  This paper considers the linear inverse problem where we wish to estimate a
structured signal $x$ from its corrupted observations. When the problem is
ill-posed, it is natural to make use of a convex function $f(\cdot)$ that
exploits the structure of the signal. For example, $\ell_1$ norm can be used
for sparse signals. To carry out the estimation, we consider two well-known
convex programs: 1) Second order cone program (SOCP), and, 2) Lasso. Assuming
Gaussian measurements, we show that, if precise information about the value
$f(x)$ or the $\ell_2$-norm of the noise is available, one can do a
particularly good job at estimation. In particular, the reconstruction error
becomes proportional to the "sparsity" of the signal rather than the ambient
dimension of the noise vector. We connect our results to existing works and
provide a discussion on the relation of our results to the standard
least-squares problem. Our error bounds are non-asymptotic and sharp, they
apply to arbitrary convex functions and do not assume any distribution on the
noise.
