An efficient iterative method to find the Moore-Penrose inverse of tensors: Applications in image processing and data mining

Matrices and tensors serve as fundamental tools in mathematical modelling, enabling applications such as linear transformations, systems of equations, and multivariate data analysis. This work introduces a computational framework for determining the Moore-Penrose (MP) inverse of tensors using the Einstein product (EP), along with a detailed theoretical analysis. The proposed method builds on an iterative method designed for solving nonlinear equations. Numerical comparisons with existing methods demonstrate that the proposed method requires fewer iterations, performs fewer EPs, and consumes significantly less CPU time. To highlight practical applications, we consider partial and fractional differential equations, particularly those resulting in sparse matrices, as representative cases. The iterates generated by the proposed method are utilized as pre-conditioners in tensor form to solve multilinear systems of the form:$B*X=C.$Finally, we present various practical numerical examples to demonstrate the efficiency and accuracy of the proposed method. The results highlight the robustness and effectiveness of the method in computing the MP inverse of tensors. This method provides significant computational advantages and proves highly applicable across diverse domains, including mathematics, physics, image processing, and data mining.