  We consider the problem of recovering linear image $Bx$ of a signal $x$ known
to belong to a given convex compact set ${\cal X}$ from indirect observation
$\omega=Ax+\xi$ of $x$ corrupted by random noise $\xi$ with finite covariance
matrix. It is shown that under some assumptions on ${\cal X}$ (satisfied, e.g.,
when ${\cal X}$ is the intersection of $K$ concentric ellipsoids/elliptic
cylinders, or the unit ball of the spectral norm in the space of matrices) and
on the norm $\|\cdot\|$ used to measure the recovery error (satisfied, e.g., by
$\|\cdot\|_p$-norms, $1\leq p\leq 2$, on ${\mathbf{R}}^m$ and by the nuclear
norm on the space of matrices), one can build, in a computationally efficient
manner, a "presumably good" linear in observations estimate, and that in the
case of zero mean Gaussian observation noise, this estimate is near-optimal
among all (linear and nonlinear) estimates in terms of its worst-case, over
$x\in {\cal X}$, expected $\|\cdot\|$-loss. These results form an essential
extension of those in our paper arXiv:1602.01355, where the assumptions on
${\cal X}$ were more restrictive, and the norm $\|\cdot\|$ was assumed to be
the Euclidean one. In addition, we develop near-optimal estimates for the case
of "uncertain-but-bounded" noise, where all we know about $\xi$ is that it is
bounded in a given norm by a given $\sigma$. Same as in arXiv:1602.01355, our
results impose no restrictions on $A$ and $B$.
  This arXiv paper slightly strengthens the journal publication Juditsky, A.,
Nemirovski, A. "Near-Optimality of Linear Recovery from Indirect Observations,"
Mathematical Statistics and Learning 1:2 (2018), 171-225.
