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On the Optimal Integer-Forcing Precoding: A Geometric Perspective and a Polynomial-Time Algorithm

Junren Qin, Fan Jiang, Tao Yang, Shanxiang Lyu, Rongke Liu, Shi Jin

TL;DR

The main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of $\mathcal{O}\left(K^4\log K\log_2(r_0)$, which is polynomial in the number of users $K$.

Abstract

The joint optimization of the integer matrix $\mathbf{A}$ and the power scaling matrix $\mathbf{D}$ is central to achieving the capacity-approaching performance of Integer-Forcing (IF) precoding. This problem, however, is known to be NP-hard, presenting a fundamental computational bottleneck. In this paper, we reveal that the solution space of this problem admits a intrinsic geometric structure: it can be partitioned into a finite number of conical regions, each associated with a distinct full-rank integer matrix $\mathbf{A}$. Leveraging this decomposition, we transform the NP-hard problem into a search over these regions and propose the Multi-Cone Nested Stochastic Pattern Search (MCN-SPS) algorithm. Our main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of $\mathcal{O}\left(K^4\log K\log_2(r_0)\right)$, which is polynomial in the number of users $K$. Numerical simulations corroborate the theoretical analysis and demonstrate the algorithm's efficacy.

On the Optimal Integer-Forcing Precoding: A Geometric Perspective and a Polynomial-Time Algorithm

TL;DR

The main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of , which is polynomial in the number of users .

Abstract

The joint optimization of the integer matrix and the power scaling matrix is central to achieving the capacity-approaching performance of Integer-Forcing (IF) precoding. This problem, however, is known to be NP-hard, presenting a fundamental computational bottleneck. In this paper, we reveal that the solution space of this problem admits a intrinsic geometric structure: it can be partitioned into a finite number of conical regions, each associated with a distinct full-rank integer matrix . Leveraging this decomposition, we transform the NP-hard problem into a search over these regions and propose the Multi-Cone Nested Stochastic Pattern Search (MCN-SPS) algorithm. Our main theoretical result is that MCN-SPS finds a near-optimal solution with a computational complexity of , which is polynomial in the number of users . Numerical simulations corroborate the theoretical analysis and demonstrate the algorithm's efficacy.
Paper Structure (29 sections, 13 theorems, 153 equations, 8 figures, 2 tables, 4 algorithms)

This paper contains 29 sections, 13 theorems, 153 equations, 8 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Model
  4. Integer-Forcing (IF) Precoding
  5. Transmitter
  6. Receivers
  7. Joint Optimization
  8. Lattice Basis Reduction
  9. The Contraction Mapping Approach to Perron-Frobenius Theory
  10. Geometric Structure of the Solution Space
  11. General Model of Joint Optimization
  12. Model on Fixed $\mathbf{A}$
  13. Optimal IF Precoder Design
  14. Optimization for Fixed $\mathbf{A}$
  15. Alternating Optimization for $(\mathbf{A},\mathbf{D})$
  16. ...and 14 more sections

Key Result

Theorem 1

Considering a non-singular generator matrix $\mathbf{G}_{A}$ of lattice $\Lambda_A$ and a positive diagonal matrix $\mathbf{D}_{A}$ with constraint $\det(\mathbf{D}_{A})=1$, the product $\mathbf{G}_{B}=\mathbf{G}_{A}\mathbf{D}_{A}$ is the generator matrix of lattice $\Lambda_B$, and the solution set and so we have

Figures (8)

  • Figure 1: Block diagram of the IF precoding architecture he2018uplinkqiu2024lattice
  • Figure 2: $3$-dimensional sample for generalized optimization model in fixed $\mathbf{A}$
  • Figure 3: $3$-dimensional sample for the convergence of Eq. (\ref{['Eq. iterative function']}) versus SNR
  • Figure 4: Flowchart of the MCN-SPS method
  • Figure 5: Trajectory in iterations for the sample $\mathbf{H}$ in: (a) SNR$=0$dB, (b) SNR$=5$dB, and (c) SNR$=10$dB. The trend beside trajectory depicts the Hilbert metric changed in iterations.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Definition 1: SIVP nguyen2010lll
  • Definition 2: Successive Minima nguyen2010lll
  • Definition 3: LLL reduced Lyu2017boost
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • ...and 23 more