Fractional Programming for Kullback-Leibler Divergence in Hypothesis Testing

Maximizing the Kullback-Leibler divergence (KLD) is a fundamental problem in waveform design for active sensing and hypothesis testing, as it directly relates to the error exponent of detection probability. However, the associated optimization problem is highly nonconvex due to the intricate coupling of log-determinant and matrix trace terms. Existing solutions often suffer from high computational complexity, typically requiring matrix inversion at every iteration. In this paper, we propose a computationally efficient optimization framework based on fractional programming (FP). Our key idea is to reformulate the KLD maximization problem into a sequence of tractable quadratic subproblems using matrix FP. To further reduce complexity, we introduce a nonhomogeneous relaxation technique that replaces the costly linear system solver with a simple closed-form update, thereby reducing the per-iteration complexity to quadratic order. To compensate for the convergence speed trade-off caused by relaxation, we employ an acceleration method called STEM by interpreting the iterative scheme as a fixed-point mapping. The resulting algorithm achieves significantly faster convergence rates with low per-iteration cost. Numerical results demonstrate that our approach reduces the total runtime by orders of magnitude compared to a state-of-the-art benchmark. Finally, we apply the proposed framework to a multiple random access scenario and a joint integrated sensing and communication scenario, validating the efficacy of our framework in such applications.

The MIMO-ME-MS Channel: Analysis and Algorithm for Secure MIMO Integrated Sensing and Communications

This paper studies precoder design for secure MIMO integrated sensing and communications (ISAC) by introducing the MIMO-ME-MS channel, where a multi-antenna transmitter serves a legitimate multi-antenna receiver in the presence of a multi-antenna eavesdropper while simultaneously enabling sensing via a multi-antenna sensing receiver. Using sensing mutual information as the sensing metric, we formulate a nonconvex weighted objective that jointly captures secure communication (via secrecy rate) and sensing performance. A high-SNR analysis based on subspace decomposition characterizes the maximum achievable weighted degrees of freedom and reveals that a quasi-optimal precoder must span a "useful subspace," highlighting why straightforward extensions of classical wiretap/ISAC precoders can be suboptimal in this tripartite setting. Motivated by these insights, we develop a practical two-stage iterative algorithm that alternates between sequential basis construction and power allocation via a difference-of-convex program. Numerical results show that the proposed approach captures the desirable precoding structure predicted by the analysis and yields substantial gains in the MIMO-ME-MS channel.