Clifford Algebra of the Vector Space of Conics for decision boundary Hyperplanes in m-Euclidean Space

TL;DR

This paper introduces a novel geometric algebra framework for decision boundaries using hyperconic sections, generalizing perceptrons to hyperconic separators in m-dimensional Euclidean space.

Contribution

It extends the concept of perceptrons to hyperconic decision boundaries using Clifford algebra, enabling dimension-independent separation methods.

Findings

Successfully separated data with hyperconic boundaries in 2D experiments.

Generalized perceptron to hyperconic separators, including elliptical perceptron.

Applicable to high-dimensional data regardless of input dimension.

Abstract

In this paper we embed $m$-dimensional Euclidean space in the geometric algebra $Cl_m $ to extend the operators of incidence in ${R^m}$ to operators of incidence in the geometric algebra to generalize the notion of separator to a decision boundary hyperconic in the Clifford algebra of hyperconic sections denoted as ${Cl}({Co}_{2})$. This allows us to extend the concept of a linear perceptron or the spherical perceptron in conformal geometry and introduce the more general conic perceptron, namely the {elliptical perceptron}. Using Clifford duality a vector orthogonal to the decision boundary hyperplane is determined. Experimental results are shown in 2-dimensional Euclidean space where we separate data that are naturally separated by some typical plane conic separators by this procedure. This procedure is more general in the sense that it is independent of the dimension of the input data and hence we can speak of the hyperconic elliptic perceptron.