The Rate-Distortion-Perception Trade-off with Side Information

In image compression, with recent advances in generative modeling, the existence of a trade-off between the rate and the perceptual quality has been brought to light, where the perception is measured by the closeness of the output distribution to the source. This leads to the question: how does a perception constraint impact the trade-off between the rate and traditional distortion constraints, typically quantified by a single-letter distortion measure? We consider the compression of a memoryless source $X$ in the presence of memoryless side information $Z,$ studied by Wyner and Ziv, but elucidate the impact of a perfect realism constraint, which requires the output distribution to match the source distribution. We consider two cases: when $Z$ is available only at the decoder or at both the encoder and the decoder. The rate-distortion trade-off with perfect realism is characterized for sources on general alphabets when infinite common randomness is available between the encoder and the decoder. We show that, similarly to traditional source coding with side information, the two cases are equivalent when $X$ and $Z$ are jointly Gaussian under the squared error distortion measure. We also provide a general inner bound in the case of limited common randomness.