C-Product Toolbox: A Computational Package for Third-Order Tensor Operations Based on the Reduced c-Product

Abstract

This paper introduces the C-Product Toolbox, a new computational package available for MATLAB and Python, designed to perform operations on third-order tensors using a tensor product known as the reduced c-product. The reduced c-product is a variant of the known c-product, a tensor product based on the discrete cosine transform and belonging to a family of tensor products defined by invertible linear transformations. This work presents the theoretical development of the reduced c-product used and provides a detailed explanation of each tensor operation implemented in the computational package. Additionally, numerical experiments compare the C-Product Toolbox with an existing MATLAB toolbox that employs the t-product, a tensor product based on the discrete Fourier transform. The numerical experiments in this paper demonstrate that the C-Product Toolbox offers superior computational efficiency, achieving faster execution times and lower memory consumption. Furthermore, the practical advantages of the proposed methods are highlighted through an application in video denoising, showcasing the effectiveness of the toolbox in real-world scenarios.