  This paper considers learning of the graphical structure of a $p$-dimensional
random vector $X \in R^p$ using both parametric and non-parametric methods.
Unlike the previous works which observe $x$ directly, we consider the indirect
observation scenario in which samples $y$ are collected via a sensing matrix $A
\in R^{d\times p}$, and corrupted with some additive noise $w$, i.e, $Y = AX +
W$. For the parametric method, we assume $X$ to be Gaussian, i.e., $x\in
R^p\sim N(\mu, \Sigma)$ and $\Sigma \in R^{p\times p}$. For the first time, we
show that the correct graphical structure can be correctly recovered under the
indefinite sensing system ($d < p$) using insufficient samples ($n < p$). In
particular, we show that for the exact recovery, we require dimension $d =
\Omega(p^{0.8})$ and sample number $n = \Omega(p^{0.8}\log^3 p)$. For the
nonparametric method, we assume a nonparanormal distribution for $X$ rather
than Gaussian. Under mild conditions, we show that our graph-structure
estimator can obtain the correct structure. We derive the minimum sample number
$n$ and dimension $d$ as $n\gtrsim (deg)^4 \log^4 n$ and $d \gtrsim p +
(deg\cdot\log(d-p))^{\beta/4}$, respectively, where deg is the maximum Markov
blanket in the graphical model and $\beta > 0$ is some fixed positive constant.
Additionally, we obtain a non-asymptotic uniform bound on the estimation error
of the CDF of $X$ from indirect observations with inexact knowledge of the
noise distribution. To the best of our knowledge, this bound is derived for the
first time and may serve as an independent interest. Numerical experiments on
both real-world and synthetic data are provided confirm the theoretical
results.
