Consider signal estimation from non-linear measurements. A rough heuristic often used in practice postulates that "non-linear measurements may be treated as noisy linear measurements" and the signal may be reconstructed accordingly. We give a rigorous backing to this idea, with a focus on low-dimensional signals buried in a high-dimensional spaces. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear measurements with linear measurements will be greatly reduced when using the signal structure in the reconstruction. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown, and ill-conditioned, covariance matrix. As a special case of our results, we give theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors.