A note on the complex and bicomplex valued neural networks

Bicomplex Neural Networks with Hypergeometric Activation Functions

“…Moreover, proceeding similarly as in the proof of Theorem 3.3 of [4], we have the following result: Theorem 4.2. Let T ⊂ C 2 (i) a bounded domain and f : T → C a threshold function and (w 0 , 0) a weighting vector of f (z 1 , z 2 ).…”

“…In the context of bicomplex convolutional neural networks, there are some activation functions proposed in the literature involving bicomplex numbers. For example, in [4] the authors considered the activation function P (z) = l in T ⊂ C n , where = exp 2πi k be the root of the unity of order k, and whenever 2πil k ≤ arg (z) < 2πi(l+1) k .…”

“…Taking into account the properties of the Bessel functions and the ideas presented in [4] we can introduce the concept of threshold function associated to our activation function (4.1).…”

“…In our approach, the structure of the bicomplex algebra is essential in providing a convergent algorithm. For more details about the theory of bicomplex numbers and its applications we refer the interested reader to [4,9,10,12].…”

An extension of the complex–real (C–R) calculus to the bicomplex setting, with applications

“…These new approaches have been especially successful in using particular examples of hypercomplex numbers such as bicomplex numbers, quaternions, and Clifford ones to develop different machine learning techniques, see, for example, [18–20, 23, 25], and references therein. The authors have also extended the complex perceptron algorithm to the bicomplex case in [5, 6]. As we have already discussed, the LMS algorithm discovered by Widrow and Hoff was extended to the complex domain in [37] for the first time and the gradient descent technique was derived with respect to the real and imaginary part.…”