Fitting Aggregation Functions to Data: Part I - Linearization and Regularization

  • Maciej Bartoszuk
  • Gleb Beliakov
  • Marek Gagolewski
  • Simon James
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 611)

Abstract

The use of supervised learning techniques for fitting weights and/or generator functions of weighted quasi-arithmetic means – a special class of idempotent and nondecreasing aggregation functions – to empirical data has already been considered in a number of papers. Nevertheless, there are still some important issues that have not been discussed in the literature yet. In the first part of this two-part contribution we deal with the concept of regularization, a quite standard technique from machine learning applied so as to increase the fit quality on test and validation data samples. Due to the constraints on the weighting vector, it turns out that quite different methods can be used in the current framework, as compared to regression models. Moreover, it is worth noting that so far fitting weighted quasi-arithmetic means to empirical data has only been performed approximately, via the so-called linearization technique. In this paper we consider exact solutions to such special optimization tasks and indicate cases where linearization leads to much worse solutions.

Keywords

Aggregation functions Weighted quasi-arithmetic means Least squares fitting Regularization Linearization 

Notes

Acknowledgments

This study was supported by the National Science Center, Poland, research project 2014/13/D/HS4/01700.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Maciej Bartoszuk
    • 1
  • Gleb Beliakov
    • 2
  • Marek Gagolewski
    • 1
    • 3
  • Simon James
    • 2
  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland
  2. 2.School of Information TechnologyDeakin UniversityBurwoodAustralia
  3. 3.Systems Research InstitutePolish Academy of SciencesWarsawPoland

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