A constructive arbitrary-degree Kronecker product decomposition of matrices

We propose a constructive algorithm, called the tensor-based Kronecker product (KP) singular value decomposition (TKPSVD), that decomposes an arbitrary real matrix A into a finite sum of KP terms with an arbitrary number of d factors, namely A = ∑R j=1 σj A dj ⊗ · · · ⊗A1j . The algorithm relies on reshaping and permuting the original matrix into a d-way tensor, after which its tensor-train rank-1 (TTr1) decomposition is computed. The TTr1 decomposition exhibits a singular value profile as with the SVD, allowing for a low-rank truncated series whenever the singular value decay is prominent. It also permits a straightforward relative approximation error measure without explicitly computing the approximant. We move on to show that for many different structured matrices, the KP factor matrices are guaranteed to inherit this structure. In providing these proofs we generalize the notion of symmetric matrices to general symmetric matrices. We also provide two application examples demonstrating the versatility of this KP-decomposed form.