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A Double Maximization Approach for Optimizing the LM Rate of Mismatched Decoding

Lingyi Chen, Shitong Wu, Xinwei Li, Huihui Wu, Hao Wu, Wenyi Zhang

TL;DR

A novel dual form of the LM rate is proposed, thereby transforming the max-min formulation into an equivalent double maximization formulation, and an alternating maximization algorithm is established to solve the resultant maximization problem setup.

Abstract

An approach is established for maximizing the Lower bound on the Mismatch capacity (hereafter abbreviated as LM rate), a key performance bound in mismatched decoding, by optimizing the channel input probability distribution. Under a fixed channel input probability distribution, the computation of the corresponding LM rate is a convex optimization problem. When optimizing the channel input probability distribution, however, the corresponding optimization problem adopts a max-min formulation, which is generally non-convex and is intractable with standard approaches. To solve this problem, a novel dual form of the LM rate is proposed, thereby transforming the max-min formulation into an equivalent double maximization formulation. This new formulation leads to a maximization problem setup wherein each individual optimization direction is convex. Consequently, an alternating maximization algorithm is established to solve the resultant maximization problem setup. Each step of the algorithm only involves a closed-form iteration, which is efficiently implemented with standard optimization procedures. Numerical experiments show the proposed approach for optimizing the LM rate leads to noticeable rate gains.

A Double Maximization Approach for Optimizing the LM Rate of Mismatched Decoding

TL;DR

A novel dual form of the LM rate is proposed, thereby transforming the max-min formulation into an equivalent double maximization formulation, and an alternating maximization algorithm is established to solve the resultant maximization problem setup.

Abstract

An approach is established for maximizing the Lower bound on the Mismatch capacity (hereafter abbreviated as LM rate), a key performance bound in mismatched decoding, by optimizing the channel input probability distribution. Under a fixed channel input probability distribution, the computation of the corresponding LM rate is a convex optimization problem. When optimizing the channel input probability distribution, however, the corresponding optimization problem adopts a max-min formulation, which is generally non-convex and is intractable with standard approaches. To solve this problem, a novel dual form of the LM rate is proposed, thereby transforming the max-min formulation into an equivalent double maximization formulation. This new formulation leads to a maximization problem setup wherein each individual optimization direction is convex. Consequently, an alternating maximization algorithm is established to solve the resultant maximization problem setup. Each step of the algorithm only involves a closed-form iteration, which is efficiently implemented with standard optimization procedures. Numerical experiments show the proposed approach for optimizing the LM rate leads to noticeable rate gains.
Paper Structure (14 sections, 4 theorems, 43 equations, 2 figures, 1 algorithm)

This paper contains 14 sections, 4 theorems, 43 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Double Maximization Model and Alternating Maximization Algorithm
  4. Double Maximization Model
  5. Alternating Maximization Algorithm
  6. Fix $\boldsymbol{T}(\boldsymbol{\phi}, \widetilde{\boldsymbol{\psi}}, \zeta)$ and update $\boldsymbol{p}$
  7. Fix $\boldsymbol{p}$ and update $\boldsymbol{\phi}, \widetilde{\boldsymbol{\psi}}, \zeta$
  8. Numerical Results
  9. Convergence behavior
  10. Comparison with the LM Rate
  11. Conclusion
  12. Proof of Proposition \ref{['prop:equi']}
  13. Derivation Details of \ref{['update_p']} and $F(\lambda)$
  14. Derivation Details of \ref{['update_dual']} and $G(\zeta)$

Key Result

Theorem 1

For a fixed $Q_X\in\mathcal{P}(\mathcal{X})$, a dual form of the LM rate problem LM_def can be written as: where $\boldsymbol{\phi}\in\mathbb{R}^{M}, \boldsymbol{\psi}\in\mathbb{R}^{N}$ and $\zeta\in\mathbb{R}^{+}$.

Figures (2)

  • Figure 1: The convergent trajectories of the residual errors $r_{\phi}$ (Red), $r_{{\psi}}$ (Blue), $r_{\zeta}$ (Green), and $r_{\lambda}$ (Purple). Upper Left: The QPSK modulation scheme. Upper Right: The 16QAM modulation scheme. Lower Left: The 64QAM modulation scheme. Lower Right: The 256QAM modulation scheme.
  • Figure 2: $C_\mathrm{LM}$ (solid) and $I_\mathrm{LM}$ (dashed) versus SNR under different mismatched cases. Upper Left: The QPSK modulation scheme. Upper Right: The 16QAM modulation scheme. Lower Left: The 64QAM modulation scheme. Lower Right: The 256QAM modulation scheme.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • proof