  Compressed Sensing refers to extracting a low-dimensional structured signal
of interest from its incomplete random linear observations. A line of recent
work has studied that, with the extra prior information about the signal, one
can recover the signal with much fewer observations. For this purpose, the
general approach is to solve weighted convex function minimization problem. In
such settings, the convex function is chosen to promote the low-dimensional
structure and the optimal weights are so chosen to reduce the number of
measurements required for the optimization problem. In this paper, we consider
a generalized non-uniform model in which the structured signal falls into some
partitions, with entries of each partition having a definite probability to be
an element of the structure support. Given these probabilities and regarding
the recent developments in conic integral geometry, we provide a method to
choose the unique optimal weights for any general low-dimensional signal model.
This class of low-dimensional signal model includes many popular examples such
as $\ell_1$ analysis (entry-wise sparsity in an arbitrary redundant
dictionary), $\ell_{1,2}$ norm (block sparsity) and total variation semi-norm
(for piece-wise constant signals). We show through precise analysis and
simulations that the weighted convex optimization problem significantly
improves the regular convex optimization problem as we choose the unique
optimal weights.
