  In this paper, we study high-dimensional estimation from truncated samples.
We focus on two fundamental and classical problems: (i) inference of sparse
Gaussian graphical models and (ii) support recovery of sparse linear models.
  (i) For Gaussian graphical models, suppose $d$-dimensional samples ${\bf x}$
are generated from a Gaussian $N(\mu,\Sigma)$ and observed only if they belong
to a subset $S \subseteq \mathbb{R}^d$. We show that ${\mu}$ and ${\Sigma}$ can
be estimated with error $\epsilon$ in the Frobenius norm, using
$\tilde{O}\left(\frac{\textrm{nz}({\Sigma}^{-1})}{\epsilon^2}\right)$ samples
from a truncated $\mathcal{N}({\mu},{\Sigma})$ and having access to a
membership oracle for $S$. The set $S$ is assumed to have non-trivial measure
under the unknown distribution but is otherwise arbitrary.
  (ii) For sparse linear regression, suppose samples $({\bf x},y)$ are
generated where $y = {\bf x}^\top{{\Omega}^*} + \mathcal{N}(0,1)$ and $({\bf
x}, y)$ is seen only if $y$ belongs to a truncation set $S \subseteq
\mathbb{R}$. We consider the case that ${\Omega}^*$ is sparse with a support
set of size $k$. Our main result is to establish precise conditions on the
problem dimension $d$, the support size $k$, the number of observations $n$,
and properties of the samples and the truncation that are sufficient to recover
the support of ${\Omega}^*$. Specifically, we show that under some mild
assumptions, only $O(k^2 \log d)$ samples are needed to estimate ${\Omega}^*$
in the $\ell_\infty$-norm up to a bounded error.
  For both problems, our estimator minimizes the sum of the finite population
negative log-likelihood function and an $\ell_1$-regularization term.
