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.",