  A new synthesis scheme is proposed to effectively generate a random vector
with prescribed joint density that induces a (latent) Gaussian tree structure.
The quality of synthesis is measured by total variation distance between the
synthesized and desired statistics. The proposed layered and successive
encoding scheme relies on the learned structure of tree to use minimal 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 simulate 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.
