A Hypercomplex Tensor-SVD and Its Application

Expressing multidimensional information as a value in hypercomplex number systems (e.g., quaternion, octonion, etc.) has great potential in signal processing applications. The benefits from the algebraically natural operations of hypercomplex numbers can be inherited as well in hypercomplex tensors, which can be widely used for representing huge volume of multi-relational datasets. In this paper, we present a new algebraic real translations of hypercomplex tensors in order to extend tensor-SVD (t-SVD) and related mathematical tools such as tensor tubal rank and tensor low tubal rank approximation into general (Cayley-Dickson) hypercomplex domain. We then propose an algorithmic solution to the hypercomplex tensor principal component pursuit based on a proximal splitting technique. Numerical experiments are performed in quaternion domain and show that the proposed algorithm outperforms a part-wise state-of-art real and complex algorithm.