Randomized Kaczmarz for Tensor Linear Systems

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.

Regularized Kaczmarz Algorithms for Tensor Recovery

“…To solve this linear constrained minimization problem, we propose a general regularized Kaczmarz tensor algorithm, which involves random projection onto the hyperplane and subgradient descent. The very recent work by Ma and Molitor [32] also uses the Kaczmarz algorithm, but it focuses on solving the system A * X = B without minimizing an objective function. We discuss convergence guarantees of the proposed algorithm with a deterministic or random control sequence.…”