Polynomial dendritic neural networks

Although many artificial neural networks have achieved success in practical applications, there is still a concern among many over their “black box” nature. Why and how do they work? Recently, some interesting interpretations have been made through polynomial regression as an alternative to neural networks. Polynomial networks have thus received more and more attention as generators of polynomial regression. Furthermore, some special polynomial works, such as dendrite net (DD) and Kileel et al.’s deep polynomial neural networks, showed that some single neurons have powerful computability. This agrees with a recent discovery on biological neurons, that is, a single biological neuron can perform XOR operations. Inspired by such works, we propose a new model called the polynomial dendritic neural network (PDN) in this article. The PDN achieves powerful computability on a single neuron in a neural network. The output of a PDN is a high degree polynomial of the inputs. To obtain its parameter values, we took PDN as a neural network and employed the back-propagation method. As shown in this context, PDN contains more polynomial outputs than DD and deep polynomial neural networks. We deliberately studied two special PDNs called the exponential PDN (EPDN) and asymptotic PDN (APDN). For interpretability, we proposed a feature analysis method based on the coefficients of the polynomial outputs of such PDNs. The EPDN and APDN showed satisfactory accuracy, precision, recall, F1 score, and AUC in several experiments. Furthermore, we found the coefficient-based interpretability to be effective on some actual health cases.