## 3 Information-Theoretic Results
### The Rate-Distortion-Perception Frontier  
The rate-distortion function in the traditional (additive) distortion setting has been successful in characterizing the minimum rate required to describe a source within a specifieddistortion. For a (not necessarily finite) set of distortion measures \(\{\Delta_{\lambda}(\cdot,\cdot)\}_{\lambda\in\Lambda}\), it is shown in [49] (Exercise 7.14, Page 117) that the rate-distortion function is  
\[R(\{D_{\lambda}\})=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad\text{s.t.}\qquad\mathbb{ E}[\Delta_{\lambda}(X,\hat{X})]\leq D_{\lambda}\qquad\forall\lambda\in\Lambda, \tag{17}\]  
assuming that \(\inf_{D_{\lambda}\
eq 0}D_{\lambda}>0\). The information rate-distortion-perception (I-RDP) function was first proposed in analogy to the traditional RD function within the machine learning community. **Definition 6** (I-Rdp). : _Given a random variable \(X\) that represents the source distribution, the information rate-distortion-perception function [4] is defined as_  
\[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]  
When the perceptual constraint **(P)** is relaxed, i.e., when \(P=\infty\), the RDP function reduces to the classic RD function \(R(D,\infty)=R(D)\). Efficient computation of the I-RDP function has been explored in [50; 51; 52]. Based on the I-RDP function, ref. [8] demonstrated that the cost of achieving perfect realism is a doubling of the lowest achievable MSE distortion, i.e.,  
\[R^{(I)}(D,0)\leq R^{(I)}(\frac{1}{2}D,\infty).\]  
Specifically, [53] considered the MSE distortion measure and the 2-Wasserstein distance (see Figure 1b), showing that the distortion-perception function using these measures is quadratic. In [47], the authors first established that the I-RDP function provides a lower bound on the one-shot achievable rate using the Poisson functional representation lemma [48]. Let \(R^{*}(D,P)\) denote the one-shot RDP function. The study in [9] subsequently established the relationship between the information function and the operational functions using the same strategy, resulting in the following upper and lower bounds:  
\[R^{(I)}(D,P)\leq R^{*}(D,P)\leq R^{(I)}(D,P)+\log(1+R^{(I)}(D,P))+5.\]  
In the asymptotic setting, the rate region of the general system illustrated in Figure 2 is given by \(\mathcal{R}_{D,0}\) as presented in [27]:  
\[\mathcal{R}_{D,0}=\{(R,R_{c},R_{d},D):\ \exists\ P_{XYY} \in\mathcal{A}\quad\text{s.t.}\] \[R\geq I_{P}(X;V)\] \[R+R_{c} \geq I_{P}(Y;V)\] \[R_{d} \geq H_{P}(Y|V)\] \[D\geq\mathbb{E}_{P}[\delta(X,Y)]\},\] where \[\mathcal{A}:=\{P_{XVY}:\ X \sim P_{X},\ Y\sim\ P_{X}\] \[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]0 \[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]1  
The RDP function can be derived from the achievability and converse of the coding theorems. **Corollary 1**. : _When common randomness and decoder randomness are unlimited, the RDP function induced by Equation (20) is given by the following:_  
\[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]2  
This result is obtained by setting \(R_{c}=R_{d}=\infty\). Consequently, the rate region becomes  
\[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]3 \[R^{(I)}(D,P):=\inf_{P_{\hat{X}|X}}I(X;\hat{X})\qquad s.t.\qquad\mathbb{E}[ \Delta(X,\hat{X})]\leq D,\quad\mathcal{D}(P_{X},P_{\hat{X}})\leq P. \tag{18}\]4  
where the equality in Equation (20) follows from the data processing inequality \(I_{P}(X;V)\geq I_{P}(X;Y)\), given that \(X\perp Y\mid V\) in \(\mathcal{A}\).