A Copula Graphical Model for Multi-Attribute Data using Optimal
  Transport
Qi Zhang Bing Li Lingzhou Xue
Motivated by modern data forms such as images and multi-view data, the
multi-attribute graphical model aims to explore the conditional independence
structure among vectors. Under the Gaussian assumption, the conditional
independence between vectors is characterized by blockwise zeros in the
precision matrix. To relax the restrictive Gaussian assumption, in this paper,
we introduce a novel semiparametric multi-attribute graphical model based on a
new copula named Cyclically Monotone Copula. This new copula treats the
distribution of the node vectors as multivariate marginals and transforms them
into Gaussian distributions based on the optimal transport theory. Since the
model allows the node vectors to have arbitrary continuous distributions, it is
more flexible than the classical Gaussian copula method that performs
coordinatewise Gaussianization. We establish the concentration inequalities of
the estimated covariance matrices and provide sufficient conditions for
selection consistency of the group graphical lasso estimator. For the setting
with high-dimensional attributes, a {Projected Cyclically Monotone Copula}
model is proposed to address the curse of dimensionality issue that arises from
solving high-dimensional optimal transport problems. Numerical results based on
synthetic and real data show the efficiency and flexibility of our methods.