Hypercomplex Tensor Completion via Convex Optimization

Abstract

Expressing multidimensional information as a value in hypercomplex number systems (e.g., quaternion, octonion, etc.) has great potential, in data sciences, e.g., signal processing, to enjoy their nontrivial algebraic benefits which are not available in standard real or complex vector systems. Strategic utilizations of such benefits would include, e.g., hypercomplex singular value decomposition (SVD) and low-rank approximation of matrices. In this paper, as powerful mathematical tools for wider signal processing applications of hypercomplex number systems, we first propose novel definitions of SVD and best low-rank approximation of matrices based on algebraic translations of Cayley–Dickson (C–D) number systems. We then derive an algorithmic solution to hypercomplex tensor completion problem based on a convex optimization technique. Numerical experiments in a scenario of color tensor completion problem show that the proposed algorithm recovers much more faithfully the original color information, masked randomly by noise, than part-wise tensor completion algorithms.