Nonparametric Envelopes for Flexible Response Reduction

Envelope methods improve the estimation efficiency in multivariate linear regression by identifying and separating the material and immaterial parts of the responses or the predictors and estimating the regression coefficients using only the material part. Though envelopes have been extended to other models, such as GLMs, these extensions still largely fall under the restrictive parametric modeling framework. In this paper, we introduce a flexible, nonparametric extension of response envelopes for improving efficiency in nonlinear multivariate regressions. We propose the kernel envelope (KENV) estimator for simultaneously estimating the response envelope subspace and the enveloped nonparametric conditional mean function in a reproducing kernel Hilbert space, with a novel penalty that accounts for the envelope structure. We prove that the prediction risk for KENV converges to the optimal risk as the sample size diverges and show that KENV achieves a lower in-sample prediction risk than kernel ridge regression when the response has a non-trivial immaterial component. We compare the prediction performance of KENV with other envelope methods and kernel regression methods in simulations and a real data example, finding that KENV delivers more accurate predictions than both the envelope-based and kernel-based alternatives in both low and high dimensions.