Symbolic Regression of Implicit Equations

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.