Bayes-Optimal Unsupervised Learning for Channel Estimation in Near-Field Holographic MIMO

TL;DR

The paper tackles the challenge of estimating high-dimensional HMIMO channels in unknown near-field EM environments without priors or supervision. It derives a Bayes-optimal MMSE estimator based on the Stein score of the observed pilots and a PCA-based noise-level estimate, implemented via an unsupervised denoising score matching framework. The approach yields closed-form estimators for fully-digital HMIMO and integrates into an OAMP-based strategy for hybrid systems, with rigorous complexity analysis and extensive simulations showing near-oracle performance, strong robustness, and rapid online adaptation. This work enables practical, scalable, and environment-agnostic channel estimation for next-generation HMIMO systems.

Abstract

Holographic MIMO (HMIMO) is being increasingly recognized as a key enabling technology for 6G wireless systems through the deployment of an extremely large number of antennas within a compact space to fully exploit the potentials of the electromagnetic (EM) channel. Nevertheless, the benefits of HMIMO systems cannot be fully unleashed without an efficient means to estimate the high-dimensional channel, whose distribution becomes increasingly complicated due to the accessibility of the near-field region. In this paper, we address the fundamental challenge of designing a low-complexity Bayes-optimal channel estimator in near-field HMIMO systems operating in unknown EM environments. The core idea is to estimate the HMIMO channels solely based on the Stein's score function of the received pilot signals and an estimated noise level, without relying on priors or supervision that is not feasible in practical deployment. A neural network is trained with the unsupervised denoising score matching objective to learn the parameterized score function. Meanwhile, a principal component analysis (PCA)-based algorithm is proposed to estimate the noise level leveraging the low-rank near-field spatial correlation. Building upon these techniques, we develop a Bayes-optimal score-based channel estimator for fully-digital HMIMO transceivers in a closed form. The optimal score-based estimator is also extended to hybrid analog-digital HMIMO systems by incorporating it into a low-complexity message passing algorithm. The (quasi-) Bayes-optimality of the proposed estimators is validated both in theory and by extensive simulation results. In addition to optimality, it is shown that our proposal is robust to various mismatches and can quickly adapt to dynamic EM environments in an online manner thanks to its unsupervised nature, demonstrating its potential in real-world deployment.

Figures (6)

• Figure 1: A ULA-shaped HMIMO BS in the cartesian coordinate system, where the transmitter and the scatterers locate in the near field of the HMIMO array.
• Figure 2: Comparison of the heat map of different spatial correlation matrices. The BS is equipped with a ULA with 1024 antennas that operates at a carrier frequency of $f_{c}=3.5$ GHz, with $S=20$ m, $R=3$ m, $\Psi=\frac{\pi}{3}$, $\mu=\frac{\pi}{4}$, and $\kappa=0$. The following four cases are plotted: (a) Near-field HMIMO with $d_{a}=\frac{\lambda_{c}}{10}$, (b) Far-field HMIMO with $d_{a}=\frac{\lambda_{c}}{10}$, (c) Near-field MIMO with $d_{a}=\frac{\lambda_{c}}{2}$, (d) Far-field MIMO with $d_{a}=\frac{\lambda_{c}}{2}$. The ranks of different spatial correlation matrices are also labeled in the caption.
• Figure 3: (a) The heat map of the received pilot signals of the near-field HMIMO channel when the received SNR is 0 dB. The other parameters are set the same as Fig. \ref{['fig:Difference-FF-NF-correlation']}(a). (b) Eigenvalues of the covariance matrices of the virtual subarray channels $\{\mathbf{h}_{t}\in\mathbb{R}^{2d\times1}\}_{t=1}^{s}$ and the pilots $\{\mathbf{y}_{t}\in\mathbb{R}^{2d\times1}\}_{t=1}^{s}$, in which the principal and redundant dimensions are separated by a dotted line.
• Figure 5: Online adaptation in dynamic environments. (a) The positions of the scatterer rings with different values of direction $\Psi$, radius $R$, and distance $S$. (b) NMSE versus the number of pilot transmission slots during the online adaptation process, when the received SNR is 10 dB.
• Figure 6: Simulation results in hybrid analog-digital systems. (a) NMSE versus the received SNR when the under-sampling ratio is $KN_{\text{RF}}/N=0.3$. (b) NMSE as a function of the iteration number when the under-sampling ratio is $KN_{\text{RF}}/N=0.5$. (c) NMSE as a function of different under-sampling ratio $KN_{\text{RF}}/N$ when the received SNR is set as 0 dB.

Theorems & Definitions (3)

• Remark 1
• Theorem 2: Alain-Bengio 2014Alain
• Theorem 3