A Stable Jacobi polynomials based least squares regression estimator associated with an ANOVA decomposition model

TL;DR

This paper introduces a stable, efficient non-parametric regression estimator using multivariate Jacobi polynomials and ANOVA decomposition, with theoretical guarantees and numerical validation.

Contribution

It develops a novel stable estimator based on Jacobi polynomials and ANOVA decomposition for multidimensional regression, with stability and error bounds proven under specific sampling conditions.

Findings

Estimator is stable under Beta-distributed sampling points.

Provides bounds for L^2-risk error of the estimator.

Numerical simulations support theoretical results.

Abstract

In this work, we construct a stable and fairly fast estimator for solving non-parametric multidimensional regression problems. The proposed estimator is based on the use of multivariate Jacobi polynomials that generate a basis for a reduced size of $d-$variate finite dimensional polynomial space. An ANOVA decomposition trick has been used for building this later polynomial space. Also, by using some results from the theory of positive definite random matrices, we show that the proposed estimator is stable under the condition that the i.i.d. random sampling points for the different covariates of the regression problem, follow a $d-$dimensional Beta distribution. Also, we provide the reader with an estimate for the $L^2-$risk error of the estimator. Moreover, a more precise estimate of the quality of the approximation is provided under the condition that the regression function belongs to some weighted Sobolev space. Finally, the various theoretical results of this work are supported by numerical simulations.