtains this by establishing nonsingularity of the layer factors.Source: Appendix E · pp. 12–13

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