AN ORDER-P TENSOR MULTIPLICATION WITH CIRCULANT STRUCTURE
Abstract
Research on mathematical operations involving multidimensional arrays or tensors has increased along with the growing applications involving multidimensional data analysis. The -product of order- tensor is one of tensor multiplications. The -product is defined using two operations that transform the multiplication of two tensors into the multiplication of two block matrices, then the result is a block matrix which is further transformed back into a tensor. The composition of both operations used in the definition of -product can transform a tensor into a block circulant matrix. This research discusses the -product of tensors based on their circulant structure. First, we present a theorem of the -product of tensors involving circulant matrices. Second, we use the definition of identity, transpose, and inverse tensors under -product operation and investigate their relationship with circulant matrices. Third, we manifest the computation of the -product involving circulant matrices. The results of the discussion show that the -product of tensors fundamentally involves circulant matrix multiplication, which means that the operation at its core relies on multiplying circulant matrices. This implies the -product operation of tensors having properties analogous to standard matrix multiplication. Furthermore, since the -product of tensors fundamentally involves circulant matrix multiplication, its computation can be simplified by diagonalizing the circulant matrix first using the discrete Fourier transform matrix. Finally, based on the obtained results, an algorithm is constructed in MATLAB to calculate the -product.
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