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In this paper, we consider the problem of sparse recovery from nonlinearmeasurements, which has applications in state estimation and bad data detectionfor power networks. An iterative mixed $\ell_1$ and $\ell_2$ convex program isused to estimate the true state by locally linearizing the nonlinearmeasurements. When the measurements are linear, through using the almostEuclidean property for a linear subspace, we derive a new performance bound forthe state estimation error under sparse bad data and additive observationnoise. As a byproduct, in this paper we provide sharp bounds on the almostEuclidean property of a linear subspace, using the "escape-through-the-mesh"theorem from geometric functional analysis. When the measurements arenonlinear, we give conditions under which the solution of the iterativealgorithm converges to the true state even though the locally linearizedmeasurements may not be the actual nonlinear measurements. We numericallyevaluate our iterative convex programming approach to perform bad datadetections in nonlinear electrical power networks problems. We are able to usesemidefinite programming to verify the conditions for convergence of theproposed iterative sparse recovery algorithms from nonlinear measurements.

Sparse Recovery from Nonlinear Measurements with Applications in Bad Data Detection for Power Networks