Analytic Geometry in Machine Learning

Abstract

The applications of analytic geometry in machine learning courses are summarized, covering the definition of hyperplanes, the computation of distances to hyperplanes, their application in support vector machines, and the geometric properties of hyperellipsoids—specifically their centers, axis orientations, and semi‑axis lengths—along with the occurrence of such ellipsoids in linear regression. By explicitly addressing these topics, it facilitates both teaching and learning for instructors and students. Moreover, these geometric interpretations render abstract high-dimensional optimization problems more intuitive and help students develop the ability to analyze machine learning algorithms from a geometric perspective. Furthermore, the intrinsic connections between these concepts—such as the role of normal vectors in defining margins and the interpretation of level sets as ellipsoids—can deepen understanding and inspire further exploration.