Efficient iterative techniques for solving tensor problems with the T-product
Malihe Nobakht Kooshkghazi, Salman Ahmadi-Asl, Hamidreza Afshin
TL;DR
This work develops finite-convergence iterative methods for solving tensor equations under the T-product, avoiding explicit matrix construction. It introduces three algorithms: one for T-symmetric positive definite coefficients, one for consistent equations, and one for inconsistent (least-squares) problems, all exploiting orthogonal tensor sequences and normal-equation reductions. Theoretical results guarantee convergence in a finite number of steps, and minimal-norm solutions can be obtained with appropriate initialization. Numerical experiments on synthetic data, images, and videos demonstrate accurate recovery and practical efficiency, highlighting the methods' relevance to high-dimensional tensor problems in signal processing and related fields.
Abstract
This paper presents iterative methods for solving tensor equations involving the T-product. The proposed approaches apply tensor computations without matrix construction. For each initial tensor, these algorithms solve related problems in a finite number of iterations, with negligible errors. The theoretical analysis is validated by numerical examples that demonstrate the practicality and effectiveness of these algorithms.