Summary
This paper explores how much information you need to send to recreate or generate data on curved spaces, like spheres, while keeping the details and reliability you want. The authors study this problem when the data points lie on complicated curved surfaces (called Riemannian manifolds) and introduce a theory combining rate, accuracy, and appearance preservation. They find exact formulas that say how communication amount relates to allowed distortion and show how properties of the curved space, like its dimension and curvature, impact these formulas. For special shapes like spheres, they give clear numbers that tell how much communicating a few extra bits can greatly improve the precision of the generated outputs.
What this means in practice
- •For network system designers: Determine minimum communication rates needed to generate high-fidelity correlated outputs transmitted over curved data spaces without common randomness.
- •For robotics control engineers: Model communication constraints for generating precise distributions of states or sensor readings on curved state spaces in distributed robotic systems.
A theory result. No direct application yet.
Authors
Tong Wu, Zhiyong Chen, Meixia Tao, Dazhi He, Feng Yang, Wenjun Zhang
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 $d\ge2$, increasing the communication rate by $d$ bits per source symbol can reduce the distortion radius by more than half.