Optimal Estimation of Slope Vector in High-dimensional Linear Transformation Model

Abstract

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.

arXiv comment25 pages, 7 figures, 1 table

References in corpus (19)

• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Rejoinder to "Least angle regression" by Efron et al
• Least Angle Regression
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Simultaneous analysis of Lasso and Dantzig selector
• Robust rank correlation based screening
• High-dimensional estimation with geometric constraints
• Sparse CCA: Adaptive Estimation and Computational Barriers
• Sparse single-index model
• The generalized Lasso with non-linear observations
• On consistency and sparsity for sliced inverse regression in high dimensions
• Kernel Additive Principal Components
• A Provable Smoothing Approach for High Dimensional Generalized Regression with Applications in Genomics