On the relaxed greedy randomized Kaczmarz methods with momentum acceleration for solving matrix equation AXB=C

TL;DR

This paper introduces momentum-accelerated relaxed greedy randomized Kaczmarz methods for efficiently solving consistent matrix equations, demonstrating linear convergence and effectiveness through numerical experiments and a real-world application.

Contribution

It extends the ME-RGRK method by incorporating Polyak's and Nesterov's momentum to accelerate convergence for solving matrix equations.

Findings

Methods converge linearly to least-squares solutions.

Numerical experiments confirm improved convergence speed.

Application to tensor surface fitting demonstrates practical utility.

Abstract

With the growth of data, it is more important than ever to develop an efficient and robust method for solving the consistent matrix equation AXB=C. The randomized Kaczmarz (RK) method has received a lot of attention because of its computational efficiency and low memory footprint. A recently proposed approach is the matrix equation relaxed greedy RK (ME-RGRK) method, which greedily uses the loss of the index pair as a threshold to detect and avoid projecting the working rows onto that are too far from the current iterate. In this work, we utilize the Polyak's and Nesterov's momentums to further speed up the convergence rate of the ME-RGRK method. The resulting methods are shown to converge linearly to a least-squares solution with minimum Frobenius norm. Finally, some numerical experiments are provided to illustrate the feasibility and effectiveness of our proposed methods. In addition, a real-world application, i.e., tensor product surface fitting in computer-aided geometry design, has also been presented for explanatory purpose.