  The information rate-distortion-perception (RDP) function characterizes the
three-way trade-off between description rate, average distortion, and
perceptual quality measured by discrepancy between probability distributions.
We study several variants of the RDP functions through the lens of optimal
transport. By transforming the information RDP function into a Wasserstein
Barycenter problem, we identify the critical transitions when one of the
constraints becomes inactive and demonstrate several critical transition
properties of the RDP variants. Further, the non-strictly convexity brought by
the perceptual constraint can be regularized by an entropy regularization term.
We prove that the entropy regularized model converges to the original problem
and propose an alternating iteration method based on the Sinkhorn algorithm to
numerically solve the regularized optimization problem. Experimental results
demonstrate the effectiveness and accuracy of the proposed algorithms. As a
practical application of our theory, we incorporate our numerical method into a
reverse data hiding problem, where a secret message is imperceptibly embedded
into the image with guarantees of the perceptual fidelity.
