  This work is about recovering an analysis-sparse vector, i.e. sparse vector
in some transform domain, from under-sampled measurements. In real-world
applications, there often exist random analysis-sparse vectors whose
distribution in the analysis domain are known. To exploit this information, a
weighted $\ell_1$ analysis minimization is often considered. The task of
choosing the weights in this case is however challenging and non-trivial. In
this work, we provide an analytical method to choose the suitable weights.
Specifically, we first obtain a tight upper-bound expression for the expected
number of required measurements. This bound depends on two critical parameters:
support distribution and expected sign of the analysis domain which are both
accessible in advance. Then, we calculate the near-optimal weights by
minimizing this expression with respect to the weights. Our strategy works for
both noiseless and noisy settings. Numerical results demonstrate the
superiority of our proposed method. Specifically, the weighted $\ell_1$
analysis minimization with our near-optimal weighting design considerably needs
fewer measurements than its regular $\ell_1$ analysis counterpart.
