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Near-Optimal MIMO Detection Using Gradient-Based MCMC in Discrete Spaces

Xingyu Zhou, Le Liang, Jing Zhang, Chao-Kai Wen, Shi Jin

TL;DR

This work tackles near-optimal MIMO detection in high-dimensional, discrete symbol spaces by introducing DMALA, a gradient-based MCMC sampler tailored for discrete spaces. DMALA leverages gradients from the continuous relaxation to accelerate sampling, while preserving exact Metropolis–Hastings corrections to guarantee convergence to the target posterior distribution; a preconditioned variant further improves convergence rates. The detector uses converged samples to compute posterior (soft) decisions via an IS-based LLR method, offering theoretical guarantees through the law of large numbers and proven convergence. Empirical results demonstrate near-optimal BER performance with robust behavior across diverse channel models, including imperfect CSI and 3GPP massive MIMO channels, and highlight scalable parallelization that makes DMALA practical for next-generation wireless systems.

Abstract

The discrete nature of transmitted symbols poses challenges for achieving optimal detection in multiple-input multiple-output (MIMO) systems associated with a large number of antennas. Recently, the combination of two powerful machine learning methods, Markov chain Monte Carlo (MCMC) sampling and gradient descent, has emerged as a highly efficient solution to address this issue. However, existing gradient-based MCMC detectors are heuristically designed and thus are theoretically untenable. To bridge this gap, we introduce a novel sampling algorithm tailored for discrete spaces. This algorithm leverages gradients from the underlying continuous spaces for acceleration while maintaining the validity of probabilistic sampling. We prove the convergence of this method and also analyze its convergence rate using both MCMC theory and empirical diagnostics. On this basis, we develop a MIMO detector that precisely samples from the target discrete distribution and generates posterior Bayesian estimates using these samples, whose performance is thereby theoretically guaranteed. Furthermore, our proposed detector is highly parallelizable and scalable to large MIMO dimensions, positioning it as a compelling candidate for next-generation wireless networks. Simulation results show that our detector achieves near-optimal performance, significantly outperforms state-of-the-art baselines, and showcases resilience to various system setups.

Near-Optimal MIMO Detection Using Gradient-Based MCMC in Discrete Spaces

TL;DR

This work tackles near-optimal MIMO detection in high-dimensional, discrete symbol spaces by introducing DMALA, a gradient-based MCMC sampler tailored for discrete spaces. DMALA leverages gradients from the continuous relaxation to accelerate sampling, while preserving exact Metropolis–Hastings corrections to guarantee convergence to the target posterior distribution; a preconditioned variant further improves convergence rates. The detector uses converged samples to compute posterior (soft) decisions via an IS-based LLR method, offering theoretical guarantees through the law of large numbers and proven convergence. Empirical results demonstrate near-optimal BER performance with robust behavior across diverse channel models, including imperfect CSI and 3GPP massive MIMO channels, and highlight scalable parallelization that makes DMALA practical for next-generation wireless systems.

Abstract

The discrete nature of transmitted symbols poses challenges for achieving optimal detection in multiple-input multiple-output (MIMO) systems associated with a large number of antennas. Recently, the combination of two powerful machine learning methods, Markov chain Monte Carlo (MCMC) sampling and gradient descent, has emerged as a highly efficient solution to address this issue. However, existing gradient-based MCMC detectors are heuristically designed and thus are theoretically untenable. To bridge this gap, we introduce a novel sampling algorithm tailored for discrete spaces. This algorithm leverages gradients from the underlying continuous spaces for acceleration while maintaining the validity of probabilistic sampling. We prove the convergence of this method and also analyze its convergence rate using both MCMC theory and empirical diagnostics. On this basis, we develop a MIMO detector that precisely samples from the target discrete distribution and generates posterior Bayesian estimates using these samples, whose performance is thereby theoretically guaranteed. Furthermore, our proposed detector is highly parallelizable and scalable to large MIMO dimensions, positioning it as a compelling candidate for next-generation wireless networks. Simulation results show that our detector achieves near-optimal performance, significantly outperforms state-of-the-art baselines, and showcases resilience to various system setups.
Paper Structure (23 sections, 4 theorems, 54 equations, 12 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 4 theorems, 54 equations, 12 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. System Model
  4. Basics of MCMC
  5. DMALA-Based MIMO Detection
  6. Gradient-Based MCMC Sampling in Discrete Spaces
  7. Convergence of the DMALA Sampler
  8. LLR Computation Methods
  9. Computational Complexity
  10. Sampling Complexity
  11. LLR Computation Complexity
  12. Simulation Results
  13. Simulation Setups
  14. Small- and Medium-Sized MIMO
  15. Large MIMO
  16. ...and 8 more sections

Key Result

Lemma 1

Assume that the probability density function of the noise $\mathbf{n}$ satisfies $0< p_{\bf n}(\cdot)<+\infty$, i.e., $\sigma^2 > 0$. Then, the transition kernel of the Markov chain induced by DMALA ($\alpha>0$) is irreducible and aperiodic. Moreover, the target posterior distribution $\pi$ is a uni

Figures (12)

  • Figure 1: Visual comparison between existing gradient-based MCMC methods gowdaMetropolisHastingsRandomWalk2021zhou2023gradient and the proposed DMALA for the 16-QAM lattice. The black cross represents the current state of the Markov chain. In (a), the candidate sample is generated via a sequential process of gradient descent, random walk, and QAM mapping. In (b), the left subplot shows the underlying continuous distribution of the target discrete distribution, with colors indicating probability levels. Gradients from the logarithm of this continuous distribution are employed to construct the proposal distribution, as shown by color dots in the right subplot.
  • Figure 2: Total variation distance as a function of the number of sampling iterations within a ${\text{2}\times\text{2}}$ MIMO system featuring Rayleigh fading channels, QPSK modulation, and ${\text{SNR} = \text{8}\;\text{dB}}$.
  • Figure 3: Comparison between the sampled and true posterior distributions within the ${\text{2}\times\text{2}}$ MIMO-QPSK space, which encompasses 16 possible states. States exhibiting a probability lower than $10^{-3}$ are omitted for clarity.
  • Figure 4: Boxplots illustrating the convergence rates in a ${\text{2}\times\text{2}}$ MIMO system employing QPSK modulation under Rayleigh fading channels. The convergence rates for naive and preconditioned DMALA configurations are depicted in panels (a) and (b), respectively.
  • Figure 5: BER performance of DMALA across varying numbers of sampling iterations within a ${\text{4}\times\text{4}}$ MIMO system employing 16-QAM modulation under Rayleigh fading channels. Both MHGD and DMALA are configured with $N_{\rm p}=128$ parallel samplers.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Remark 3
  • Remark 4