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An LiGME Regularizer of Designated Isolated Minimizers -- An Application to Discrete-Valued Signal Estimation

Satoshi Shoji, Wataru Yata, Keita Kume, Isao Yamada

TL;DR

This work tackles discrete-valued signal estimation from noisy linear measurements by introducing the LiGME regularizer, a nonconvex Generalized Moreau Enhanced refinement of the convex SOAV prior that enforces designated isolated minimizers while preserving overall convexity. The authors formulate a cLiGME-based model and prove that, with a suitable choice of tuning matrices, the cost remains convex and a global minimizer can be reached via Krasnosel’skiĭ–Mann iterations; they further offer iterative reweighting and generalized superiorization to boost practical accuracy. The approach is validated in MIMO detection scenarios, where cLiGME outperforms SOAV, and the enhancements IW and GS provide additional BER reductions, highlighting the method’s potential for reliable discrete-valued estimation in communications. Overall, the paper contributes a principled, tractable framework for contrastive nonconvex regularization that enforces discrete alphabets while guaranteeing convergence under convexity constraints, with tangible gains in signal-detection performance.

Abstract

For a regularized least squares estimation of discrete-valued signals, we propose a Linearly involved Generalized Moreau Enhanced (LiGME) regularizer, as a nonconvex regularizer, of designated isolated minimizers. The proposed regularizer is designed as a Generalized Moreau Enhancement (GME) of the so-called sum-of-absolute-values (SOAV) convex regularizer. Every candidate vector in the discrete-valued set is aimed to be assigned to an isolated local minimizer of the proposed regularizer while the overall convexity of the regularized least squares model is maintained. Moreover, a global minimizer of the proposed model can be approximated iteratively by using a variant of the constrained LiGME (cLiGME) algorithm. To enhance the accuracy of the proposed estimation, we also propose a pair of simple modifications, called respectively an iterative reweighting and a generalized superiorization. Numerical experiments demonstrate the effectiveness of the proposed model and algorithms in a scenario of multiple-input multiple-output (MIMO) signal detection.

An LiGME Regularizer of Designated Isolated Minimizers -- An Application to Discrete-Valued Signal Estimation

TL;DR

This work tackles discrete-valued signal estimation from noisy linear measurements by introducing the LiGME regularizer, a nonconvex Generalized Moreau Enhanced refinement of the convex SOAV prior that enforces designated isolated minimizers while preserving overall convexity. The authors formulate a cLiGME-based model and prove that, with a suitable choice of tuning matrices, the cost remains convex and a global minimizer can be reached via Krasnosel’skiĭ–Mann iterations; they further offer iterative reweighting and generalized superiorization to boost practical accuracy. The approach is validated in MIMO detection scenarios, where cLiGME outperforms SOAV, and the enhancements IW and GS provide additional BER reductions, highlighting the method’s potential for reliable discrete-valued estimation in communications. Overall, the paper contributes a principled, tractable framework for contrastive nonconvex regularization that enforces discrete alphabets while guaranteeing convergence under convexity constraints, with tangible gains in signal-detection performance.

Abstract

For a regularized least squares estimation of discrete-valued signals, we propose a Linearly involved Generalized Moreau Enhanced (LiGME) regularizer, as a nonconvex regularizer, of designated isolated minimizers. The proposed regularizer is designed as a Generalized Moreau Enhancement (GME) of the so-called sum-of-absolute-values (SOAV) convex regularizer. Every candidate vector in the discrete-valued set is aimed to be assigned to an isolated local minimizer of the proposed regularizer while the overall convexity of the regularized least squares model is maintained. Moreover, a global minimizer of the proposed model can be approximated iteratively by using a variant of the constrained LiGME (cLiGME) algorithm. To enhance the accuracy of the proposed estimation, we also propose a pair of simple modifications, called respectively an iterative reweighting and a generalized superiorization. Numerical experiments demonstrate the effectiveness of the proposed model and algorithms in a scenario of multiple-input multiple-output (MIMO) signal detection.

Paper Structure

This paper contains 18 sections, 6 theorems, 47 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Selected elements of convex analysis and characterization of local minimizers
  4. Brief Introduction to cLiGME model
  5. Proposed LiGME regularizer with designated isolated local minimizers
  6. Proposed LiGME regularizer
  7. Proposed model and algorithm for discrete-valued signal estimation
  8. Numerical experiments
  9. An application of the proposed model to MIMO signal detection problem
  10. Experimental setting and results on MIMO detection
  11. Conclusion
  12. Known facts
  13. Proof of Lemma \ref{['lemma:separable']}
  14. Proof for Lemma \ref{['lem:relative_interior']}
  15. Proof of Proposition \ref{['prop:cLiGME_algorithm']}
  16. ...and 3 more sections

Key Result

Lemma 1

Let $f:\mathcal{H}\coloneqq \mathbb{R}^{N} \to (-\infty,\infty]$ be a proper function. Assume that $f$ is the sum of univariate proper functions $f_{n}:\mathbb{R} \to (-\infty,\infty]\ (n=1,2,\ldots,N)$, i.e., Then, the following hold:

Figures (12)

  • Figure 1: $\Theta_{\text{SOAV}}$
  • Figure 2: $\Theta_{\text{LiGME}}$ with $\mathbf{B}^{\left\langle{l}\right\rangle}=(1/\sqrt{L})\mathbf{I}_{50}$
  • Figure 4: Overall view
  • Figure 5: Enlarged view around $a_1$
  • Figure 7: 4-QAM ($N=50$, $M=35$)
  • ...and 7 more figures

Theorems & Definitions (21)

  • Definition 1: Characterizations of minimizers nocedal_numerical_2006
  • Lemma 1: Equivalent conditions of isolated local minimizers
  • proof
  • Definition 2: Tools of convex analysis
  • Example 1: Expression of proximity operators
  • Lemma 2: A relaxation of abe_linearly_2020 without even symmetric condition of $\Psi$
  • proof
  • Proposition 1: A cLiGME algorithm without even symmetric condition of $\Psi$ in abe_linearly_2020yata_constrained_2022yata_imposing_2024kitahara_multi-contrast_2021 for \ref{['eq:cLiGME_model']} in Problem \ref{['prob:cLiGME_model']}
  • proof
  • Theorem 1: Isolated local minimizers at designated points
  • ...and 11 more