# Graph Estimation From Multi-Attribute Data

Mladen Kolar, Han Liu, Eric P. Xing.

Year: 2014, Volume: 15, Issue: 51, Pages: 1713−1750

#### Abstract

Undirected graphical models are important in a number of modern applications that involve exploring or exploiting dependency structures underlying the data. For example, they are often used to explore complex systems where connections between entities are not well understood, such as in functional brain networks or genetic networks. Existing methods for estimating structure of undirected graphical models focus on scenarios where each node represents a scalar random variable, such as a binary neural activation state or a continuous mRNA abundance measurement, even though in many real world problems, nodes can represent multivariate variables with much richer meanings, such as whole images, text documents, or multi-view feature vectors. In this paper, we propose a new principled framework for estimating the structure of undirected graphical models from such multivariate (or multi-attribute) nodal data. The structure of a graph is inferred through estimation of non-zero partial canonical correlation between nodes. Under a Gaussian model, this strategy is equivalent to estimating conditional independencies between random vectors represented by the nodes and it generalizes the classical problem of covariance selection (Dempster, 1972). We relate the problem of estimating non-zero partial canonical correlations to maximizing a penalized Gaussian likelihood objective and develop a method that efficiently maximizes this objective. Extensive simulation studies demonstrate the effectiveness of the method under various conditions. We provide illustrative applications to uncovering gene regulatory networks from gene and protein profiles, and uncovering brain connectivity graph from positron emission tomography data. Finally, we provide sufficient conditions under which the true graphical structure can be recovered correctly.