Randomized Kaczmarz for Tensor Linear Systems

TL;DR

This paper extends the randomized Kaczmarz method to efficiently solve large tensor linear systems using the tensor-tensor t-product, providing convergence guarantees and demonstrating faster convergence than naive matrix approaches.

Contribution

The paper introduces a tensor-specific randomized Kaczmarz algorithm with proven convergence guarantees, extending the method from matrices to tensors.

Findings

Tensor randomized Kaczmarz converges faster than naive matrix approaches.

Convergence guarantees are analogous to the matrix case.

Experimental results confirm improved efficiency.

Abstract

Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend the randomized Kaczmarz method to solve multi-linear (tensor) systems under the tensor-tensor t-product. We provide convergence guarantees for the proposed tensor randomized Kaczmarz that are analogous to those of the randomized Kaczmarz method for matrix linear systems. We demonstrate experimentally that the tensor randomized Kaczmarz method converges faster than traditional randomized Kaczmarz applied to a naively matricized version of the linear system. In addition, we draw connections between the proposed algorithm and a previously known extension of the randomized Kaczmarz algorithm for matrix linear systems.