Conjugational Stability and Symmetry-Invariant Transformations in Artificial Neural Networks

This paper examines the properties of algebraically stabilized linear transformations in artificial neural networks (ANNs) under a change of basis and their compatibility with symmetry-invariant architectures. We investigate the conditions under which a stabilizing transformation of the form W^k=I+μD, defined over integer matrices, preserves its properties under conjugation, i.e., when passing to an equivalent form W'=B^(-1) WB in a new basis. It is shown that such transformations form a closed class with respect to conjugational equivalence and can be employed in environments with diverse geometric and symmetry requirements. Additionally, we analyze the role of symmetric actions on the input space and define conditions for the invariance of transformations with respect to specific groups of transformations. Both formal algebraic properties and the geometric and semantic interpretation of stability in this context are considered. Different types of stability are presented – algebraic, conjugational, and symmetry-based – along with the relationships between them. The results are relevant to the design of ANN architectures with predictable behavior, robustness under transformations, and compatibility with predefined symmetric structures.