  This paper is concerned with the problem of recovering a structured signal
from a relatively small number of corrupted random measurements. Sharp phase
transitions have been numerically observed in practice when different convex
programming procedures are used to solve this problem. This paper is devoted to
presenting theoretical explanations for these phenomenons by employing some
basic tools from Gaussian process theory. Specifically, we identify the precise
locations of the phase transitions for both constrained and penalized recovery
procedures. Our theoretical results show that these phase transitions are
determined by some geometric measures of structure, e.g., the spherical
Gaussian width of a tangent cone and the Gaussian (squared) distance to a
scaled subdifferential. By utilizing the established phase transition theory,
we further investigate the relationship between these two kinds of recovery
procedures, which also reveals an optimal strategy (in the sense of Lagrange
theory) for choosing the tradeoff parameter in the penalized recovery
procedure. Numerical experiments are provided to verify our theoretical
results.
