Symbolic Regression of Implicit Equations

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  author =       "Michael Schmidt and Hod Lipson",
  title =        "Symbolic Regression of Implicit Equations",
  booktitle =    "Genetic Programming Theory and Practice {VII}",
  year =         "2009",
  editor =       "Rick L. Riolo and Una-May O'Reilly and 
                 Trent McConaghy",
  series =       "Genetic and Evolutionary Computation",
  address =      "Ann Arbor",
  month =        "14-16 " # may,
  publisher =    "Springer",
  chapter =      "5",
  pages =        "73--85",
  keywords =     "genetic algorithms, genetic programming, Symbolic
                 Regression, Implicit Equations, Unsupervised Learning",
  isbn13 =       "978-1-4419-1653-2",
  DOI =          "doi:10.1007/978-1-4419-1626-6_5",
  abstract =     "Traditional Symbolic Regression applications are a
                 form of supervised learning, where a label y is
                 provided for every x and an explicit symbolic
                 relationship of the form y=f(x) is sought. This chapter
                 explores the use of symbolic regression to perform
                 unsupervised learning by searching for implicit
                 relationships of the form f(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.",
  notes =        "Entered 2010 HUMIES GECCO 2010

                 part of \cite{Riolo:2009:GPTP}",

Genetic Programming entries for Michael D Schmidt Hod Lipson