  In a linear transformation model, there exists an unknown monotone nonlinear
transformation function such that the transformed response variable and the
predictor variables satisfy a linear regression model. In this paper, we
present CENet, a new method for estimating the slope vector and simultaneously
performing variable selection in the high-dimensional sparse linear
transformation model. CENet is the solution to a convex optimization problem
and can be computed efficiently from an algorithm with guaranteed convergence
to the global optimum. We show that under a pairwise elliptical distribution
assumption on each predictor-transformed-response pair and some regularity
conditions, CENet attains the same optimal rate of convergence as the best
regression method in the high-dimensional sparse linear regression model. To
the best of our limited knowledge, this is the first such result in the
literature. We demonstrate the empirical performance of CENet on both simulated
and real datasets. We also discuss the connection of CENet with some nonlinear
regression/multivariate methods proposed in the literature.
