Layered Synthesis of Latent Gaussian Trees

TL;DR

This paper introduces a layered synthesis method for generating Gaussian tree-structured data with prescribed joint densities, utilizing learned structures and random sources to optimize synthesis rates.

Contribution

It presents a novel layered synthesis scheme for latent Gaussian trees, characterizes the achievable rate region, and shows how sign ambiguity can reduce synthesis rates.

Findings

Achieves vanishing total variation distance between synthesized and target distributions.

Characterizes the rate region for multi-layer latent Gaussian trees.

Demonstrates that sign ambiguity reduces synthesis rates.

Abstract

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.