Generative Rate-Distortion-Perception Theory on Riemannian Manifolds

Generative communication uses transmitted information to control stochastic generation at the receiver. A fundamental question is how much source-specific information must be communicated so that outputs generated at the receiver satisfy prescribed fidelity and reliability requirements while preserving the conditional source distribution. We address this question for sources on compact Riemannian manifolds by developing a generative rate-distortion-perception (GRDP) framework without common randomness. We formulate an asymptotic operational GRDP function under a geodesic distortion radius and a failure-probability constraint, and establish its single-letter characterization through an equivalent informational formulation. Furthermore, for the informational formulation with the high-reliability limit, we derive a geometric lower bound determined by the largest source probability mass of a region with bounded geodesic diameter, together with an upper bound based on probabilistic coverings of the source manifold. Small-radius analysis of the lower bound reveals the roles of intrinsic dimension and scalar curvature in the required communication rate. For uniformly distributed sources on compact homogeneous Riemannian manifolds, we prove that the bounds coincide, yielding an exact characterization of the limiting GRDP function. We further specialize this result to unit spheres, obtaining an explicit expression that relates the minimum communication rate to the geodesic distortion radius and intrinsic dimension, revealing that for the $d$-dimensional unit sphere with $d\ge2$, increasing the communication rate by $d$ bits per source symbol can reduce the distortion radius by more than half.