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Symbolic Regression of Implicit Equations

  • Michael Schmidt
  • Hod Lipson
Chapter
Part of the Genetic and Evolutionary Computation book series (GEVO)

Abstract

Traditional Symbolic Regression applications are a form of supervised learning, where a label y is provided for every \(\vec{x}\) and an explicit symbolic relationship of the form \(y = f(\vec{x})\) is sought. This chapter explores the use of symbolic regression to perform unsupervised learning by searching for implicit relationships of the form \(f(\vec{x}, y) = 0\). Implicit relationships are more general and more expressive than explicit equations in that they can also represent closed surfaces, as well as continuous and discontinuous multi-dimensional manifolds. However, searching these types of equations is particularly challenging because an error metric is difficult to define. We studied several direct and indirect techniques, and present a successful method based on implicit derivatives. Our experiments identified implicit relationships found in a variety of datasets, such as equations of circles, elliptic curves, spheres, equations of motion, and energy manifolds.

Keywords

Symbolic Regression Implicit Equations Unsupervised Learning 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Michael Schmidt
    • 1
  • Hod Lipson
    • 2
    • 3
  1. 1.Computational BiologyCornell UniversityIthacaUSA
  2. 2.School of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA
  3. 3.Computing and Information ScienceCornell UniversityIthacaUSA

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