Representation Similarity and Transformation-Dependent Stability: Distinction and Boundary Conditions in Neural Networks

Representation similarity metrics such as CKA are widely used to compare neural representations across models and layers by quantifying geometric alignment in representation space. In contrast, transformation-dependent stability functionals such as $S(M)$ characterize sensitivity to structured input perturbations. This paper provides a unified conceptual analysis of the relationship between these two notions. We first clarify their conceptual distinction and then identify the boundary conditions under which they become approximately equivalent or fundamentally decoupled. We argue that representation similarity and transformation-dependent stability capture complementary aspects of neural network behavior: one geometric, the other functional. We further clarify the status of $S(M)$ as a transformation-dependent functional and outline its basic properties. The goal of this work is not to propose a new metric, but to clarify the structural relationship between existing approaches.