  Many applications have benefited remarkably from low-dimensional models in
the recent decade. The fact that many signals, though high dimensional, are
intrinsically low dimensional has given the possibility to recover them stably
from a relatively small number of their measurements. For example, in
compressed sensing with the standard (synthesis) sparsity prior and in matrix
completion, the number of measurements needed is proportional (up to a
logarithmic factor) to the signal's manifold dimension.
  Recently, a new natural low-dimensional signal model has been proposed: the
cosparse analysis prior. In the noiseless case, it is possible to recover
signals from this model, using a combinatorial search, from a number of
measurements proportional to the signal's manifold dimension. However, if we
ask for stability to noise or an efficient (polynomial complexity) solver, all
the existing results demand a number of measurements which is far removed from
the manifold dimension, sometimes far greater. Thus, it is natural to ask
whether this gap is a deficiency of the theory and the solvers, or if there
exists a real barrier in recovering the cosparse signals by relying only on
their manifold dimension. Is there an algorithm which, in the presence of
noise, can accurately recover a cosparse signal from a number of measurements
proportional to the manifold dimension? In this work, we prove that there is no
such algorithm. Further, we show through numerical simulations that even in the
noiseless case convex relaxations fail when the number of measurements is
comparable to the manifold dimension. This gives a practical counter-example to
the growing literature on compressed acquisition of signals based on manifold
dimension.
