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Success probability of the $L_0$-regularized box-constrained Babai point and column permutation strategies

Xiao-Wen Chang, Yingzi XU

TL;DR

The paper derives exact formulas for the success probability of the $L_0$-regularized box-constrained Babai point in a linear model with sparse priors, and establishes that the optimal regularization parameter $\lambda^*$ maximizes SP relative to the unregularized case. It proves monotonicity properties of the SP with respect to QR-diagonal elements and provides bounds that depend on spectral-like quantities ${\mu_1},{\mu_2}$. The authors then analyze LLL-P and present three SP-based column permutation strategies (LSP, GSP, MSP) to enhance detector performance, with theoretical guarantees and extensive numerical validation. The work demonstrates that regularization and carefully designed permutations can yield substantial SP gains, especially for sparse signals or ill-conditioned matrices, offering practical guidance for detectors in MIMO-like settings. Overall, the contributions advance understanding of suboptimal solvers for $L_0$-RBILS and provide actionable algorithms to improve detection performance.

Abstract

We consider the success probability of the $L_0$-regularized box-constrained Babai point, which is a suboptimal solution to the $L_0$-regularized box-constrained integer least squares problem and can be used for MIMO detection. First, we derive formulas for the success probability of both $L_0$-regularized and unregularized box-constrained Babai points. Then we investigate the properties of the $L_0$-regularized box-constrained Babai point, including the optimality of the regularization parameter, the monotonicity of its success probability, and the monotonicity of the ratio of the two success probabilities. A bound on the success probability of the $L_0$-regularized Babai point is derived. After that, we analyze the effect of the LLL-P permutation strategy on the success probability of the $L_0$-regularized Babai point. Then we propose some success probability based column permutation strategies to increase the success probability of the $L_0$-regularized box-constrained Babai point. Finally, we present numerical tests to confirm our theoretical results and to show the advantage of the $L_0$ regularization and the effectiveness of the proposed column permutation algorithms compared to existing strategies.

Success probability of the $L_0$-regularized box-constrained Babai point and column permutation strategies

TL;DR

The paper derives exact formulas for the success probability of the -regularized box-constrained Babai point in a linear model with sparse priors, and establishes that the optimal regularization parameter maximizes SP relative to the unregularized case. It proves monotonicity properties of the SP with respect to QR-diagonal elements and provides bounds that depend on spectral-like quantities . The authors then analyze LLL-P and present three SP-based column permutation strategies (LSP, GSP, MSP) to enhance detector performance, with theoretical guarantees and extensive numerical validation. The work demonstrates that regularization and carefully designed permutations can yield substantial SP gains, especially for sparse signals or ill-conditioned matrices, offering practical guidance for detectors in MIMO-like settings. Overall, the contributions advance understanding of suboptimal solvers for -RBILS and provide actionable algorithms to improve detection performance.

Abstract

We consider the success probability of the -regularized box-constrained Babai point, which is a suboptimal solution to the -regularized box-constrained integer least squares problem and can be used for MIMO detection. First, we derive formulas for the success probability of both -regularized and unregularized box-constrained Babai points. Then we investigate the properties of the -regularized box-constrained Babai point, including the optimality of the regularization parameter, the monotonicity of its success probability, and the monotonicity of the ratio of the two success probabilities. A bound on the success probability of the -regularized Babai point is derived. After that, we analyze the effect of the LLL-P permutation strategy on the success probability of the -regularized Babai point. Then we propose some success probability based column permutation strategies to increase the success probability of the -regularized box-constrained Babai point. Finally, we present numerical tests to confirm our theoretical results and to show the advantage of the regularization and the effectiveness of the proposed column permutation algorithms compared to existing strategies.
Paper Structure (24 sections, 14 theorems, 169 equations, 1 figure, 10 tables, 4 algorithms)

This paper contains 24 sections, 14 theorems, 169 equations, 1 figure, 10 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Babai points
  4. Column permutation strategies
  5. SP of the $L_0$-regularized Babai point ${\mathbf{x}}^{ \mathrm{RB}}$
  6. Some properties of the $L_0$-regularized Babai point.
  7. Some basic properties of $\rho_k^{ \mathrm{RB}}$
  8. The superiority of ${\mathbf{x}}^{ \mathrm{RB}}$
  9. Monotonicity properties
  10. A bound on the SP of the $L_0$-regularized Babai point
  11. Effects of LLL-P on SP of ${\mathbf{x}}^{ \mathrm{RB}}$
  12. Success probability based column permutation strategies
  13. A local success probability based strategy LSP
  14. A global success probability based strategy GSP
  15. A mixed success probability based strategy MSP
  16. ...and 9 more sections

Key Result

Lemma 1

Suppose that $\lambda\geq 0$ and $M\geq 2$. Let $\alpha_k^{(j)}$ be defined by eq:alphakj. Then and $\alpha_k^{(j)}$ is strictly decreasing when $j$ is increasing from 1 to $M$. Let If $j_k \geq 2$, then and if $j_k \leq M-1$, then

Figures (1)

  • Figure 1: Theoretical and experimental success probability of the Babai points

Theorems & Definitions (46)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof
  • Remark 4
  • ...and 36 more