Reviews: Statistical-Computational Tradeoff in Single Index Models

The paper first introduces first-order and second-order Stein's identity and then defines two function sets, C1 and C2, characterized by the covariance between f and X T\beta *. Further, authors define a common function set C(psi), which includes all link functions such that the second-order Stein's identity does not vanish under transformation psi. Then, authors propose a mixed model in 2.6 using two link functions f1\in C1\cap C(\psi) and f2\in C2\cap C(\psi). This model is finally used to derive lower bound. This is reasonable since true beta with link function f1 is easy to estimate (using first-order Stein's identity), while true beta with f2 is indistinguishable. The minimax rate is established in Prop 3.1.