The Generally Unitarily Decomposable Partially Symmetric Tensors and Their Approximation

In this paper, we introduce the class of generally unitarily decomposable partially symmetric tensors, which extend conventional orthogonal tensors while retaining many of their desirable properties. We show that the best low-rank approximation of such tensors is directly determined by the largest coefficients in their decompositions. To compute these decompositions, we propose a successive rank-one approximation algorithm and a matrix-based method. Moreover, we study the approximation of general partially symmetric tensors using the proposed tensors and establish Comon’s conjecture for partially symmetric tensors whose canonical polyadic rank is smaller than their order. Numerical experiments demonstrate the effectiveness of the proposed algorithms.