Geometry-Aware Oracle-Preserving Flows on Spherical Latent Spaces

Oracle-preserving latent flows learn continuous transformations that locally preserve a task-specific oracle, but their long-horizon behavior depends strongly on the geometry of the learned representation of the data. The 2D rotation group SO(2) is inherently continuous and periodic, so representing it with standard unbounded Euclidean latent spaces may not provide the most natural geometric match. Building on this observation, we evaluate whether a compact spherical latent manifold, which prevents radial escape, can improve the long-horizon stability and continuity of the rotational flows. Specifically, we evaluate four representation models, including an autoencoder (AE), a Gaussian variational autoencoder (VAE), a spherical AE (S-AE), and a spherical VAE (S-VAE), and show through a series of experiments on a rotation-augmented MNIST dataset that six-coordinate product-of-spheres representations, particularly when combined with a VAE, support longer and more continuous rotation-like trajectories while preserving oracle outputs. The anonymized code repository is available at \url{https://anonymous.4open.science/r/symmetries-44C4/}.