Adaptive estimation of multivariate piecewise polynomials and bounded variation functions by optimal decision trees

Multivariate piecewise polynomials

This article was supposed to be on `multivariate splines'. An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a `multivariate spline', resulted in the answer that a multivariate spline is a possibly smooth, piecewise polynomial function of several arguments. In particular, the potentially very useful thin-plate spline was thought to belo...

Multivariate Piecewise Polynomials

Simple Learning Algorithms for Decision Trees and Multivariate Polynomials

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over nite elds under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an-approximation of t...

On the Dimension of Multivariate Piecewise Polynomials

Lower bounds are given on the dimension of piecewise polynomial C 1 and C 2 functions de ned on a tessellation of a polyhedral domain into Tetrahedra. The analysis technique consists of embedding the space of interest into a larger space with a simpler structure, and then making appropriate adjustments. In the bivariate case, this approach reproduces the well-known lower bounds derived by Schum...

Bounded-degree factors of lacunary multivariate polynomials

In this paper, we present a new method for computing boundeddegree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in ...