On the Impact of Dimension Reduction on Graphical Structures

Statisticians and quantitative neuroscientists have actively promoted the use of independence relationships for investigating brain networks, genomic networks, and other measurement technologies. Estimation of these graphs depends on two steps. First is a feature extraction by summarizing measurements within a parcellation, regional or set definition to create nodes. Secondly, these summaries are then used to create a graph representing relationships of interest. In this manuscript we study the impact of dimension reduction on graphs that describe different notions of relations among a set of random variables. We are particularly interested in undirected graphs that capture the random variables' independence and conditional independence relations. A dimension reduction procedure can be any mapping from high dimensional spaces to low dimensional spaces. We exploit a general framework for modeling the raw data and advocate that in estimating the undirected graphs, any acceptable dimension reduction procedure should be a graph-homotopic mapping, i.e., the graphical structure of the data after dimension reduction should inherit the main characteristics of the graphical structure of the raw data. We show that, in terms of inferring undirected graphs that characterize the conditional independence relations among random variables, many dimension reduction procedures, such as the mean, median, or principal components, cannot be theoretically guaranteed to be a graph-homotopic mapping. The implications of this work are broad. In the most charitable setting for researchers, where the correct node definition is known, graphical relationships can be contaminated merely via the dimension reduction. The manuscript ends with a concrete example, characterizing a subset of graphical structures such that the dimension reduction procedure using the principal components can be a graph-homotopic mapping.