Convex Quaternion Optimization for Signal Processing: Theory and Applications

Shuning Sun, Qiankun Diao, Dongpo Xu, Pauline Bourigault, Danilo P. Mandic

Abstract: Convex optimization methods have been extensively used in communications and signal processing. However, the theory of quaternion optimization is currently not as fully developed and systematic as that of complex and real optimization. To this end, we establish the convex optimization theory in quaternion variables based on the generalized Hamilton-real (GHR) calculus. This is achieved in a way that conforms with traditional complex and real optimization theory. We present several discriminant theorems for convex quaternion functions analogous to their complex counterparts. We also provide several discriminant criteria for strongly convex functions by the theorems of convex quaternion functions. Furthermore, we prove that the quaternion Newton method can converge in one step for positive definite quadratic quaternion functions and provide two applications in quaternion signal processing. These results provide a solid theoretical foundation for convex quaternion optimization and open avenues for further developments in quaternion signal processing applications.