We propose a computationally efficient estimator, formulated as a convex program, for a broad class of nonlinear regression problems that involve difference of convex(DC) nonlinearities. The proposed method can be viewed as a significant extension of the “anchored regression” method formulated and analyzed in  for regression with convex nonlinearities. Our main assumption, in addition to other mild statistical and computational assumptions, is availability of a certain approximation oracle for the average of the gradients of the observation functions at a ground truth. Under this assumption and using a PAC-Bayesian analysis we show that the proposed estimator produces an accurate estimate with high probability. As a concrete example, we study the proposed framework in the bilinear regression problem with Gaussian factors and quantify a sufficient sample complexity for exact recovery. Furthermore, we describe a computationally tractable scheme that provably produces the required approximation oracle in the considered bilinear regression problem.
"Estimation from nonlinear observations via convex programming with application to bilinear regression." Electron. J. Statist. 13 (1) 1978 - 2011, 2019. https://doi.org/10.1214/19-EJS1567