Computation and Critical Transitions of Rate-Distortion-Perception Functions With Wasserstein Barycenter

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.