  Consider a linear model $Y=X\beta+z$, $z\sim N(0,I_n)$. Here, $X=X_{n,p}$,
where both $p$ and $n$ are large, but $p>n$. We model the rows of $X$ as i.i.d.
samples from $N(0,\frac{1}{n}\Omega)$, where $\Omega$ is a $p\times p$
correlation matrix, which is unknown to us but is presumably sparse. The vector
$\beta$ is also unknown but has relatively few nonzero coordinates, and we are
interested in identifying these nonzeros. We propose the Univariate
Penalization Screeing (UPS) for variable selection. This is a screen and clean
method where we screen with univariate thresholding and clean with penalized
MLE. It has two important properties: sure screening and separable after
screening. These properties enable us to reduce the original regression problem
to many small-size regression problems that can be fitted separately. The UPS
is effective both in theory and in computation.
