  This paper treats the problem of screening for variables with high
correlations in high dimensional data in which there can be many fewer samples
than variables. We focus on threshold-based correlation screening methods for
three related applications: screening for variables with large correlations
within a single treatment (autocorrelation screening); screening for variables
with large cross-correlations over two treatments (cross-correlation
screening); screening for variables that have persistently large
auto-correlations over two treatments (persistent-correlation screening). The
novelty of correlation screening is that it identifies a smaller number of
variables which are highly correlated with others, as compared to identifying a
number of correlation parameters. Correlation screening suffers from a phase
transition phenomenon: as the correlation threshold decreases the number of
discoveries increases abruptly. We obtain asymptotic expressions for the mean
number of discoveries and the phase transition thresholds as a function of the
number of samples, the number of variables, and the joint sample distribution.
We also show that under a weak dependency condition the number of discoveries
is dominated by a Poisson random variable giving an asymptotic expression for
the false positive rate. The correlation screening approach bears tremendous
dividends in terms of the type and strength of the asymptotic results that can
be obtained. It also overcomes some of the major hurdles faced by existing
methods in the literature as correlation screening is naturally scalable to
high dimension. Numerical results strongly validate the theory that is
presented in this paper. We illustrate the application of the correlation
screening methodology on a large scale gene-expression dataset, revealing a few
influential variables that exhibit a significant amount of correlation over
multiple treatments.
