A Copula Graphical Model for Multi-Attribute Data using Optimal Transport

Introducing the Cyclically Monotone Copula for Semiparametric Multi-Attribute Graphical Model

Overview

The development of graphical models for multivariate data has largely concentrated on Gaussian Graphical Models (GGMs) due to the elegant mathematical properties of normal distributions. However, reality often deviates from normality, particularly in the case of complex data structures inherent in modern applications, such as gene expression profiles or color image data. Addressing this challenge, this paper introduces a novel semiparametric model named the Cyclically Monotone Copula Gaussian Graphical Model (CMC-GGM), designed for multi-attribute data that relaxes the Gaussian assumption by incorporating a new form of copula based on cyclically monotone maps derived from optimal transport theory. This approach is more flexible than conventional coordinatewise Gaussianization, allowing for arbitrary continuous distributions of node vectors.

Semiparametric Multi-Attribute Graphical Model

The classical approach to graphical models assumes scalar variables on nodes, limiting its application to complex multi-attribute or vector-valued data. The CMC-GGM extends current methodologies by considering nodes represented by vectors, encoding the conditional dependence structure among these vectors. Traditional Gaussian copula methods transform each vector's coordinates to Gaussian, assuming the entire vector joint distribution becomes Gaussian. The paper argues this assumption is too restrictive for multi-attribute settings, motivating the need for a model that operates on the distribution of node vectors as multivariate marginals.

Cyclically Monotone Copula

The core innovation in this paper is the introduction of the Cyclically Monotone Copula that operates based on cyclically monotone functions guided by optimal transport theory. This copula transforms node vector distributions into Gaussian distributions, assuming optimal transport maps make the transformed entire vector joint Gaussian. The key strength of this copula lies in its flexibility; it allows for arbitrary continuous distributions, including non-Gaussian joint distributions among vectors, thus accommodating more complex dependencies than possible with traditional copula approaches.

Technical Contribution and Theoretical Implications

The paper rigorously establishes the concentration inequalities of estimated covariance matrices and sufficient conditions for selection consistency of the group graphical lasso estimator. For high-dimensional attribute settings, it proposes a Projected Cyclically Monotone Copula Model (PCMC-GGM) addressing the curse of dimensionality in solving high-dimensional optimal transport problems. These contributions are significant as they provide a robust theoretical foundation for the practical application of the proposed model, ensuring consistent estimation and model selection in various settings.

Practical Applications and Future Directions

Utilizing synthetic and real data, the numerical results demonstrate the proposed methods' efficiency and flexibility. The application to gene and protein regulatory networks and color image graphs exemplifies the model's capability to uncover complex dependence structures in multi-attribute data, highlighting its potential impact on fields requiring in-depth analysis of multi-dimensional data.

Speculatively, the study opens pathways to further research, particularly in optimizing the computation of cyclically monotone maps for larger datasets and exploring the extension of this copula approach to other types of graphical models. Moreover, the potential for integrating this model with machine learning frameworks offers exciting prospects for enhancing model performance in predictive analytics and understanding complex data structures in high-dimensional settings.

In sum, the Cyclically Monotone Copula Gaussian Graphical Model represents a significant advancement in statistical methodology for graphical models, offering a flexible and theoretically sound approach to analyzing multi-attribute data.