Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

A new layered encoding scheme is proposed to effectively generate a random vector with prescribed joint density that induces a latent Gaussian tree structure. The encoding algorithm relies on the learned structure of tree to use minimal number of common random variables to synthesize the desired density, which we argue such algorithm is also computationally efficient. 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 along with Bernoulli sign inputs, which capture the correlation signs between the variables. In latent Gaussian trees the pairwise correlation signs between the variables are intrinsically unrecoverable. Such information is vital since it completely determines the direction in which two variables are associated. As a by-product of determining the achievable rate region, we quantify the amount of information loss due to unrecoverable sign information. It is shown that maximizing the achievable rate-region is equivalent to finding the worst case density for Bernoulli sign inputs where maximum amount of sign information is lost.