A Constrained Talagrand Transportation Inequality with Applications to Rate-Distortion-Perception Theory

A constrained version of Talagrand's transportation inequality is established, which reveals an intrinsic connection between the Gaussian distortion-rate-perception functions with limited common randomness under the Kullback-Leibler divergence-based and squared Wasserstein-2 distance-based perception measures. This connection provides an organizational framework for assessing existing bounds on these functions. In particular, we show that the best-known bounds of Xie et al. are nonredundant when examined through this connection.