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Optimal Low-Dimensional Structures of ISAC Beamforming: Theory and Efficient Algorithms

Xiaotong Zhao, Mian Li, Ya-Feng Liu, Qingjiang Shi, Anthony Man-Cho So

TL;DR

The paper addresses CRB-based ISAC beamforming for massive MIMO by proving that the optimal solution lies in a low-dimensional subspace tied to the number of users, not the number of antennas. This insight enables an equivalent reformulation with reduced dimension and paves the way for a Reduced IPM (R-IPM) giving complexity $\mathcal{O}(K^{10})$, plus a Balanced Augmented Lagrangian (BAL) based solver (R-BAL) with $\mathcal{O}(K^4)$ per-iteration cost. The authors also show how to extract rank-one beamformers from the reduced problem, preserving optimality. Extensive simulations demonstrate that the low-dimensional approach yields the same global optimum as the original high-dimensional formulation while delivering massive speedups (up to $10^4\times$) in massive MIMO regimes, with R-BAL offering the best practical performance and scalability.

Abstract

Transmit beamforming design is a fundamental problem in integrated sensing and communication (ISAC) systems. Numerous methods have been proposed to jointly optimize key performance metrics such as the signal-to-interference-plus-noise ratio and Cramér-Rao bound. However, the computational complexity of these methods often grows rapidly with the number of transmit antennas at the base station (BS). To tackle this challenge, we prove a fundamental structural property of the ISAC beamforming problem, i.e., there exists an optimal solution exhibiting a low-dimensional structure. This leads to an equivalent reformulation of the problem with dimension related to the number of users rather than the number of BS antennas, thereby enabling the development of low-complexity algorithms. When applying the interior-point method to the reformulated problem, we achieve up to six orders of magnitude in complexity reduction when the number of antennas exceeds the number of users by an order of magnitude. To further reduce the complexity, we develop a balanced augmented Lagrangian method to solve the reformulated problem. The proposed algorithm maintains optimality while achieving a computational complexity that scales quartically with the number of users. Our simulation results demonstrate that the proposed R-BAL method can achieve a speedup of more than 10000$\times$ over the conventional IPM in massive MIMO scenarios.

Optimal Low-Dimensional Structures of ISAC Beamforming: Theory and Efficient Algorithms

TL;DR

The paper addresses CRB-based ISAC beamforming for massive MIMO by proving that the optimal solution lies in a low-dimensional subspace tied to the number of users, not the number of antennas. This insight enables an equivalent reformulation with reduced dimension and paves the way for a Reduced IPM (R-IPM) giving complexity , plus a Balanced Augmented Lagrangian (BAL) based solver (R-BAL) with per-iteration cost. The authors also show how to extract rank-one beamformers from the reduced problem, preserving optimality. Extensive simulations demonstrate that the low-dimensional approach yields the same global optimum as the original high-dimensional formulation while delivering massive speedups (up to ) in massive MIMO regimes, with R-BAL offering the best practical performance and scalability.

Abstract

Transmit beamforming design is a fundamental problem in integrated sensing and communication (ISAC) systems. Numerous methods have been proposed to jointly optimize key performance metrics such as the signal-to-interference-plus-noise ratio and Cramér-Rao bound. However, the computational complexity of these methods often grows rapidly with the number of transmit antennas at the base station (BS). To tackle this challenge, we prove a fundamental structural property of the ISAC beamforming problem, i.e., there exists an optimal solution exhibiting a low-dimensional structure. This leads to an equivalent reformulation of the problem with dimension related to the number of users rather than the number of BS antennas, thereby enabling the development of low-complexity algorithms. When applying the interior-point method to the reformulated problem, we achieve up to six orders of magnitude in complexity reduction when the number of antennas exceeds the number of users by an order of magnitude. To further reduce the complexity, we develop a balanced augmented Lagrangian method to solve the reformulated problem. The proposed algorithm maintains optimality while achieving a computational complexity that scales quartically with the number of users. Our simulation results demonstrate that the proposed R-BAL method can achieve a speedup of more than 10000 over the conventional IPM in massive MIMO scenarios.
Paper Structure (34 sections, 10 theorems, 86 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 34 sections, 10 theorems, 86 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Related Work
  4. Motivation and Contributions
  5. System Model and Problem Formulation
  6. System Model
  7. Problem Formulation
  8. Communication metric
  9. Sensing metric
  10. Optimal Low-Dimensional Beamforming Structure
  11. Feasibility Analysis
  12. Optimal Solution Structure Analysis
  13. Rank-One Optimal Solution via IPM
  14. Balanced Augmented Lagrangian-Based Low-Complexity Algorithm
  15. Brief Introduction of BAL Method and Challenges
  16. ...and 19 more sections

Key Result

Proposition 1

Problem eq_extended_mu_isac_problem_sdr (or equivalently, problem eq_extended_mu_isac_problem) is infeasible if and only if $P_T< P_{\operatorname{low}}$, where $P_{\operatorname{low}}$ is given by with each $\lambda_k$ computable via the fixed-point system Here, $\mathbf{H}=[\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{K}]$ and $\accentset{{\cc@style\underline{ }}}{\mathbf{h}}_i= \mathbf{H}

Figures (3)

  • Figure 1: CRB performance under two scenarios. (a) $N_t=64$ and varying $K$; (b) $K=8$ and varying $N_t$.
  • Figure 2: Optimality gap versus wall-clock time under different $N_t$, with $K=8$ and $P_T=15$ dBm.
  • Figure 3: Constraint violation versus wall-clock time under different $N_t$, with $K=8$ and $P_T=15$ dBm.

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 3
  • Theorem 3
  • Theorem 4
  • Proposition 2
  • ...and 3 more