On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression

TL;DR

This paper characterizes the degrees of freedom for multivariate nonparametric regression estimators derived from quadratic optimization, enabling better tuning and understanding of various estimators including isotonic, convex, and penalized regressions.

Contribution

It provides explicit formulas for degrees of freedom of a broad class of nonparametric estimators, unifying and extending existing results in the literature.

Findings

Explicit degrees of freedom formulas for various regression methods

Connection between bounded isotonic regression and general partial orders

Facilitates tuning parameter selection via Stein's unbiased risk estimate

Abstract

In this paper, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures, namely, minimizers of the least squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, e.g., bounded isotonic regression, multivariate (penalized) convex regression, and additive total variation regularization. Our theory also yields, as special cases, known results on the degrees of freedom of many well-studied estimators in the statistics literature, such as ridge regression, Lasso and generalized Lasso. Our results can be readily used to choose the tuning parameter(s) involved in the estimation procedure by minimizing the Stein's unbiased risk estimate. As a by-product of our analysis we derive an interesting connection between bounded isotonic regression and isotonic regression on a general partially ordered set, which is of independent interest.