Constructive Approximation

, Volume 26, Issue 2, pp 127–152

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


DOI: 10.1007/s00365-006-0658-z

Cite this article as:
Binev, P., Cohen, A., Dahmen, W. et al. Constr Approx (2007) 26: 127. doi:10.1007/s00365-006-0658-z


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.

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