Block Matrix and Tensor Randomized Kaczmarz Methods for Linear Feasibility Problems

TL;DR

This paper introduces new block and tensor variants of the randomized Kaczmarz method for solving large-scale linear feasibility problems, demonstrating linear convergence and effectiveness in applications like image deblurring.

Contribution

It proposes novel block and tensor randomized Kaczmarz algorithms with proven linear convergence for linear feasibility problems, extending applicability to tensor data and mixed constraints.

Findings

B-MRK converges linearly in expectation for matrix problems.

TRK-L and TRK-LB methods converge linearly for tensor problems.

Numerical experiments show effectiveness in Gaussian data and image deblurring.

Abstract

The randomized Kaczmarz methods are a popular and effective family of iterative methods for solving large-scale linear systems of equations, which have also been applied to linear feasibility problems. In this work, we propose a new block variant of the randomized Kaczmarz method, B-MRK, for solving linear feasibility problems defined by matrices. We show that B-MRK converges linearly in expectation to the feasible region.Furthermore, we extend the method to solve tensor linear feasibility problems defined under the tensor t-product. A tensor randomized Kaczmarz (TRK) method, TRK-L, is proposed for solving linear feasibility problems that involve mixed equality and inequality constraints. Additionally, we introduce another TRK method, TRK-LB, specifically tailored for cases where the feasible region is defined by linear equality constraints coupled with bound constraints on the variables. We show that both of the TRK methods converge linearly in expectation to the feasible region. Moreover, the effectiveness of our methods is demonstrated through numerical experiments on various Gaussian random data and applications in image deblurring.