Generative Rate-Distortion-Perception Theory on Riemannian Manifolds
Abstract
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≥2, increasing the communication rate by d bits per source symbol can reduce the distortion radius by more than half.
Related
- An Overview of Rate-Distortion-Perception Theory
- Semantic Transferability of Generative Rate-Distortion-Perception Systems
- How Geometry Shapes Generative Model Evaluation on Riemannian Manifolds
- Rate-Distortion-Perception Tradeoff for Vector Gaussian Sources with Limited Common Randomness: Exact Characterization and Approximate Universality
- A Self-Coupling Gaussian Extremal Inequality for Information-Constrained Optimal Transport with Applications to Rate-Distortion-Perception
BibTeX
@misc{wu2026generative,
title = {{Generative Rate-Distortion-Perception Theory on Riemannian Manifolds}},
author = {Tong Wu and Zhiyong Chen and Meixia Tao and Dazhi He and Feng Yang and Wenjun Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39024},
eprint = {2610.39024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39024},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generative Rate-Distortion-Perception Theory on Riemannian Manifolds
AU - Tong Wu
AU - Zhiyong Chen
AU - Meixia Tao
AU - Dazhi He
AU - Feng Yang
AU - Wenjun Zhang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39024
UR - https://arxiv.org/abs/2610.39024
AB - 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≥2, increasing the communication rate by d bits per source symbol can reduce the distortion radius by more than half.
ER -