## 2 Distortion and Perceptual Measures
### Interplay Between Distortion and Perception Constraints  
Distortion and perception measures, though distinct, are closely intertwined through a series of distance-divergence inequalities. For instance, in the product space \(\mathcal{X}^{n}\), the transportation-cost inequalities pioneered by [PERSON] [34] can be employed to derive concentration results.  
A notable bound relates the expected values of bounded functions to the TV distance:  
\[|\mathrm{E}_{P_{X}}[f(X)]-\mathrm{E}_{P_{Y}}[f(X)]|\leq\sup_{x}|f(x)|\; \mathcal{D}_{\text{TV}}(P_{X},P_{Y}), \tag{9}\]  
which facilitates the analysis of expected values using the TV distance. This relationship is particularly useful in machine learning scenarios where the objective is to minimize loss functions that depend on expected values. In such cases, the optimization problem can be reformulated as minimizing the TV distance. Specifically, given that the distortion is bounded by \(d_{\text{max}}\), and assuming that \(P_{X^{n}Y^{n}}\) and \(Q_{X^{n}Y^{n}}\) are i.i.d. distributions, we have the following:  
\[|\mathrm{E}_{\mathbb{P}}[D(X^{n},Y^{n})]-\mathrm{E}_{\mathbb{Q}}[D(X^{n},Y^{n} )]|\leq d_{\text{max}}\mathcal{D}_{\text{TV}}(P,Q). \tag{10}\]  
Similarly, the bounded distortion measure exhibits a property elucidated in Lemma 5 of [36] with respect to the TV distance: if \(P_{XY},Q_{XY}\) are probability mass functions with \(\mathrm{E}_{P_{XY}}[\delta(X,Y)]=A\) and \(\mathcal{D}_{\text{TV}}(P_{XY},Q_{XY})<\epsilon\), then  
\[\mathrm{E}_{Q_{XY}}[\delta(X,Y)]\leq A+\epsilon d_{\text{max}}. \tag{11}\]Perceptual measures can also be related through a set of distance-divergence inequalities. A fundamental relationship between the KL divergence and the TV distance, originally due to [PERSON] [37], states the following:  
\[\mathcal{D}_{\text{TV}}(P_{X},P_{Y})\leq\sqrt{\frac{1}{2}\mathcal{D}_{\text{KL} }(P_{X},P_{Y})}. \tag{12}\]  
For a finite alphabet, the reverse Pinsker inequality is available (see Lemma 6.3 in [38]):  
\[\mathcal{D}_{\text{KL}}(P_{X},P_{Y})\leq\frac{\mathcal{D}_{\text{TV}}^{2}(P_{ X},P_{Y})}{\min_{y\in\mathcal{X}}P_{Y}(y)}. \tag{13}\]  
Moreover, for i.i.d. distributions \(P_{Y^{n}}\) and \(P_{X^{n}}\ll P_{Y^{n}}\), we have (see Equation (30) in [39])  
\[\mathcal{D}_{\text{KL}}(P_{X^{n}},P_{Y^{n}})=O((n-\log(\mathcal{D}_{\text{TV} }(P_{X^{n}},P_{Y^{n}})))\mathcal{D}_{\text{TV}}(P_{X^{n}},P_{Y^{n}})). \tag{14}\]  
Assuming that the diameter of \(\mathcal{X}\) is bounded by \(B\), we also have (see Theorem 6.15 in [40]) the following:  
\[W_{1}(P_{X},P_{Y})\leq B\cdot\mathcal{D}_{\text{TV}}(P_{X},P_{Y}). \tag{15}\]  
For coding blocks on \(\text{Prob}(\mathcal{X}^{n})\), Wasserstein distances in product spaces can be naturally related to the distortion measure:  
\[\Delta_{\beta,n}(P_{X^{n}},P_{Y^{n}}):=\inf_{\pi\in\Pi(P_{X^{n}},P_{Y^{n}})} \;(\mathbb{E}_{\pi}[\sum_{i=1}^{n}\delta^{p}(X_{i},Y_{i})])^{\frac{1}{p}},\]  
and, according to [35], under certain conditions,  
\[W_{1,n}(P_{X^{n}},P_{Y^{n}})\leq\text{Const.}\sqrt{\frac{1}{2}\mathcal{D}_{ \text{KL}}(P_{X^{n}},P_{Y^{n}})}. \tag{16}\]  
These relationships connect common perceptual measures to the TV distance, which is frequently considered in information-theoretic settings. The perceptual measures discussed--TV distance, KL divergence, and Wasserstein distance--each offer unique advantages depending on the application context. TV distance provides a straightforward measure of distributional similarity, KL divergence captures directional divergence between distributions, and Wasserstein distance offers a geometrically intuitive measure of the transport cost between probability distributions.