Relate diffeomorphism sensitivity to hierarchical representation formation

Establish how the empirically observed correlation between a network’s test error and its sensitivity to diffeomorphisms relates to the formation of hierarchical representations in deep neural networks trained on high-dimensional visual data, clarifying the mechanisms that connect deformation stability and hierarchical feature abstraction.

Background

The paper highlights a strong empirical correlation: networks with lower sensitivity to diffeomorphisms tend to achieve better test performance. Despite this observation, prior work did not explain how deformation stability connects to the emergence of hierarchical representations, a hallmark of deep networks.

The authors introduce the Sparse Random Hierarchy Model to unify these viewpoints, but the introduction frames the general relationship as unexplained at the time. Providing a comprehensive theoretical account that directly links diffeomorphism stability and hierarchical representation learning beyond the specific model would address this broader uncertainty.