  This paper studies the rate-distortion-perception (RDP) tradeoff for a
Gaussian vector source coding problem where the goal is to compress the
multi-component source subject to distortion and perception constraints. The
purpose of imposing a perception constraint is to ensure visually pleasing
reconstructions. This paper studies this RDP setting with either the
Kullback-Leibler (KL) divergence or Wasserstein-2 metric as the perception loss
function, and shows that for Gaussian vector sources, jointly Gaussian
reconstructions are optimal. We further demonstrate that the optimal tradeoff
can be expressed as an optimization problem, which can be explicitly solved. An
interesting property of the optimal solution is as follows. Without the
perception constraint, the traditional reverse water-filling solution for
characterizing the rate-distortion (RD) tradeoff of a Gaussian vector source
states that the optimal rate allocated to each component depends on a constant,
called the water-level. If the variance of a specific component is below the
water-level, it is assigned a {zero} compression rate. However, with active
distortion and perception constraints, we show that the optimal rates allocated
to the different components are always {positive}. Moreover, the water-levels
that determine the optimal rate allocation for different components are
unequal. We further treat the special case of perceptually perfect
reconstruction and study its RDP function in the high-distortion and
low-distortion regimes to obtain insight to the structure of the optimal
solution.
