A copula graphical model for multi-attribute data using optimal transport

Abstract Motivated by modern data types such as images and multi-view data, the multi-attribute graphical model aims to uncover conditional independence structures among vector-valued nodes. Under the Gaussian assumption, such independence is encoded in blockwise zeros of the precision matrix. To relax the restrictive Gaussian assumption, we propose a semiparametric multi-attribute graphical model leveraging a newly introduced cyclically monotone copula. This copula treats the distribution of node vectors as multivariate marginals and transforms them into Gaussian distributions using optimal transport. Since our approach supports arbitrary continuous distributions over node vectors, it is significantly more flexible than existing copula Gaussian graphical models that only perform coordinatewise Gaussianization. We establish concentration inequalities for the estimated covariance matrices and provide sufficient conditions for the selection consistency of the group graphical lasso estimator. To address the curse of dimensionality when handling high-dimensional attributes, we further introduce a projected cyclically monotone copula model. Numerical experiments on both synthetic and real-world datasets demonstrate the effectiveness and flexibility of our methods.