An innovative fractional order LMS based on variable initial value and gradient order

This article presents a novel fractional order LMS (FOLMS) algorithm, which involves a variable gradient order scheme. The fractional order gradient descent method is revisited firstly. A variable initial value scheme is proposed to attenuate the non-locality of fractional order calculus and to ensure the convergence of the proposed FOLMS algorithm. Furthermore, it is noticed that a contradiction between rapidity and accuracy always appears together with the advancement of FOLMS algorithm; namely, a larger value of the gradient order can not only give a faster convergence speed, but also correspond to a larger estimation error. For the purpose of removing the contradiction between rapidity and accuracy, a variable gradient order scheme is designed for the FOLMS algorithm. Based on a sufficient large number of independent runs, the efficiency and superiority of the proposed algorithm are demonstrated in numerical examples finally.