  This paper introduces a general regularized thresholded least-square
procedure estimating a structured signal $\theta_*\in\mathbb{R}^d$ from the
following observations: $y_i = f(\langle\mathbf{x}_i, \theta_*\rangle,
\xi_i),~i\in\{1,2,\cdots,N\}$, with i.i.d. heavy-tailed measurements
$\{(\mathbf{x}_i,y_i)\}_{i=1}^N$. A general framework analyzing the
thresholding procedure is proposed, which boils down to computing three
critical radiuses of the bounding balls of the estimator. Then, we demonstrate
these critical radiuses can be tightly bounded in the following two scenarios:
(1) The link function $f(\cdot)$ is linear, i.e. $y =
\langle\mathbf{x},\theta_*\rangle + \xi$, with $\theta_*$ being a sparse vector
and $\{\mathbf{x}_i\}_{i=1}^N$ being general heavy-tailed random measurements
with bounded $(20+\epsilon)$-moments. (2) The function $f(\cdot)$ is arbitrary
unknown (possibly discontinuous) and $\{\mathbf{x}_i\}_{i=1}^N$ are
heavy-tailed elliptical random vectors with bounded $(4+\epsilon)$-moments. In
both scenarios, we show under these rather minimal bounded moment assumptions,
such a procedure and corresponding analysis lead to optimal sample and error
bounds with high probability in terms of the structural properties of
$\theta_*$.
