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Accelerating Quadratic Transform and WMMSE

Kaiming Shen, Ziping Zhao, Yannan Chen, Zepeng Zhang, Hei Victor Cheng

TL;DR

The paper addresses FP problems with multiple (matrix) ratios by advancing the quadratic transform (QT) and establishing a gradient-projection interpretation. It introduces a nonhomogeneous QT that eliminates costly matrix inversions and an extrapolated QT (EQT) based on Nesterov momentum to achieve $O(1/k^2)$ convergence locally, improving over the standard $O(1/k)$. Theoretical results prove convergence to stationary points for QT, GQT, and EQT and characterize iteration-time trade-offs. The approach is demonstrated in ISAC and massive MIMO applications, where inversion-free or reduced-dimension updates enable scalable optimization in large antenna arrays, underscoring practical impact for next-generation wireless networks.

Abstract

Fractional programming (FP) arises in various communications and signal processing problems because several key quantities in the field are fractionally structured, e.g., the Cramér-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).

Accelerating Quadratic Transform and WMMSE

TL;DR

The paper addresses FP problems with multiple (matrix) ratios by advancing the quadratic transform (QT) and establishing a gradient-projection interpretation. It introduces a nonhomogeneous QT that eliminates costly matrix inversions and an extrapolated QT (EQT) based on Nesterov momentum to achieve convergence locally, improving over the standard . Theoretical results prove convergence to stationary points for QT, GQT, and EQT and characterize iteration-time trade-offs. The approach is demonstrated in ISAC and massive MIMO applications, where inversion-free or reduced-dimension updates enable scalable optimization in large antenna arrays, underscoring practical impact for next-generation wireless networks.

Abstract

Fractional programming (FP) arises in various communications and signal processing problems because several key quantities in the field are fractionally structured, e.g., the Cramér-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).
Paper Structure (19 sections, 8 theorems, 119 equations, 7 figures, 3 algorithms)

This paper contains 19 sections, 8 theorems, 119 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Main Results
  5. Paper Organization and Notation
  6. Problem Statement
  7. Quadratic Transform
  8. Preliminary
  9. Connection with Gradient Projection
  10. Accelerated Quadratic Transform
  11. Convergence Analysis
  12. Extension to Other FP Problems
  13. Log-Ratio Case
  14. General Utility Case
  15. Matrix Ratio Case
  16. ...and 4 more sections

Key Result

Proposition 1

The sum-of-weighted-ratios FP problem prob:MFP is equivalent to with the new objective function in the sense that $\underline\bm{x}^\star$ is a solution of problem prob:MFP if and only if $(\underline\bm{x}^\star,\underline\bm{y}^\star)$ is a solution of problem QT:new_prob with some optimal auxiliary variable $\underline\bm{y}^\star$.

Figures (7)

  • Figure 1: The conventional quadratic transform amounts to the projection onto an ellipsoid and incurs matrix inverse operation. In contrast, the new quadratic transform avoids matrix inverse by computing the projection onto a sphere.
  • Figure 2: Algorithm \ref{['algorithm:QT']} approximates $f_o(\underline\bm{x})$ as $f_q(\underline\bm{x},\underline\bm{y})$ while Algorithm \ref{['algorithm:GQT']} approximates $f_o(\underline\bm{x})$ as $f_t(\underline\bm{x},\underline\bm{y},\underline\bm{z})$. By the MM procedure, for the current solution $\underline\bm{x}^k$, Algorithm \ref{['algorithm:QT']} updates it to $a$, while Algorithm \ref{['algorithm:GQT']} updates it to $b$. Algorithm \ref{['algorithm:QT']} converges faster in iterations because its approximation is tighter.
  • Figure 3: Average performance of solving 100 random examples of problem \ref{['prob:MFP']}. Let each $\omega_i=1$, let each $\mathcal{X}_i=\{\bm{X}\in\mathbb C^{d\times\ell}:\text{tr}(\bm{X}\bm{X}^\mathrm{H})\le10\}$, and randomly generate each entry of $\bm{A}_i$ and $\bm{B}_{ij}$ i.i.d. according to $\mathcal{CN}(0,1)$; further, add $\bm{I}$ to each matrix denominator to ensure its positive definiteness.
  • Figure 4: Two BSs serve one downlink user each. One BS performs ISAC while the other BS only performs transmission; the aim of sensing is to recover the angle $\theta$. The dashed arrows represent the interference.
  • Figure 5: Maximizing a weighted sum of the Fisher information and the SINRs, $J_\theta+\omega_1\mathrm{SINR}_1+\omega_2\mathrm{SINR}_2$, for an ISAC system.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Lemma 1
  • Lemma 2: Nonhomogeneous Bound sun2016majorization
  • Remark 1: Geometric Interpretation
  • Remark 2
  • Proposition 2
  • Proposition 3: Convergence Rates of Algorithm \ref{['algorithm:QT']} and Algorithm \ref{['algorithm:GQT']}
  • Proposition 4: Convergence Rate of Algorithm \ref{['algorithm:EQT']}
  • Proposition 5
  • Remark 3
  • ...and 1 more