RiCoTA: Riemannian Companding Token Autoencoder

Fast generation on Riemannian manifolds must preserve the intrinsic structure of data distributions with low sampling cost. We introduce the Riemannian Companding Token Autoencoder (RiCoTA), which learns a joint distribution over discrete latent codes for continuous manifold generation. Its design addresses a geometric mismatch: equal-width latent intervals can induce very different geodesic displacements after decoding. RiCoTA therefore organizes latent coordinates by average decoder sensitivity and allocates finer resolution where local changes have larger intrinsic effects, keeping the number of intervals per coordinate fixed. Training and generation share a uniform within-cell reconstruction rule. At inference, an autoregressive prior selects cells, continuous values are sampled within them, and one decoder evaluation produces manifold samples without trajectory integration. For a fixed representation, our asymptotic analysis derives a coordinatewise bound on intrinsic distortion and identifies the sensitivity-based allocation that minimizes its leading term. A complementary population bound controls generation error in terms of quantized reconstruction error and joint code-distribution mismatch. Held-out diagnostics show that estimated sensitivity predicts geometric risk and adaptive allocation reduces coordinatewise intrinsic distortion at a fixed budget. Experiments across 14 manifold benchmarks demonstrate strong support fidelity at low end-to-end sampling cost.