# On the Rate-Distortion Theory for Task-Specific Semantic Communication
## 4 Theoretical Results
### Generation Task  
Our proposed semantic communication architecture can also accommodate the generation task. In this context, the semantic source \(S\) represents the ground-truth data distribution, while the indirect observation \(U\) constitutes a noisy or compressed version of \(S\). For instance, in image generation scenarios, \(S\) might correspond to the distribution of high-resolution natural images, while \(U\) could represent their quantized or downsampled versions. The decoder learns a mapping function that reconstructs an approximation \(\hat{S}\) of the original semantic source, where both \(S\) and \(\hat{S}\) share the same alphabet \(\mathcal{S}\)  
Inspired by the existing literature on generative models [11; 31], we can adopt Wasserstein distance and reverse KL divergence for exact realizations of the semantic distance that measures the perceptual quality of generative samples. Using the above formulation, the semantic rate-distortion problem in (7) for generation task under Wasserstein distance is then specified by:  
\[R_{\text{gen}}(D,P_{\text{gen}})=\inf_{P_{\hat{S}|U}}I(U,\hat{S}|Y)\] (12) \[\text{s.t.}\;E[d(S^{n},\hat{S}^{n})]\leq D,\] \[\;\;\  
We now present the main results of the Gaussian semantic rate-distortion functions as stated in the following two theorems:  
**Theorem 1**. : _The semantic rate-distortion function for Gaussian semantic source \(\mathcal{N}(0,1)\) under Wasserstein distance is_  
\[R_{WD}(D,P)= \begin{cases}\dfrac{1}{2}\log\dfrac{1-\eta^{2}}{D},&\text{if }0<D<D_{1} ^{L}\\ \dfrac{1}{2}\log\dfrac{1-\eta^{2}}{1-\left(\frac{1+\sigma_{1}^{2}-D}{2\sigma_{ 1}}\right)^{2}},&\text{if }D_{1}^{L}\leq D<D_{1}^{R},\\ 0,&\text{if }D\geq D_{1}^{R}\end{cases}  \tag{15}\]  
_where \(D_{1}^{L}:=\sqrt{P}\), \(D_{1}^{R}:=1+\sigma_{1}^{2}-2\eta\sigma_{1}\) and \(\sigma_{1}=1-\sqrt{P}\)._  
Proof. : See Appendix A. **Theorem 2**. : _The semantic rate-distortion function for Gaussian semantic source \(\mathcal{N}(0,1)\) under KL divergence is_  
\[R_{KL}(D,P)= \begin{cases}\dfrac{1}{2}\log\dfrac{1-\eta^{2}}{D},&\text{if }0<D<D_{2} ^{L}\\ \dfrac{1}{2}\log\dfrac{1-\eta^{2}}{1-\left(\frac{1+\sigma_{2}^{2}-D}{2\sigma_ {2}}\right)^{2}},&\text{if }D_{2}^{L}\leq D<D_{2}^{R},\\ 0,&\text{if }D\geq D_{2}^{R},\end{cases}  \tag{16}\]  
_and that under reverse KL divergence is_  
\[R_{RKL}(D,P)= \begin{cases}\dfrac{1}{2}\log\dfrac{1-\eta^{2}}{D},&\text{if }0<D<D_{ 3}^{L}\\ \dfrac{1}{2}\log\dfrac{1-\eta^{2}}{1-\left(\frac{1+\sigma_{3}^{2}-D}{2\sigma_ {3}}\right)^{2}},&\text{if }D_{3}^{L}\leq D<D_{3}^{R},\\ 0,&\text{if }D\geq D_{3}^{R},\end{cases}  \tag{17}\]  
_where \(D_{r}^{L}:=1-\sigma_{2}^{2},D_{2}^{R}:=1+\sigma_{2}^{2}-2\eta\sigma_{2}\), and \(D_{3}^{L}:=1-\sigma_{3}^{2},D_{3}^{R}:=1+\sigma_{3}^{2}-2\eta\sigma_{3}\). Here, \(\sigma_{2},\sigma_{3}\in[0,1]\) are the unique solutions to_  
\[\ln\sigma_{2}^{2}+\frac{1}{\sigma_{2}^{2}}-1-2P=0,\] \[\ln\sigma_{3}^{2}-\sigma_{3}^{2}+1+2P=0, \tag{18}\]  
_respectively._  
Proof. : See Appendix A. We note that, taking \(P\to\infty\), we have  
\[R_{WD}(D,\infty)=R_{KL}(D,\infty)=R_{RKL}(D,\infty)=R(D)=\frac{1}{2}\log\frac {1-\eta^{2}}{D}, \tag{19}\]  
where \(R(D)\) is the classical Gaussian rate-distortion function with side information (see the Wyner-Ziv theorem in [32]). This indicates that, when the constraint on statistical difference is inactive, the semantic rate-distortion functions under Wasserstein distance,KL divergence, and reverse KL divergence all degenerate into traditional rate-distortion functions. When the side information is independent of the source \(S\) (i.e., taking \(\eta=0\)), the semantic rate-distortion function \(R_{W}(D,P)\) in Theorem 1 reduces to the rate-distortion-perception function derived in [16]. The above consistency of our results with prior works confirms both the generalization capability and the correctness of our theoretical results. Based on Theorems 1 and 2, we can quantify the rate difference between the Gaussian semantic rate-distortion function and the traditional rate-distortion function.