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Accelerated and Deep Expectation Maximization for One-Bit MIMO-OFDM Detection

Mingjie Shao, Wing-Kin Ma, Junbin Liu, Zihao Huang

TL;DR

This work analyzes expectation maximization for one-bit MIMO-OFDM detection (OMOD), establishing a convergence rate of at least ${1/k}$ and revealing that EM can slow at high SNR due to problem conditioning. By leveraging a proximal-gradient perspective, the authors derive accelerated EM with a ${1/k^2}$ rate and an inexact EM variant with convergence guarantees, then instantiate these ideas as AIEM and DIEM for efficient, scalable OMOD inference. The paper also demonstrates deep unfolding of the inexact EM framework to yield a structured neural detector that maintains model-based guarantees while delivering competitive performance in detection and runtime. Collectively, these contributions provide a family of fast, scalable EM-based detectors for CQ MIMO-OFDM with ONE-BIT ADCs, supported by theory and extensive simulations.

Abstract

In this paper we study the expectation maximization (EM) technique for one-bit MIMO-OFDM detection (OMOD). Arising from the recent interest in massive MIMO with one-bit analog-to-digital converters, OMOD is a massive-scale problem. EM is an iterative method that can exploit the OFDM structure to process the problem in a per-iteration efficient fashion. In this study we analyze the convergence rate of EM for a class of approximate maximum-likelihood OMOD formulations, or, in a broader sense, a class of problems involving regression from quantized data. We show how the SNR and channel conditions can have an impact on the convergence rate. We do so by making a connection between the EM and the proximal gradient methods in the context of OMOD. This connection also gives us insight to build new accelerated and/or inexact EM schemes. The accelerated scheme has faster convergence in theory, and the inexact scheme provides us with the flexibility to implement EM more efficiently, with convergence guarantee. Furthermore we develop a deep EM algorithm, wherein we take the structure of our inexact EM algorithm and apply deep unfolding to train an efficient structured deep net. Simulation results show that our accelerated exact/inexact EM algorithms run much faster than their standard EM counterparts, and that the deep EM algorithm gives promising detection and runtime performances.

Accelerated and Deep Expectation Maximization for One-Bit MIMO-OFDM Detection

TL;DR

This work analyzes expectation maximization for one-bit MIMO-OFDM detection (OMOD), establishing a convergence rate of at least and revealing that EM can slow at high SNR due to problem conditioning. By leveraging a proximal-gradient perspective, the authors derive accelerated EM with a rate and an inexact EM variant with convergence guarantees, then instantiate these ideas as AIEM and DIEM for efficient, scalable OMOD inference. The paper also demonstrates deep unfolding of the inexact EM framework to yield a structured neural detector that maintains model-based guarantees while delivering competitive performance in detection and runtime. Collectively, these contributions provide a family of fast, scalable EM-based detectors for CQ MIMO-OFDM with ONE-BIT ADCs, supported by theory and extensive simulations.

Abstract

In this paper we study the expectation maximization (EM) technique for one-bit MIMO-OFDM detection (OMOD). Arising from the recent interest in massive MIMO with one-bit analog-to-digital converters, OMOD is a massive-scale problem. EM is an iterative method that can exploit the OFDM structure to process the problem in a per-iteration efficient fashion. In this study we analyze the convergence rate of EM for a class of approximate maximum-likelihood OMOD formulations, or, in a broader sense, a class of problems involving regression from quantized data. We show how the SNR and channel conditions can have an impact on the convergence rate. We do so by making a connection between the EM and the proximal gradient methods in the context of OMOD. This connection also gives us insight to build new accelerated and/or inexact EM schemes. The accelerated scheme has faster convergence in theory, and the inexact scheme provides us with the flexibility to implement EM more efficiently, with convergence guarantee. Furthermore we develop a deep EM algorithm, wherein we take the structure of our inexact EM algorithm and apply deep unfolding to train an efficient structured deep net. Simulation results show that our accelerated exact/inexact EM algorithms run much faster than their standard EM counterparts, and that the deep EM algorithm gives promising detection and runtime performances.
Paper Structure (52 sections, 13 theorems, 164 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 52 sections, 13 theorems, 164 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Problem of Interest: CQ MIMO-OFDM Detection
  3. This Work
  4. Related Works
  5. Organization and Notations
  6. A Review of EM for Quantized Regression
  7. A Review of the MIMO-OFDM Background
  8. MIMO-OFDM Model
  9. Subcarrier Decoupling in Unquantized MIMO-OFDM
  10. OMOD Problem Statement and Prior Study
  11. One-Bit MIMO-OFDM Detection (OMOD)
  12. Problem Statement
  13. Existing Method: Proximal Gradient
  14. Existing Method: Expectation Maximization
  15. Convergence Analysis and New Insights
  16. ...and 37 more sections

Key Result

Proposition 1

The function $f$ in eq:ana_P has $L_f$-Lipschitz continuous gradient on $\mathbb{R}^n$, where $L_f = \sigma_{\rm max}({\bm B})^2$. Consequently, the PG method eq:ana_PG with step size $\eta= 1/\sigma_{\rm max}({\bm B})^{2}$ has the following convergence result for problem eq:ana_P:

Figures (8)

  • Figure 1: BERs of the PG and EM algorithms. $16$-QAM. Dashed: GMAP-based algorithms; solid: box-based algorithms.
  • Figure 2: Average number of iterations of the PG and EM algorithms. 16-QAM. Solid lines: $(M,N,W) = (256,12,256)$, dashed lines: $(M,N,W) = (128,10,256)$.
  • Figure 3: Average runtimes of the PG and EM algorithms. 16-QAM. Solid lines: $(M,N,W) = (256,12,256)$, dashed lines: $(M,N,W) = (128,10,256)$.
  • Figure 4: BER performance of DIEM; 16-QAM.
  • Figure 5: BER performance of DIEM; 64-QAM.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 3 more