  We propose communication-efficient distributed estimation and inference
methods for the transelliptical graphical model, a semiparametric extension of
the elliptical distribution in the high dimensional regime. In detail, the
proposed method distributes the $d$-dimensional data of size $N$ generated from
a transelliptical graphical model into $m$ worker machines, and estimates the
latent precision matrix on each worker machine based on the data of size
$n=N/m$. It then debiases the local estimators on the worker machines and send
them back to the master machine. Finally, on the master machine, it aggregates
the debiased local estimators by averaging and hard thresholding. We show that
the aggregated estimator attains the same statistical rate as the centralized
estimator based on all the data, provided that the number of machines satisfies
$m \lesssim \min\{N\log d/d,\sqrt{N/(s^2\log d)}\}$, where $s$ is the maximum
number of nonzero entries in each column of the latent precision matrix. It is
worth noting that our algorithm and theory can be directly applied to Gaussian
graphical models, Gaussian copula graphical models and elliptical graphical
models, since they are all special cases of transelliptical graphical models.
Thorough experiments on synthetic data back up our theory.
