Partial correlation graphs and the neighborhood lattice

We define and study partial correlation graphs (PCGs) with variables in a general Hilbert space and their connections to generalized neighborhood regression, without making any distributional assumptions. Using operator-theoretic arguments, and especially the properties of projection operators on Hilbert spaces, we show that these neighborhood regressions have the algebraic structure of a lattice, which we call a neighborhood lattice. This lattice property significantly reduces the number of conditions one has to check when testing all partial correlation relations among a collection of variables. In addition, we generalize the notion of perfectness in graphical models for a general PCG to this Hilbert space setting, and establish that almost all Gram matrices are perfect. Under this perfectness assumption, we show how these neighborhood lattices may be "graphically" computed using separation properties of PCGs. We also discuss extensions of these ideas to directed models, which present unique challenges compared to their undirected counterparts. Our results have implications for multivariate statistical learning in general, including structural equation models, subspace clustering, and dimension reduction. For example, we discuss how to compute neighborhood lattices efficiently and furthermore how they can be used to reduce the sample complexity of learning directed acyclic graphs. Our work demonstrates that this abstract viewpoint via projection operators significantly simplifies existing ideas and arguments from the graphical modeling literature, and furthermore can be used to extend these ideas to more general nonparametric settings.