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Over-The-Air Extreme Learning Machines with Nonlinear Stacked Intelligent Metasurfaces
Kyriakos Stylianopoulos, Mattia Fabiani, Giulia Torcolacci, Davide Dardari, George C. Alexandropoulos
TL;DR
The paper addresses how to perform binary ML inference directly over wireless signals with lower receiver hardware complexity. It implements an OTA ELM using a fixed nonlinear metasurface followed by tunable linear stacked metasurfaces and a single RF chain. Across evaluated settings, the proposed approaches achieve performance close to ideal LS under the reported conditions, while theory establishes universal approximation under specified assumptions.
Problem
Prior OTA metasurface designs are constrained by linear operations, motivating evidence for nonlinear, low-complexity OTA inference architectures.
Method
A fixed nonlinear metasurface implements the ELM activation, while cascaded tunable linear metasurfaces approximate output weights in a single-RF-chain receiver using PGD or black-box GO training.
Results
Both approximate SIM solutions achieve performance close to the ideal LS case across reported system parameters, with universal approximation established theoretically under stated assumptions.
Takeaways & Limitations
The results support the viability of fully OTA ML inference using nonlinear and stacked metasurfaces with reduced reception-side hardware complexity.
Takeaways & Limitations
Training with measured nonlinear-layer signals requires one or more RF chains and is affected by measurement noise.
Abstract
from arXiv · showhide
The recently envisioned goal-oriented communications paradigm requires machine learning inference to be performed directly on wirelessly transferred data. This paper presents an eXtremely Large (XL) Multiple-Input Multiple-Output (MIMO) system that operates as an Extreme Learning Machine (ELM) to execute Over-The-Air (OTA) binary classification. To reduce hardware complexity, the receiver is equipped with cascaded metasurfaces terminating in a single radio-frequency chain. A front metasurface layer applies a fixed nonlinear response to the incoming signal, acting as the ELM's activation function. Subsequent tunable linear metasurface layers physically approximate the trained network weights directly in the wave domain. Numerical evaluations across diverse datasets showcase that our XL MIMO architecture achieves classification accuracy comparable to idealized digital models, thereby proving the viability of low-complexity, wave-domain OTA learning.
I. INTRODUCTION
The paper targets OTA machine-learning inference by using wireless propagation and metasurfaces to compute directly on transferred data. It proposes an XL-MIMO ELM architecture that adds nonlinear processing and low-complexity OTA training to address limitations of prior approaches.
- Motivation: OTA computing performs computations directly through superposed RF signals, avoiding reconstruction of input data at the receiver.This paradigm has been explored for wireless ML inference using metasurfaces that emulate or approximate neural-network layers.
- Motivation: Prior metasurface-based OTA models rely on linear operations, which significantly constrains their approximation capability.
- Prior XL-MIMO ELM: An earlier XL-MIMO ELM uses the wireless channel as random hidden-layer weights and an analog combiner as the output layer.It supports rapid training and reconfiguration while retaining universal function approximation, but has practical limitations.
- Proposed approach: The proposed XL-MIMO-ELM receiver uses a fixed nonlinear metasurface layer for activation and subsequent linear layers for trainable OTA combining.The cascaded metasurfaces terminate in a single antenna and RF chain, reducing reception-side hardware complexity.
- Proposed approach: Two OTA training strategies configure the linear metasurface layers: PGD approximates the ideal LS solution, while GO requires only the final scalar RF-chain output.
II. THE PROPOSED XL MIMO SYSTEM MODEL
The system maps transmitted inputs to binary targets through a narrowband XL-MIMO channel, a nonlinear receiving metasurface, and analog combining before a single RF chain. Its nonlinear layer preserves input phase while transforming signal envelopes, and the resulting scalar output includes receiver AWGN.
- System operation: The XL-MIMO receiver performs end-to-end OTA function approximation, estimating target values directly from transmitted inputs without reconstructing the input data.All computational processing is performed in the analog/RF domain.
- System operation: The model assumes real-valued binary classification with x(i) ∈ (0, 1)^Nt and z(i) ∈ {0, 1}.A trainable transmitter feature-extraction module is possible but left for future investigation.
- Channel model: The channel follows a quasi-static Ricean model during training, with LoS and non-LoS components controlled by the Ricean factor K.
- Nonlinear metasurface: The receiving nonlinear metasurface applies a memoryless element-wise mapping that preserves phase and transforms the input envelope through first-order harmonic extraction.Its baseband response is g(y) = C(|y|) exp(ȷarg{y}), with C(·) denoting the AM/AM characteristic.
- Nonlinear metasurface: The nonlinear response is approximated by a softplus activation, after which a controllable complex weight vector linearly combines the elements before RF-chain reception.The physical output includes additive white Gaussian noise at the receiver.
A. XL MIMO as an ELM
The XL-MIMO system can be interpreted as a complex-valued ELM: channel and bias parameters form a random hidden layer, while combining weights are trained as the output layer. The analysis establishes universal approximation under stated high-SNR and randomness assumptions.
- ELM formulation: The channel transformation and nonlinear response implement the random hidden layer, while combining weights serve as trainable ELM output weights.
- ELM formulation: The output weights are optimized by minimizing least-squares error between dataset targets and system outputs.The formulation uses the standard ELM training objective.
- ELM formulation: L2 regularization with hyperparameter ℓ > 0 is included in the closed-form weight solution to support generalization beyond the training dataset.
- Approximation guarantee: Universal approximation is established for the proposed ELM under high SNR, Ricean fading, and biases drawn from a continuous positive-support distribution.The result asserts existence of a weight vector achieving the stated approximation condition.
- Approximation guarantee: The proof reduces zero-error solvability to showing that det(G) is nonzero with probability 1 using real-analyticity and a measure-zero zero set.
B. OTA Analog Combining using SIM
The receiver implements OTA combining with a stack of diffractive linear metasurfaces placed after the nonlinear layer and before a single antenna. Controllable layer responses produce an effective transfer vector that replaces the ELM output weights.
- SIM architecture: Outputs from the nonlinear metasurface feed a stack of L diffractive linear metasurfaces terminating in a single antenna and RF chain.Each layer is modeled through propagation coefficients between adjacent metasurfaces.
- SIM architecture: The proposed architecture models inter-layer propagation with full-rank pseudo-random matrices, assuming a reverberating enclosure with non-uniform propagation diversity.
- SIM configuration: Figure 1 sketches the channel and metasurface responses as ELM components and shows the forward-pass computation flow.
- SIM configuration: Each linear metasurface layer has controllable amplitudes and phase shifts that determine its complex element responses.
- SIM configuration: The stacked metasurface response performs OTA combining by substituting the ELM weight vector w with the configured effective vector wφ.
A. Approximation of the LS Solution
The SIM is trained to approximate the optimal digital LS weights, enabling OTA combining whose accuracy is evaluated against ideal performance across datasets and channel conditions.
- A. Approximation of the LS Solution: The SIM responses are optimized so their effective weights wφ closely approximate the optimal digital weights w∗.The resulting parameter optimization can be solved via projected gradient descent (PGD).
- A. Approximation of the LS Solution: Classification accuracy is evaluated against the number of front-layer metamaterials Nr across three distinct datasets.Nr also corresponds to the number of trainable parameters at the receiver’s nonlinear metasurface layer.
- A. Approximation of the LS Solution: Classification accuracy is evaluated over different Ricean factors using the WBCD dataset.This probes sensitivity to channel scattering conditions.
B. Direct SIM Training via Genetic Optimization
A genetic-optimization approach directly searches SIM parameters without requiring per-element observations of the received signal matrix.
- B. Direct SIM Training via Genetic Optimization: The black-box genetic-optimization approach searches SIM parameters without relying on observations of G.It begins from a population of candidate weight vectors and evaluates a maximization objective.
- B. Direct SIM Training via Genetic Optimization: The method retains top-performing candidates, generates offspring through crossover, and applies complex-valued mutations across generations.Mutations use Δ = ±0.2 ± j0.2, and the best candidate is retained after kgen generations.
- B. Direct SIM Training via Genetic Optimization: The genetic-optimization latency depends on kgen × kpop objective evaluations and the SIM-layer switching time.This creates a computational and reconfiguration cost relative to direct gradient-based training.
IV. NUMERICAL RESULTS AND DISCUSSION
Experiments evaluate OTA-ELM training across datasets, model sizes, channel scattering, and measurement noise, showing close SIM-to-ideal performance within identified operating conditions.
- IV. NUMERICAL RESULTS AND DISCUSSION: The evaluation covers Parkinson’s, WBCD, and MNIST binary-classification datasets with standardized preprocessing and repeated random initializations.MNIST is converted to even/odd digit classification after subsampling to 100 pixels.
- IV. NUMERICAL RESULTS AND DISCUSSION: The experiments use two 32 × 32 linear SIM layers, receive SNR of 15 dB, and up to T = 2000 PGD iterations.The idealized LS solution serves as an upper bound for comparison.
- IV. NUMERICAL RESULTS AND DISCUSSION: As Nr increases, classification accuracy generally improves, while slight overfitting appears in the simplest WBCD case.Nr is also the number of trainable ELM parameters.
- IV. NUMERICAL RESULTS AND DISCUSSION: Both approximate SIM solutions perform close to the ideal case, but the largest Nr values degrade when training iterations are insufficient.For MNIST at Nr = 64, execution times are 0.32, 6.21, and 178.5 seconds for Ideal LS, LS & PGD, and GO, respectively.
- IV. NUMERICAL RESULTS AND DISCUSSION: At sufficiently high measurement SNR, the ideal LS and PGD-trained SIM approaches converge to noise-free performance.Performance degrades at low measurement SNR, while the SIM approximation remains comparable to the ideal LS case.
- IV. NUMERICAL RESULTS AND DISCUSSION: Both ideal and SIM-approximate methods remain stable under sufficiently diverse channels, but accuracy degrades as Ricean fading becomes LoS-dominant.LoS dominance makes columns of G linearly dependent, so the universal-approximation conditions are no longer satisfied.
V. CONCLUSION
The proposed receiver realizes complex-valued ELM inference fully over the air with a nonlinear front metasurface and cascaded linear SIM layers, achieving near-ideal performance while retaining a stated comparison gap.
- V. CONCLUSION: The XL MIMO receiver performs computations equivalent to a complex-valued ELM for fully OTA inference using properly designed metasurface components.The channel provides random hidden-layer weights, while the receiver implements the activation, bias, and trainable weights.
- V. CONCLUSION: The LS-approximation and GO approaches achieve performance close to the ideal LS case across a range of system parameters.The GO approach does not require per-element signal observations.
- V. CONCLUSION: The architecture jointly uses a fixed nonlinear metasurface for activation and a cascaded linear SIM for trainable output weights within a single-RF-chain receiver.This combination is identified as the proposed architecture’s novelty relative to prior XL-MIMO-ELM work.
- V. CONCLUSION: A comparative analysis against digital and OTA ELM implementations remains future work for clarifying hardware, energy, and training or measurement overhead benefits.The paper identifies this comparison as necessary for further positioning the NL-SIM architecture.