Exact Common Information and Exact Channel Synthesis for Correlated Gaussian Sources

2026-08-26

AI Summary

本文解决了关于相关高斯源的确切共同信息和确切信道合成的两个猜想，使用了最优传输表示、Fathi的高斯传输不等式及协方差结构中的行列式不等式。

Abstract

In this paper, we resolve two conjectures posed by Yu and Tan in 2020 (in two separate papers published in the IEEE Trans. Inf. Theory). Specifically, we establish that: 1) the exact common information for a pair of $ρ$-correlated Gaussian sources is given by the conjectured expression $\frac{1}{2}\log\frac{1+ρ}{1-ρ}+\fracρ{1+ρ}$; and 2) the admissible region for the shared randomness rate and the communication rate in exact channel synthesis is exactly the conjectured one. These results yield two important consequences. First, for any $ρ>0$, the exact common information of a correlated Gaussian pair strictly exceeds Wyner's common information. Second, for $ρ>0$, the exact channel synthesis of such a pair requires strictly higher rates than the total-variation version. The proof combines an exact optimal-transport representation of the worst-case Gaussian cross-entropy, Fathi's Gaussian transport inequality, and a determinant inequality arising from the covariance structure of the conditional means.

Problem

• Gaussian sources
• exact common information
• channel synthesis

Innovation

• Gaussian sources
• exact common information
• channel synthesis
• optimal transport
• Gaussian cross-entropy