This paper is concerned with the recovery of bandlimited signals that are distorted by non-invertible sensor nonlinearities. When the nonlinear sensor characteristic is such that the input-output relationship is invertible, the distortion caused by sensor nonlinearity may be compensated and the signals may be reproduced from the distorted sensor measurements. However, such methods may not work if the nonlinear sensor dictates ill-posed input-output behavior such as saturation and dead-band. We propose a new approach that optimizes a non-quadratic criterion to reproduce the signal from the distorted sensor data. The fact that the introduction of a non-quadratic penalty in the cost function does not alter the optimal solution as much as the quadratic penalty does, makes it a better candidate for the problem posed. However, this is a non-convex optimization problem, which implies that multiple local optimal solutions may exist. A systematic solution procedure is presented to interpret the solutions generated by non-quadratic criteria and to identify the global optimal point. Illustrative examples are presented to further explain the proposed framework.