  Many applications benefit from theory relevant to the identification of
variables having large correlations or partial correlations in high dimension.
Recently there has been progress in the ultra-high dimensional setting when the
sample size $n$ is fixed and the dimension $p$ tends to infinity. Despite these
advances, the correlation screening framework suffers from practical,
methodological and theoretical deficiencies. For instance, previous correlation
screening theory requires that the population covariance matrix be sparse and
block diagonal. This block sparsity assumption is however restrictive in
practical applications. As a second example, correlation and partial
correlation screening requires the estimation of dependence measures, which can
be computationally prohibitive. In this paper, we propose a unifying approach
to correlation and partial correlation mining that is not restricted to 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
compound Poisson finite-sample characterizations, which hold for both the
finite $p$ case and when $p$ tends to infinity. The unifying framework also
demonstrates a duality between correlation and partial correlation screening
with theoretical and practical consequences.
