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Symmetric Rank-$1$ Regularization for Iterative Inversion of Ill-Conditioned MIMO Channels

Jinfei Wang, Yi Ma, Rahim Tafazolli

TL;DR

This paper tackles the challenge of slow convergence in iterative matrix inversion for ill-conditioned MIMO channels by introducing symmetric rank-$1$ regularization (SR-$1$R), which augments the Gram matrix with a rank-$1$ symmetric term to reduce its condition number. The authors derive theoretical bounds showing the best achievable conditioning $\kappa(\mathbf{R})$ can approach $\lambda_1/\lambda_{N-2}$ and develop a practical, eigen-structure–based optimization, complemented by a power iteration–assisted (PIA) approach that obviates full EVD/SVD. An enhanced version (e-PIA) leverages parallel candidate evaluations to accelerate convergence, achieving substantial reductions in iteration counts (up to $35\%$ in some cases) while maintaining compatibility with parallel computing and approaching regularized zero-forcing performance. The work also analyzes scalability, offering a complexity comparable to direct inversion with a logarithmic algorithm-depth, and demonstrates strong performance across stationary and ELAA-MIMO channel models, including rank-$K$ extensions as future directions. The proposed SR-$1$R framework provides a practical path toward fast, accurate, and scalable channel inversion in next-generation MIMO systems.

Abstract

While iterative matrix inversion methods excel in computational efficiency, memory optimization, and support for parallel and distributed computing when managing large matrices, their limitations are also evident in multiple-input multiple-output (MIMO) fading channels. These methods encounter challenges related to slow convergence and diminished accuracy, especially in ill-conditioned scenarios, hindering their application in future MIMO networks such as extra-large aperture array. To address these challenges, this paper proposes a novel matrix regularization method termed symmetric rank-$1$ regularization (SR-$1$R). The proposed method functions by augmenting the channel matrix with a symmetric rank-$1$ matrix, with the primary goal of minimizing the condition number of the resultant regularized matrix. This significantly improves the matrix condition, enabling fast and accurate iterative inversion of the regularized matrix. Then, the inverse of the original channel matrix is obtained by applying the Sherman-Morrison transform on the outcome of iterative inversions. Our eigenvalue analysis unveils the best channel condition that can be achieved by an optimized SR-$1$R matrix. Moreover, a power iteration-assisted (PIA) approach is proposed to find the optimum SR-$1$R matrix without need of eigenvalue decomposition. The proposed approach exhibits logarithmic algorithm-depth in parallel computing for MIMO precoding. Finally, computer simulations demonstrate that SR-$1$R has the potential to reduce the required iteration by up to $35\%$ while achieving the performance of regularized zero-forcing.

Symmetric Rank-$1$ Regularization for Iterative Inversion of Ill-Conditioned MIMO Channels

TL;DR

This paper tackles the challenge of slow convergence in iterative matrix inversion for ill-conditioned MIMO channels by introducing symmetric rank- regularization (SR-R), which augments the Gram matrix with a rank- symmetric term to reduce its condition number. The authors derive theoretical bounds showing the best achievable conditioning can approach and develop a practical, eigen-structure–based optimization, complemented by a power iteration–assisted (PIA) approach that obviates full EVD/SVD. An enhanced version (e-PIA) leverages parallel candidate evaluations to accelerate convergence, achieving substantial reductions in iteration counts (up to in some cases) while maintaining compatibility with parallel computing and approaching regularized zero-forcing performance. The work also analyzes scalability, offering a complexity comparable to direct inversion with a logarithmic algorithm-depth, and demonstrates strong performance across stationary and ELAA-MIMO channel models, including rank- extensions as future directions. The proposed SR-R framework provides a practical path toward fast, accurate, and scalable channel inversion in next-generation MIMO systems.

Abstract

While iterative matrix inversion methods excel in computational efficiency, memory optimization, and support for parallel and distributed computing when managing large matrices, their limitations are also evident in multiple-input multiple-output (MIMO) fading channels. These methods encounter challenges related to slow convergence and diminished accuracy, especially in ill-conditioned scenarios, hindering their application in future MIMO networks such as extra-large aperture array. To address these challenges, this paper proposes a novel matrix regularization method termed symmetric rank- regularization (SR-R). The proposed method functions by augmenting the channel matrix with a symmetric rank- matrix, with the primary goal of minimizing the condition number of the resultant regularized matrix. This significantly improves the matrix condition, enabling fast and accurate iterative inversion of the regularized matrix. Then, the inverse of the original channel matrix is obtained by applying the Sherman-Morrison transform on the outcome of iterative inversions. Our eigenvalue analysis unveils the best channel condition that can be achieved by an optimized SR-R matrix. Moreover, a power iteration-assisted (PIA) approach is proposed to find the optimum SR-R matrix without need of eigenvalue decomposition. The proposed approach exhibits logarithmic algorithm-depth in parallel computing for MIMO precoding. Finally, computer simulations demonstrate that SR-R has the potential to reduce the required iteration by up to while achieving the performance of regularized zero-forcing.
Paper Structure (27 sections, 4 theorems, 69 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 27 sections, 4 theorems, 69 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Feasibility and optimality study of SR-$1$R
  3. Practical approaches for optimization
  4. Simulations and Performance Evaluation
  5. System Model, Prior Art, and Problem Analysis
  6. MIMO Linear Precoding
  7. Prior Art Analysis
  8. Iterative LSI
  9. Iterative matrix inversion
  10. Research Problem and Analysis
  11. SR-$1$R: Feasibility and Optimality Study
  12. Feasibility Study
  13. Optimality Study
  14. Power Iteration-Assisted Approach for The Implementation of SR-$1$R
  15. The Proposed PIA Approach
  16. ...and 12 more sections

Key Result

Lemma 1

Define the eigenvalues of the rank-$1$ modified symmetric matrix $\mathbf{\Theta}$ as $(\theta_0, \theta_1, ..., \theta_{N-1})$. When $\xi<0$, $\mathbf{\Theta}$ is a positive-definite matrix whose eigenvalues satisfies the following inequalities When $\xi>0$, the following inequalities are satisfied

Figures (6)

  • Figure 1: Block diagram of the proposed symmetric rank-$1$ regularization for iterative matrix inversion.
  • Figure 2: Results for Experiment 1: (a) CDF of $(\theta_{N-1}+\theta_{N-2})/(-\theta_{0}+\theta_{N-2})$ for the PIA approach; CDF of condition number compared to the preconditioning baselines in (b) i.i.d. Rayleigh channel, (c) i.i.d. Rician channel, (d) LoS-dominated ELAA, (e) mixed LoS/non-LoS ELAA.
  • Figure 3: Computation latency of the PIA/e-PIA approaches compared to the preconditioning baselines as $N$ increases from $16$ to $512$.
  • Figure 4: Average SER of the e-PIA ($i=46$) and the baselines ($i=50$) for the RZF and ZF w.r.t. $(\mathcal{E})/(\sigma_v^2)$ in mixed LoS/non-LoS ELAA.
  • Figure 5: Average SER of the PIA/e-PIA approaches as the iteration number increases compared to exact RZF performance in (a) i.i.d. Rayleigh channel, (b) i.i.d. Rician channel, (c) LoS-dominated ELAA, (d) mixed LoS/non-LoS ELAA.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Lemma 1: from Gu1994
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof