  A new synthesis scheme is proposed to generate a random vector with
prescribed joint density that induces a (latent) Gaussian tree structure. The
quality of synthesis is shown by vanishing total variation distance between the
synthesized and desired statistics. The proposed layered and successive
synthesis scheme relies on the learned structure of tree to use sufficient
number of common random variables to synthesize the desired density. We
characterize the achievable rate region for the rate tuples of multi-layer
latent Gaussian tree, through which the number of bits needed to synthesize
such Gaussian joint density are determined. The random sources used in our
algorithm are the latent variables at the top layer of tree, the additive
independent Gaussian noises, and the Bernoulli sign inputs that capture the
ambiguity of correlation signs between the variables. We have shown that such
ambiguity can further help in reducing the synthesis rates for the underlying
Gaussian trees.
