  A key challenge to performing effective analyses of high-dimensional data is
finding a signal-rich, low-dimensional representation. For linear subspaces,
this is generally performed by decomposing a design matrix (via eigenvalue or
singular value decomposition) into orthogonal components, and then retaining
those components with sufficient variations. This is equivalent to estimating
the rank of the matrix and deciding which components to retain is generally
carried out using heuristic or ad-hoc approaches such as plotting the
decreasing sequence of the eigenvalues and looking for the "elbow" in the plot.
While these approaches have been shown to be effective, a poorly calibrated or
misjudged elbow location can result in an overabundance of noise or an
under-abundance of signal in the low-dimensional representation, making
subsequent modeling difficult. In this article, we propose a
latent-space-construction procedure to estimate the rank of the detectable
signal space of a matrix by retaining components whose variations are
significantly greater than random matrices, of which eigenvalues follow a
universal March\u{e}nko-Pastur (MP) distribution.
