Deep Neural Network Approximation of Invariant Functions through Dynamical Systems

Qianxiao Li, Ting Lin, Zuowei Shen

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main theorem (G-UAP for Hode(F,G) when F resolves G and CH(coor(F)) contains a symmetric invariant well and a 1D well, with scaling/translation along 1 and terminal coverage) is correct and supported by a complete proof strategy via composition, coordinate zooming, relaxed point-matching, and equivariantization (Proposition 4, Theorem 5, Theorem 8). The candidate solution asserts a stronger, unproven construction of local bump translations and arbitrary-direction push maps from CH(coor(F)) and scaling/translation along 1; this step is not justified by the paper’s assumptions and conflicts with the need for only 1-directional affine invariance. Hence the model’s proof has critical gaps.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The work establishes a robust universal approximation theorem for invariant functions via equivariant dynamics under general transitive permutation groups, generalizing prior results without symmetry. The framework—equivariantization, coordinate zooming from wells, and point matching tailored to symmetry—appears correct and significant. Minor clarifications about the coor operator’s role and the resolving-G condition would further aid readers.