In this paper, we study the computation of the rate-distortion-perception function (RDPF) for a multivariate Gaussian source under mean squared error (MSE) distortion and, respectively, Kullback-Leibler divergence, geometric Jensen-Shannon divergence, squared Hellinger distance, and squared Wasserstein-2 distance perception metrics. To this end, we first characterize the analytical bounds of the scalar Gaussian RDPF for the aforementioned divergence functions, also providing the RDPF-achieving forward "test-channel" realization. Focusing on the multivariate case, we establish that, for tensorizable distortion and perception metrics, the optimal solution resides on the vector space spanned by the eigenvector of the source covariance matrix. Consequently, the multivariate optimization problem can be expressed as a function of the scalar Gaussian RDPFs of the source marginals, constrained by global distortion and perception levels. Leveraging this characterization, we design an alternating minimization scheme based on the block nonlinear Gauss-Seidel method, which optimally solves the problem while identifying the Gaussian RDPF-achieving realization. Furthermore, the associated algorithmic embodiment is provided, as well as the convergence and the rate of convergence characterization. Lastly, for the "perfect realism" regime, the analytical solution for the multivariate Gaussian RDPF is obtained. We corroborate our results with numerical simulations and draw connections to existing results.

翻译：本文研究了多元高斯源在均方误差（MSE）失真以及分别采用Kullback-Leibler散度、几何Jensen-Shannon散度、平方Hellinger距离和平方Wasserstein-2距离感知度量下的率-失真-感知函数（RDPF）计算问题。为此，我们首先刻画了上述散度函数下标量高斯RDPF的解析界限，并给出了实现RDPF的前向"测试信道"结构。聚焦于多元情形，我们证明：对于可张量化的失真与感知度量，最优解位于源协方差矩阵特征向量张成的向量空间中。由此，多元优化问题可表示为源边缘分布标量高斯RDPF的函数，并受全局失真与感知水平的约束。基于这一表征，我们设计了基于块非线性Gauss-Seidel方法的交替最小化方案，该方案在实现高斯RDPF最优解的同时，可识别其结构实现。此外，我们提供了相应的算法实现，并刻画了收敛性及其收敛速率。最后，针对"完美现实主义"情形，给出了多元高斯RDPF的解析解。我们通过数值仿真验证了理论结果，并与现有成果建立了关联。