  A sharp phase transition emerges in convex programs when solving the linear
inverse problem, which aims to recover a structured signal from its linear
measurements. This paper studies this phenomenon in theory under Gaussian
random measurements. Different from previous studies, in this paper, we
consider convex programs with multiple prior constraints. These programs are
encountered in many cases, for example, when the signal is sparse and its
$\ell_2$ norm is known beforehand, or when the signal is sparse and
non-negative simultaneously. Given such a convex program, to analyze its phase
transition, we introduce a new set and a new cone, called the prior restricted
set and prior restricted cone, respectively. Our results reveal that the phase
transition of a convex problem occurs at the statistical dimension of its prior
restricted cone. Moreover, to apply our theoretical results in practice, we
present two recipes to accurately estimate the statistical dimension of the
prior restricted cone. These two recipes work under different conditions, and
we give a detailed analysis for them. To further illustrate our results, we
apply our theoretical results and the estimation recipes to study the phase
transition of two specific problems, and obtain computable formulas for the
statistical dimension and related error bounds. Simulations are provided to
demonstrate our results.
