This study proposes a new formalized approach to the stabilization of linear transformations in artificial neural networks, based on discrete algebraic properties. In contrast to existing stability methods that rely on spectral norms, regularization techniques, or empirical heuristics, this work introduces the concept of algebraic stabilization—stability that arises from the structural properties of the matrices defining linear operators. The central object of investigation is the class of integer-valued matrices for which exponentiation to a form of the type Wk=I+μD is possible, where D∈Zn×n,μ∈Z>1. A well-known problem in group algebra is considered that guarantees the existence of such an exponent under the condition that μ is coprime with the determinant of W. Within this framework, modular arithmetic, reduction modulo μ, and the group structure of GLn(Zμ) are employed, thereby linking the proposed method to the theory of finite groups and linear automata. The advantages of the approach are discussed, including formal control over the iterative behavior of transformations, compatibility with quantized and finitely representable networks, the possibility of embedding stabilizing conditions directly into the network architecture, and the potential to improve model interpretability and reliability. At the same time, limitations are identified, particularly those related to constructive implementation, the selection of suitable hyperparameters, and generalization to broader classes of transformations.
Yotov et al. (Tue,) studied this question.