# On the Rate-Distortion Theory for Task-Specific Semantic Communication
## 4 Theoretical Results
### Generation Task
Formally, taking Wasserstein distance as an example, we have  
\[R_{WD}(D,P)-R(D)=\frac{1}{2}\log\frac{D}{1-\left(\frac{1-(P+D)/2}{1-\sqrt{P}} \right)^{2}}. \tag{20}\]  
We can observe that the above rate difference is always positive. Moreover, this rate difference increases with \(D\) and decreases with \(P\). This indicates that, under the same distortion level, the extra rate to satisfy the constraint on semantic distance increases when this constraint becomes tighter. This additional rate requirement decreases when lower bit-wise quality is acceptable. From Theorem 2, we can derive an inequality relationship between the Gaussian semantic rate-distortion functions under KL divergence and reverse KL divergence:  
\[R_{KL}(D,P)\geq R_{RKL}(D,P), \tag{21}\]  
where the equality holds if and only if \(P=\infty\). This means that, under the same \(D\) and \(P\), it generally requires a higher rate to meet the constraint on KL divergence, compared to reverse KL divergence. This also demonstrates that reverse KL divergence is a more rate-efficient metric for preserving the source distribution under the same fidelity requirements. Moreover, the equality condition \(P=\infty\) indicates that both divergence metrics converge to the same rate-distortion function when the semantic distance constraint becomes inactive. This aligns with the observation of Theorem 2 that both functions degenerate to the classical Wyner-Ziv rate-distortion function in this limit. The theoretical inequality is further supported by the experimental results in Section 5, where generation tasks using reverse KL divergence achieve better perceptual quality at lower rates. When \(D\leq D_{1}^{L}\) (resp. \(D\leq D_{2}^{L}\), \(D\leq D_{3}^{L}\)), the semantic rate-distortion functions under Wasserstein distance (resp. KL divergence, reverse KL divergence) also coincide with the classical rate-distortion function with side information \(R(D)\). This is because, when the distortion constraint is tight, the optimal coding scheme of the classical Shannon rate-distortion problem is able to satisfy the constraint on the distribution-wise semantic distance. Therefore, there is no need to require extra rate to meet the constraints on statistical divergence. In Figure 2a, we plot the curves of semantic rate-distortion functions using the results of Theorems 1 and 2. We observe a three-way tradeoff among transmission rate, bit-wise distortion, and distribution-wise semantic distance. Specifically, given the distortion level, an increase in transmission rate leads to higher semantic fidelity, i.e., smaller semantic distance. Similarly, given the transmission rate, an increase in semantic fidelity leads to lower bit-wise quality, i.e., higher MSE distortion. This observation is aligned with the existing rate-distortion-perception tradeoff [13]. We further observe that, for a given semantic distance level and MSE distortion level, the semantic rate-distortion function under KL divergence consistently exceeds that under the Wasserstein distance. This suggests that KL divergence imposes a stricter measure of statistical difference than the Wasserstein distance for Gaussian sources. However, the semantic rate-distortion function can only capture one aspect of the difference between these two divergence metrics. A more comprehensive discussion is made in the experiment section by investigating the task performance under different divergence metrics for practical sources of real-world datasets. Figure 2b illustrates the semantic rate-distortion curves under varying side information conditions. The parameter \(\eta\) represents the covariance between \(S\) and \(Y\), serving as a measure of their statistical dependence. A higher \(\eta\) indicates a stronger correlation between the semantic source \(S\) and the side information \(Y\), while \(\eta=0\) corresponds to complete independence between them. As shown in Figure 2b, the semantic rate-distortion function varies significantly with \(\eta\).