Revisiting noise in learning: A generalized fractional Tikhonov approach within the adaptive network-based fuzzy inference system for classification problems

In machine learning, noise has traditionally been regarded as a detrimental factor. However, recent studies have revealed its potential to enhance learning through mechanisms such as noise-induced regularization and data augmentation. Building upon the established connection between input noise injection and Tikhonov regularization, this study proposes a generalized fractional Tikhonov regularization within the Adaptive Network-Based Fuzzy Inference System (ANFIS) framework. We introduce a novel model, ANFIS-GT, generalizing the previous ANFIS-T scheme, which was the first ANFIS variant equipped with fractional Tikhonov regularization, in order to improve its classification performance. The proposed method employs differential quadrature rules to construct the regularization matrix in the penalty term, extending the conventional fractional Tikhonov approach that assumes an identity matrix. We formally prove the error stability of the new scheme, establishing a result that also holds for the former ANFIS-T under certain conditions, for which stability had not previously been demonstrated. Furthermore, we analytically identify the conditions under which the solution of the proposed scheme is equivalent to that obtained by small perturbations of the system's input in a non-penalized error function. Extensive evaluations on binary and multiclass classification tasks demonstrate that the proposed approach outperforms state-of-the-art techniques, validating the theoretical developments and yielding a robust, interpretable, and effective learning model.