text: abstract
  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.