  This paper studies the decentralized learning of tree-structured Gaussian
graphical models (GGMs) from noisy data. In decentralized learning, data set is
distributed across different machines (sensors), and GGMs are widely used to
model complex networks such as gene regulatory networks and social networks.
The proposed decentralized learning uses the Chow-Liu algorithm for estimating
the tree-structured GGM.
  In previous works, upper bounds on the probability of incorrect tree
structure recovery were given mostly without any practical noise for
simplification. While this paper investigates the effects of three common types
of noisy channels: Gaussian, Erasure, and binary symmetric channel. For
Gaussian channel case, to satisfy the failure probability upper bound $\delta >
0$ in recovering a $d$-node tree structure, our proposed theorem requires only
$\mathcal{O}(\log(\frac{d}{\delta}))$ samples for the smallest sample size
($n$) comparing to the previous literature \cite{Nikolakakis} with
$\mathcal{O}(\log^4(\frac{d}{\delta}))$ samples by using the positive
correlation coefficient assumption that is used in some important works in the
literature. Moreover, the approximately bounded Gaussian random variable
assumption does not appear in \cite{Nikolakakis}. Given some knowledge about
the tree structure, the proposed Algorithmic Bound will achieve obviously
better performance with small sample size (e.g., $< 2000$) comparing with
formulaic bounds. Finally, we validate our theoretical results by performing
simulations on synthetic data sets.
