A constructive arbitrary‐degree Kronecker product decomposition of tensors

Summary We generalize the matrix Kronecker product to tensors and propose the tensor Kronecker product singular value decomposition that decomposes a real k ‐way tensor A into a linear combination of tensor Kronecker products with an arbitrary number of d factors. We show how to construct A = ∑ j = 1 Rσ jA j ( d ) ⊗ ⋯ ⊗ A j ( 1 ) , where each factorA j ( i )is also a k ‐way tensor, thus including matrices ( k =2) as a special case. This problem is readily solved by reshaping and permuting A into a d ‐way tensor, followed by a orthogonal polyadic decomposition. Moreover, we introduce the new notion of general symmetric tensors (encompassing symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors, etc.) and prove that when A is structured then its factorsA j ( 1 ) , ⋯ , A j ( d )will also inherit this structure.