We consider a source coding problem with a network
scenario in mind, and formulate it as a remote vector
Gaussian Wyner-Ziv problem under covariance matrix distortions.
We define a notion of minimum for two positive-definite
matrices based on which we derive an explicit formula for the
rate-distortion function (RDF). We then study the special cases
and applications of this result. We show that two well-studied
source coding problems, i.e. remote vector Gaussian Wyner-
Ziv problems with mean-squared error and mutual information
constraints are in fact special cases of our results. Finally,
we apply our results to a joint source coding and denoising
problem. We consider a network with a centralized topology and
a given weighted sum-rate constraint, where the received signals
at the center are to be fused to maximize the output SNR while
enforcing no linear distortion. We show that one can design the
distortion matrices at the nodes in order to maximize the output
SNR at the fusion center. We thereby bridge between denoising
and source coding within this setup.