Low-Rank Tensor Decompositions for Quaternion Multiway Arrays

Abstract

Quaternion multiway arrays appear naturally as compact representations of 3D or 4D multidimensional signals. However, the non-commutativity of quaternion multiplication prevents a straightforward extension of standard tensor algebra to analyze and process quaternion multiway arrays. After reviewing the theoretical difficulties related to quaternion tensor algebra, we propose the first construction of quaternion tensors as representation of dedicated quaternion multilinear forms. This theoretical construction ensures that usual tensor algebraic properties, such as mode products properties are preserved. This novel framework enables us to generalize Tucker and canonical polyadic tensor decompositions to the quaternion case. For the latter, we carefully design a full quaternion ALS-type algorithm. Its relevance is validated numerically.