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.

💡 Research Summary

This paper addresses the challenging problem of maximizing the Kullback‑Leibler divergence (KLD) in active sensing and binary hypothesis testing, where KLD directly determines the exponential decay rate of the miss‑detection probability via the Chernoff‑Stein lemma. The underlying optimization involves a log‑determinant term and a trace term that are both coupled quadratically with the transmit waveform matrix X, rendering the problem highly non‑convex and computationally intensive. Existing approaches—such as semi‑definite relaxation, minorization‑maximization (MM), or heuristic designs—either require solving large linear systems at each iteration (cubic complexity) or impose restrictive structural assumptions, limiting scalability for modern large‑scale MIMO radar and integrated sensing‑communication (ISAC) systems.

The authors propose a novel framework built on matrix fractional programming (FP). By applying the matrix Lagrangian dual transform to the log‑det term and introducing auxiliary variables for the trace term, the original objective f(X)=log|K₀⁻¹K₁|+tr(K₁⁻¹K₀) is lower‑bounded by a surrogate that is concave in the new variables. This surrogate decouples the matrix inverse from X, turning the original non‑convex problem into a sequence of convex quadratic subproblems that can be solved efficiently.

To further reduce per‑iteration cost, a non‑homogeneous relaxation is introduced. Instead of solving a full linear system to update the auxiliary variables, the authors replace the anisotropic curvature of the surrogate with a conservative isotropic bound, yielding a closed‑form update for X that requires only matrix multiplications and a pre‑computed inverse. Consequently, the per‑iteration computational complexity drops from O(N³) to O(N²), where N denotes the dimension of the transmit antenna array.

Because the relaxation may slow convergence, the paper incorporates a fixed‑point acceleration technique called STEM (Steffensen‑type acceleration). By treating each FP iteration as a fixed‑point mapping g(·) and approximating the Jacobian using two successive iterates, STEM achieves quadratic convergence, effectively compensating for the additional iterations introduced by the relaxation.

Theoretical analysis proves monotonic ascent of the original KLD objective, guarantees convergence to a stationary point, and quantifies the computational savings. Numerical experiments on a 64‑antenna MIMO radar scenario (T=20 snapshots) demonstrate that the proposed FP‑STEM algorithm attains the same or slightly higher KLD values as the state‑of‑the‑art MM method while reducing total runtime by one to two orders of magnitude (e.g., from ~7 seconds to ~0.1 seconds). Convergence is typically achieved within 5–10 iterations, and the STEM accelerator further cuts this to 3–4 iterations.

Beyond the baseline detection problem, the framework is applied to two practical settings. In a multiple random‑access scenario, the weighted sum of per‑user KLDs retains the same algebraic structure, allowing parallel application of the FP‑STEM updates and yielding a 20 % increase in system throughput. In an ISAC scenario, the authors jointly maximize a composite objective α·MI + β·KLD, where MI (mutual information) is already amenable to FP‑based water‑filling. The combined algorithm simultaneously preserves communication rates (≤5 % loss) while boosting the sensing KLD by ≈15 %.

In summary, the paper delivers a highly efficient, provably convergent algorithm for KLD maximization that overcomes the computational bottlenecks of prior methods. It opens the door to real‑time waveform design for large‑scale sensing and joint sensing‑communication systems, and suggests future extensions to non‑Gaussian models, adaptive online implementations, and hardware‑level acceleration.