Order-Specified Quaternion CP Decomposition: Models, Algorithms, and Applications

Abstract

Tensor CP decomposition serves as a fundamental tool for representing multiway data. While quaternion tensors are well suited to modeling color visual data, extending CP decomposition to the quaternion domain is nontrivial due to the noncommutativity of quaternion multiplication. This paper develops an order-specified quaternion CP decomposition (OS-QCP) for third-order tensors, allowing all three factor matrices to be quaternion-valued and explicitly incorporating their multiplication order into the model. We show that different multiplication orders can yield different OS-QCP ranks for the same quaternion tensor, characterize the permutation and quaternion scaling indeterminacies, and establish a sufficient condition for essential uniqueness for any prescribed multiplication order. An explicit example further shows that OS-QCP rank can depend on the multiplication order even when the complex representation of the quaternion tensor admits an essentially unique CP decomposition. An alternating least-squares (ALS) algorithm is derived by expressing quaternion left and right multiplication as real linear operators, reducing each factor update to real least-squares problems. To accommodate order dependence in tensor completion, we further introduce an order-ensemble quaternion CP (OE-QCP) model that incorporates different multiplication orders through nonnegative weights. The model jointly estimates three quaternion factor matrices and the order weights, with each single-order OS-QCP model included as a special case. The resulting completion problem is solved by a parallel row-wise masked ALS algorithm. Experiments on color video inpainting demonstrate the effectiveness and competitive performance of the proposed method.