Hypercomplex Low Rank Matrix Completion with Non-negative Constraints 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 real world applications, e.g., representing color images, of hypercomplex number systems, all attributes are often restricted to be non-negative. In this paper, we formulate non-negative matrix completion problem in hypercomplex domain as a convex optimization problem in real domain. These formulation is based on algebraic translations of Cayley-Dickson (C-D) linear systems. We then derive an algorithmic solution to hypercomplex low rank matrix completion with non-negative constraint based on a proximal splitting technique. Numerical experiments are performed in a scenario of high-dimensional hypercomplex matrix completion problem and show that the proposed algorithm recovers much more faithfully the original information, masked randomly by noise, than a part-wise state-of-art algorithms.