  This work investigates the problem of signal recovery from undersampled noisy
sub-Gaussian measurements under the assumption of a synthesis-based sparsity
model. Solving the $\ell^1$-synthesis basis pursuit allows for a simultaneous
estimation of a coefficient representation as well as the sought-for signal.
However, due to linear dependencies within redundant dictionary atoms it might
be impossible to identify a specific representation vector, although the actual
signal is still successfully recovered. The present manuscript studies both
estimation problems from a non-uniform, signal-dependent perspective. By
utilizing recent results on the convex geometry of linear inverse problems, the
sampling rates describing the phase transitions of each formulation are
identified. In both cases, they are given by the conic Gaussian mean width of
an $\ell^1$-descent cone that is linearly transformed by the dictionary. In
general, this expression does not allow a simple calculation by following the
polarity-based approach commonly found in the literature. Hence, two upper
bounds involving the sparsity of coefficient representations are provided: The
first one is based on a local condition number and the second one on a
geometric analysis that makes use of the thinness of high-dimensional
polyhedral cones with not too many generators. It is furthermore revealed that
both recovery problems can differ dramatically with respect to robustness to
measurement noise -- a fact that seems to have gone unnoticed in most of the
related literature. All insights are carefully undermined by numerical
simulations.
