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

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.

💡 Research Summary

This paper tackles the computational bottleneck of transmit beamforming design in integrated sensing and communication (ISAC) systems, where conventional methods scale poorly with the number of base‑station antennas (N_t). The authors first prove a fundamental structural property: an optimal solution always lies in a low‑dimensional subspace spanned by the user channel matrix (H). Consequently, each beamforming matrix (W_k) can be expressed as (W_k = H V_k H^{\mathrm H}) with (V_k) of size (K\times K), and the auxiliary beamforming matrix occupies the null space of (H). This insight reduces the original (N_t\times N_t) problem to an equivalent formulation involving only (K\times K) variables (equation 6).

Two solution approaches are developed. The first applies a standard interior‑point method (IPM) to the reduced‑dimension problem, cutting the complexity from (O(N_t^{6.5}K^{3.5})) to (O(K^{10})). The second introduces a balanced augmented Lagrangian (B‑BAL) algorithm that alternates between updating the primal variables (V_k) and the dual multipliers. By exploiting block‑rank‑one structures, each sub‑problem admits a closed‑form solution, yielding an overall computational cost of (O(K^4)).

Complexity analysis shows that when (N_t\gg K) (e.g., massive MIMO), the proposed methods achieve up to six orders of magnitude reduction in runtime. Numerical experiments confirm that the B‑BAL algorithm attains the same optimal objective value as the full‑scale IPM while delivering speed‑ups exceeding 10 000× in scenarios such as (N_t=256) and (K=16). The designs satisfy all SINR and power constraints and minimize the Cramér‑Rao bound for extended target estimation.

In summary, the work reveals a universal low‑dimensional structure in CRB‑based ISAC beamforming, leverages it to formulate a compact optimization problem, and provides two highly efficient algorithms—one based on IPM and one on a novel balanced augmented Lagrangian scheme—thereby enabling practical, optimal beamforming for massive MIMO ISAC deployments.