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Learned Trimmed-Ridge Regression for Channel Estimation in Millimeter-Wave Massive MIMO

Pengxia Wu, Julian Cheng, Yonina C. Eldar, John M. Cioffi

TL;DR

This work tackles CSI estimation in mmWave massive MIMO with hybrid RF constraints by leveraging beamspace sparsity. It introduces a trimmed-ridge regression formulation and derives an iterative solution via DC programming, then unfolds the iterations into a trainable deep network (UTRR) with complexity-reducing configurations and a model-averaging ensemble. Empirical results show that UTRR and its ensemble achieve significantly lower NMSE and higher downlink sum rates than iterative baselines and several deep-learning competitors, while requiring far fewer iterations and offering real-time feasibility. The framework demonstrates strong potential for practical beamspace channel estimation and can extend to other sparse reconstruction problems.

Abstract

Channel estimation poses significant challenges in millimeter-wave massive multiple-input multiple-output systems, especially when the base station has fewer radio-frequency chains than antennas. To address this challenge, one promising solution exploits the beamspace channel sparsity to reconstruct full-dimensional channels from incomplete measurements. This paper presents a model-based deep learning method to reconstruct sparse, as well as approximately sparse, vectors fast and accurately. To implement this method, we propose a trimmed-ridge regression that transforms the sparse-reconstruction problem into a least-squares problem regularized by a nonconvex penalty term, and then derive an iterative solution. We then unfold the iterations into a deep network that can be implemented in online applications to realize real-time computations. To this end, an unfolded trimmed-ridge regression model is constructed using a structural configuration to reduce computational complexity and a model ensemble strategy to improve accuracy. Compared with other state-of-the-art deep learning models, the proposed learning scheme achieves better accuracy and supports higher downlink sum rates.

Learned Trimmed-Ridge Regression for Channel Estimation in Millimeter-Wave Massive MIMO

TL;DR

This work tackles CSI estimation in mmWave massive MIMO with hybrid RF constraints by leveraging beamspace sparsity. It introduces a trimmed-ridge regression formulation and derives an iterative solution via DC programming, then unfolds the iterations into a trainable deep network (UTRR) with complexity-reducing configurations and a model-averaging ensemble. Empirical results show that UTRR and its ensemble achieve significantly lower NMSE and higher downlink sum rates than iterative baselines and several deep-learning competitors, while requiring far fewer iterations and offering real-time feasibility. The framework demonstrates strong potential for practical beamspace channel estimation and can extend to other sparse reconstruction problems.

Abstract

Channel estimation poses significant challenges in millimeter-wave massive multiple-input multiple-output systems, especially when the base station has fewer radio-frequency chains than antennas. To address this challenge, one promising solution exploits the beamspace channel sparsity to reconstruct full-dimensional channels from incomplete measurements. This paper presents a model-based deep learning method to reconstruct sparse, as well as approximately sparse, vectors fast and accurately. To implement this method, we propose a trimmed-ridge regression that transforms the sparse-reconstruction problem into a least-squares problem regularized by a nonconvex penalty term, and then derive an iterative solution. We then unfold the iterations into a deep network that can be implemented in online applications to realize real-time computations. To this end, an unfolded trimmed-ridge regression model is constructed using a structural configuration to reduce computational complexity and a model ensemble strategy to improve accuracy. Compared with other state-of-the-art deep learning models, the proposed learning scheme achieves better accuracy and supports higher downlink sum rates.
Paper Structure (22 sections, 24 equations, 13 figures, 3 tables, 2 algorithms)

This paper contains 22 sections, 24 equations, 13 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Channel Estimation in a MmWave Massive MIMO System
  4. Trimmed-Ridge Regression
  5. Iterative Trimmed-Ridge Regression Algorithm
  6. Deriving an Iteration Solution Using DC Programming
  7. Relaxing Step Sizes
  8. ITRR Performance on Reconstructing a Sparse Sample
  9. Learned Trimmed-Ridge Regression Algorithm
  10. Unfolding Trimmed-Ridge Regression (UTRR) Structure
  11. Reducing Computational Complexity
  12. Training Procedure
  13. Model-Averaging Ensemble
  14. Numerical Results
  15. Simulation Setup
  16. ...and 7 more sections

Figures (13)

  • Figure 1: A mmWave massive MIMO system with a lens antenna array.
  • Figure 2: Channel magnitudes of a sparse sample $\mathbf h_s$ and the reconstruction $\hat{\mathbf h}_s$ by the ridge regression, Lasso regression and the proposed ITRR algorithm, wherein the normalized squared $\ell_2$-error is computed as $\lVert\Re(\mathbf{h}_s)-\Re(\hat{\mathbf{h}}_s)\rVert_2^2/\lVert\Re(\mathbf h_s)\rVert_2^2$.
  • Figure 3: Objective evolution when reconstructing the real part of the sparse sample $\Re(\mathbf h_s)$.
  • Figure 4: Normalized reconstruction error evolution (i.e., $\lVert\hat{\Re(\mathbf h_s)}-\Re(\mathbf h_s)\rVert_2^2 / \lVert\Re(\mathbf h_s)\rVert_2^2 )$) when reconstructing the real part of the sparse sample
  • Figure 5: The $\ell_2$-norm value evolution when reconstructing the real part of the sparse sample
  • ...and 8 more figures