On Convergence Rates of Quadratic Transform and WMMSE Methods

Fractional programming (FP) plays an important role in information science because of the Cramer-Rao bound,the Fisher information, and the signal-to-interference-plus-noise ratio (SINR). A state-of-the-art method called the quadratic transform has been extensively used to address the FP problems. This work aims to accelerate the quadratic transform-based iterative optimization via gradient projection and extrapolation. The main contributions of this work are three-fold. First, we relate the quadratic transform to the gradient projection, thereby eliminating the matrix inverse operation from the iterative optimization; our result generalizes the weighted sum-of-rates (WSR) maximization algorithm in [1] to a wide range of FP problems. Second, based on this connection to gradient projection, we incorporate Nesterov's extrapolation strategy [2] into the quadratic transform so as to accelerate the convergence of the iterative optimization. Third, from a minorization-maximization (MM) point of view, we examine the convergence rates of the conventional quadratic transform methods--which include the weighted minimum mean square error (WMMSE) algorithm as a special case--and the proposed accelerated ones. Moreover, we illustrate the practical use of the accelerated quadratic transform in two popular application cases of future wireless networks: (i) integrated sensing and communication (ISAC) and (ii) massive multiple-input multiple-output (MIMO).