Constructive Approximation

, Volume 26, Issue 2, pp 127–152 | Cite as

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

  • Peter BinevEmail author
  • Albert CohenEmail author
  • Wolfgang DahmenEmail author
  • Ronald DeVoreEmail author


This paper is concerned with estimating the regression function fρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.


Learn Theory Regression Function Besov Space Polynomial Space Piecewise Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Industrial Mathematics Institute, Department of Mathematics, University of South CarolinaColumbia, SC 29208USA
  2. 2.Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie, 175 rue du Chevaleret75013 ParisFrance
  3. 3.Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55D-52056 AachenGermany

Personalised recommendations