We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and Hoff for Adaptive Linear Neuron (ADALINE). These gradient operators will be used to formulate new learning rules for the Bicomplex Least Mean Square (BLMS) algorithms and we will also formulate these learning rules will for the case of multicomplex LMS algorithms (MLMS). This approach extends both the classical real and complex LMS algorithms.

翻译：我们研究并引入了复数与双复数框架下的新型梯度算子，其灵感源自1960年Widrow和Hoff为自适应线性神经元（ADALINE）提出的著名最小均方（LMS）算法。这些梯度算子将被用于构建双复最小均方（BLMS）算法的新学习规则，同时我们还将针对多复最小均方（MLMS）算法情形推导相应的学习规则。该方法拓展了经典的实值与复值LMS算法。