Comments on "Approximate Characterizations for the Gaussian Source Broadcast Distortion Region"

TL;DR

This paper clarifies the tightness of an outer bound on the Gaussian source broadcast distortion region, showing it is nontrivial only in bandwidth expansion scenarios and degenerate otherwise.

Contribution

It proves the outer bound's tightness for bandwidth expansion cases and demonstrates its triviality in bandwidth compression or match scenarios.

Findings

Outer bound is strictly tighter in bandwidth expansion cases.

Outer bound degenerates to trivial in bandwidth compression/match cases.

No nontrivial outer bound exists for bandwidth compression in Gaussian broadcast.

Abstract

Recently, Tian et al. [1] considered joint source-channel coding of transmitting a Gaussian source over $K$-user Gaussian broadcast channel, and derived an outer bound on the admissible distortion region. In [1], they stated "due to its nonlinear form, it appears difficult to determine whether it is always looser than the trivial outer bound in all distortion regimes with bandwidth compression". However, in this correspondence we solve this problem and prove that for the bandwidth expansion case ($K\geq2$), this outer bound is strictly tighter than the trivial outer bound with each user being optimal in the point-to-point setting; while for the bandwidth compression or bandwidth match case, this outer bound actually degenerates to the trivial outer bound. Therefore, our results imply that on one hand, the outer bound given in [1] is nontrivial only for Gaussian broadcast communication ($K\geq2$) with bandwidth expansion; on the other hand, unfortunately, no nontrivial outer bound exists so far for Gaussian broadcast communication ($K\geq2$) with bandwidth compression.