Tree Structured Synthesis of Gaussian Trees

TL;DR

This paper introduces a layered synthesis scheme for generating Gaussian tree-structured random vectors with prescribed joint densities, optimizing the use of random sources and characterizing the minimal bit rates needed.

Contribution

It presents a novel layered encoding method for Gaussian trees, utilizing minimal common randomness and characterizing the achievable rate region for synthesis.

Findings

Achieves effective synthesis with minimal randomness

Characterizes the rate region for multi-layer Gaussian trees

Quantifies the bits needed for accurate joint density simulation

Abstract

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.