  In this paper, we consider the problem of sparse recovery from nonlinear
measurements, which has applications in state estimation and bad data detection
for power networks. An iterative mixed $\ell_1$ and $\ell_2$ convex program is
used to estimate the true state by locally linearizing the nonlinear
measurements. When the measurements are linear, through using the almost
Euclidean property for a linear subspace, we derive a new performance bound for
the state estimation error under sparse bad data and additive observation
noise. As a byproduct, in this paper we provide sharp bounds on the almost
Euclidean property of a linear subspace, using the "escape-through-the-mesh"
theorem from geometric functional analysis. When the measurements are
nonlinear, we give conditions under which the solution of the iterative
algorithm converges to the true state even though the locally linearized
measurements may not be the actual nonlinear measurements. We numerically
evaluate our iterative convex programming approach to perform bad data
detections in nonlinear electrical power networks problems. We are able to use
semidefinite programming to verify the conditions for convergence of the
proposed iterative sparse recovery algorithms from nonlinear measurements.
