Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

The paper clearly describes the SAE pipeline (STN for F(s), Siamese branches for G, argmin L2 branch selection, and inverse transforms) and empirically argues invariance and improved model reduction, but it does not provide a formal proof that the latent is orbit-invariant or that s-PCA is exactly PCA on a quotient dataset; these are presented as architectural design and observations (including that STN outputs can be linear and resemble trigonometric weights, and that predicted shifts match first-mode slicing up to a phase) . The candidate solution supplies a precise argument establishing orbit-invariance of the selected-branch latent and the equivalence of s-PCA to PCA on the canonical representatives (under explicit assumptions such as L2-isometries, F–G equivariance, consistent tie-breaking, and zero branch error), and gives an explicit linear-STN construction via first Fourier coefficients—details not proven in the paper but consistent with its architecture and empirical findings .