Fractional programming (FP) arises in various communications and signal processing problems because several key quantities in the field are fractionally structured, e.g., the Cram\'{e}r-Rao bound, the Fisher information, and the signal-to-interference-plus-noise ratio (SINR). A recently proposed method called the quadratic transform has been applied to the FP problems extensively. The main contributions of the present paper are two-fold. First, we investigate how fast the quadratic transform converges. To the best of our knowledge, this is the first work that analyzes the convergence rate for the quadratic transform as well as its special case the weighted minimum mean square error (WMMSE) algorithm. Second, we accelerate the existing quadratic transform via a novel use of Nesterov's extrapolation scheme [1]. Specifically, by generalizing the minorization-maximization (MM) approach in [2], we establish a nontrivial connection between the quadratic transform and the gradient projection, thereby further incorporating the gradient extrapolation into the quadratic transform to make it converge more rapidly. Moreover, the paper showcases the practical use of the accelerated quadratic transform with two frontier wireless applications: integrated sensing and communications (ISAC) and massive multiple-input multiple-output (MIMO).

翻译：分数规划（FP）在通信与信号处理领域广泛出现，因为该领域中若干关键量具有分数结构，例如克拉美-罗界、费舍尔信息以及信干噪比（SINR）。近期提出的二次变换方法已被广泛应用于求解FP问题。本文的主要贡献有两点：首先，我们研究了二次变换的收敛速度。据我们所知，这是首次对二次变换及其特例——加权最小均方误差（WMMSE）算法——进行收敛速率分析的工作。其次，我们通过创新性地运用Nesterov外推方案[1]来加速现有二次变换。具体而言，通过推广文献[2]中的最小化-最大化（MM）方法，我们建立了二次变换与梯度投影之间的非平凡联系，从而进一步将梯度外推融入二次变换以加速其收敛。此外，本文通过两个前沿无线应用——集成感知与通信（ISAC）和大规模多输入多输出（MIMO）——展示了加速二次变换的实际应用价值。