  We discuss a general notion of "sparsity structure" and associated recoveries
of a sparse signal from its linear image of reduced dimension possibly
corrupted with noise. Our approach allows for unified treatment of (a) the
"usual sparsity" and "usual $\ell_1$ recovery," (b) block-sparsity with
possibly overlapping blocks and associated block-$\ell_1$ recovery, and (c)
low-rank-oriented recovery by nuclear norm minimization. The proposed recovery
routines are natural extensions of the usual $\ell_1$ minimization used in
Compressed Sensing. Specifically we present nullspace-type sufficient
conditions for the recovery to be precise on sparse signals in the noiseless
case. Then we derive error bounds for imperfect (nearly sparse signal, presence
of observation noise, etc.) recovery under these conditions. In all of these
cases, we present efficiently verifiable sufficient conditions for the validity
of the associated nullspace properties.
