The question of when zeros (i.e., sparsity) in a positive definite matrix $A$
are preserved in its Cholesky decomposition, and vice versa, was addressed by
Paulsen et al. in the Journal of Functional Analysis (85, pp151-178). In
particular, they prove that for the pattern of zeros in $A$ to be retained in
the Cholesky decomposition of $A$, the pattern of zeros in $A$ has to
necessarily correspond to a chordal (or decomposable) graph associated with a
specific type of vertex ordering.

This paper treats the problem of screening a p-variate sample for strongly
and multiply connected vertices in the partial correlation graph associated
with the the partial correlation matrix of the sample. This problem, called hub
screening, is important in many applications ranging from network security to
computational biology to finance to social networks. In the area of network
security, a node that becomes a hub of high correlation with neighboring nodes
might signal anomalous activity such as a coordinated flooding attack.

Positive definite (p.d.) matrices arise naturally in many areas within
mathematics and also feature extensively in scientific applications. In modern
high-dimensional applications, a common approach to finding sparse positive
definite matrices is to threshold their small off-diagonal elements. This
thresholding, sometimes referred to as hard-thresholding, sets small elements
to zero. Thresholding has the attractive property that the resulting matrices
are sparse, and are thus easier to interpret and work with.

Gaussian covariance graph models encode marginal independence among the
components of a multivariate random vector by means of a graph $G$.

Standard statistical techniques often require transforming data to have mean
0 and standard deviation 1. Typically, this process of "standardization" or
"normalization" is applied across subjects when each subject produces a single
number. High throughput genomic and financial data often come as rectangular
arrays where each coordinate in one direction concerns subjects who might have
different status (case or control, say), and each coordinate in the other
designates "outcome" for a specific feature, for example, "gene," "polymorphic
site" or some aspect of financial profile.