  An important problem in large scale inference is the identification of
variables that have large correlations or partial correlations. Recent work has
yielded breakthroughs in the ultra-high dimensional setting when the sample
size $n$ is fixed and the dimension $p \rightarrow \infty$ ([Hero, Rajaratnam
2011, 2012]). Despite these advances, the correlation screening framework
suffers from some serious practical, methodological and theoretical
deficiencies. For instance, theoretical safeguards for partial correlation
screening requires that the population covariance matrix be block diagonal.
This block sparsity assumption is however highly restrictive in numerous
practical applications. As a second example, results for correlation and
partial correlation screening framework requires the estimation of dependence
measures or functionals, which can be highly prohibitive computationally. In
this paper, we propose a unifying approach to correlation and partial
correlation mining which specifically goes beyond the block diagonal
correlation structure, thus yielding a methodology that is suitable for modern
applications. By making connections to random geometric graphs, the number of
highly correlated or partial correlated variables are shown to have novel
compound Poisson finite-sample characterizations, which hold for both the
finite $p$ case and when $p \rightarrow \infty$. The unifying framework also
demonstrates an important duality between correlation and partial correlation
screening with important theoretical and practical consequences.
