# On the Rate-Distortion Theory for Task-Specific Semantic Communication
## 3 System Model and Problem Formulation  
### System Model  
We adopt the system model of semantic communication introduced in [2; 5], where the primary objective is to effectively convey implicit semantic information to end users by leveraging background knowledge, as depicted in Figure 1. Formally, let the semantic source signal \(S^{n}\) represent an \(n\)-length i.i.d. random variable containing intrinsic information that remains unobservable to the encoder. The encoder accesses only a limited set of \(k\) indirect observations \(U^{k}\) about the semantic source, which may comprise noisy samples or partial observations. The relationship between the source and the observations is characterized by the conditional probability \(p_{U|S}\) of obtaining observation \(U\) given source \(S\). Due to the limited channel capacity, the encoder produces an encoded signal sequence \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Divergence** & **Source** & \(R(D,P)\) & **SI** & **Work** \\ \hline \multirow{4}{*}{TV} & \multirow{4}{*}{Bernoulli} & \(H_{b}(\pi)-H_{b}(D)\), & \(D\in[0,D_{1})\) & \multirow{4}{*}{✗} & \multirow{4}{*}{[13]} \\  & & \(2H_{b}(\pi)+H_{b}(\pi-P)-H_{t}(\frac{D-P}{2},\pi)\) & & \\ \cline{1-1} \cline{5-5}  & & \(-H_{t}(\frac{D+P}{2},1-\pi)\), & \(D\in[D_{1},D_{2})\) & & \\ \cline{1-1} \cline{5-5}  & & \(D\in[D_{2},\infty]\) & & \\ \hline \multirow{4}{*}{WD} & \multirow{4}{*}{Gaussian} & \(\begin{cases}\frac{1}{2}\log\frac{1}{1-\left(\frac{(-\sqrt{9}2,1-D)}{1-2\sqrt{ 9}}\right)^{2}},&\sqrt{P}\leq 1-\sqrt{|1-D|}\\ \max\{\frac{1}{2}\log\frac{1}{D},0\},&\sqrt{P}>1-\sqrt{|1-D|}\end{cases}\) & & \\ \cline{1-1} \cline{5-5}  & & \(\text{Equations}\) (15)–(17) & & \(\check{\check{\mathcal{V}}}\) & Proposed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A summary of information-theoretic results in RDP theory. For Bernoulli case, the source follows a Bernoulli distribution with parameter \(\pi\). For Gaussian case, the source follows a Gaussian distribution \(\mathcal{N}(0,1)\). Here \(D_{1}=\frac{\check{P}}{1-2\pi-2\check{P}}\), \(D_{2}=2\pi(1-\pi)-(1-2\pi)P\). \(X^{m}\) with its length being \(m\). The decoder then receives a potentially corrupted version \(\hat{X}^{m}\) of this signal through a discrete memoryless channel modeled by \(p_{\hat{X}|X}\). Moreover, we assume that both the encoder and the decoder have access to side information, denoted as \(Y\). This may correspond to some background knowledge shared between the source and the destination. For example, in human communication scenarios [5], different users may employ different languages or word sequences to convey the same semantic concept. In such cases, the side information may include the user's language background and lexical preferences when transforming the semantic information source into observable signals. The main objective of the decoder is to infer the implicit semantic source \(\hat{S}\) with a certain level of signal fidelity, based on the signal received from the channel and the available side information. It can be observed that the above model is similar to the indirect source coding problem with side information in the classic information theory setup. In this paper, we extend the above problem to a more general scenario in which, in addition to MSE distortion, we also consider the impact of task-specific semantic distances on the minimum rate required to recover the implicit signal source. We define the semantic distance as a much broader range of signal fidelity measures, including not only classical distribution divergences such as KL and Wasserstein distance but also the state-of-the-art perceptual quality measures such as the Inception score and SSIM. We should clarify that the semantic source \(S\) and its recovery \(\hat{S}\) can indeed take traditional forms whether discrete or continuous.