Accelerating Quadratic Transform and WMMSE

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).