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.
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Schmidt, M., Lipson, H. (2010). Symbolic Regression of Implicit Equations. In: Riolo, R., O'Reilly, UM., McConaghy, T. (eds) Genetic Programming Theory and Practice VII. Genetic and Evolutionary Computation. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1626-6_5
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DOI: https://doi.org/10.1007/978-1-4419-1626-6_5
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