Quaternion LMS for Graph Signal Recovery

Abstract

This letter generalizes the Graph Signal Recovery (GSR) problem in Graph Signal Processing (GSP) to the Quaternion domain. It extends the Quaternion Least Mean Square (QLMS) in adaptive filtering literature, and Graph LMS (GLMS) algorithm in GSP literature, to an algorithm called Quaternion GLMS (QGLMS). The basic adaptation formula using Quaternion-based algebra is derived. Moreover, mean convergence analysis and mean-square convergence analysis are mathematically performed. Hence, a sufficient condition on the step-size parameter of QGLMS is suggested. Also, simulation results demonstrate the effectiveness of the proposed algorithm in graph signal reconstruction.