The Rate-Distortion-Perception Trade-off: The Role of Private Randomness

In image compression, with recent advances in generative modeling, the existence of a trade-off between the rate and the perceptual quality (realism) has been brought to light, where the realism is measured by the closeness of the output distribution to the source. It has been shown that randomized codes can be strictly better under a number of formulations. In particular, the role of common randomness has been well studied. We elucidate the role of private randomness in the compression of a memoryless source $X^n=(X_1,...,X_n)$ under two kinds of realism constraints. The near-perfect realism constraint requires the joint distribution of output symbols $(Y_1,...,Y_n)$ to be arbitrarily close the distribution of the source in total variation distance (TVD). The per-symbol near-perfect realism constraint requires that the TVD between the distribution of output symbol $Y_t$ and the source distribution be arbitrarily small, uniformly in the index $t.$ We characterize the corresponding asymptotic rate-distortion trade-off and show that encoder private randomness is not useful if the compression rate is lower than the entropy of the source, however limited the resources in terms of common randomness and decoder private randomness may be.