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

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.

💡 Research Summary

The paper proposes a geometric‑algebraic framework for extending binary classifiers beyond linear and spherical decision surfaces to general second‑order hypersurfaces (hyperconics) by exploiting Clifford algebras. Starting from the standard Euclidean space ℝ^m, the authors embed the space into the Clifford algebra Cl_m and introduce a mapping τ that identifies the space of symmetric m×m matrices M_s with the real vector space ℝ^{½m(m+1)}. This identification enables the definition of a new Clifford algebra Cl(V₂), where V₂ denotes the vector space of hyperconic sections.

An explicit embedding i : ℝ^m → M_s is defined by i(x)=x′^T x′ with x′=(x,1). Composing i with τ yields an embedding of points into Cl(V₂). Lemma 8 shows that the incidence condition “point x lies on hyperconic defined by symmetric matrix A” can be expressed simply as the Clifford inner product x·a=0, where a=τ(A). Thus, geometric incidence is captured by algebraic null‑products.

The authors then connect projective duality with Clifford duality. Proposition 11 and Corollary 12 establish that the dual hyperplane of a hyperconic in projective space corresponds exactly to the Clifford dual of the associated multivector. This provides a systematic way to compute the normal (or separating) vector of a decision hyperplane that is a hyperconic. Lemma 5 and Lemma 6 prove that the embedding i is injective and that τ∘i preserves the necessary algebraic structure.

With this theoretical foundation, the paper revisits the spherical perceptron in conformal geometric algebra PK_m, where a sphere is represented by a null‑vector S and classification is performed via the sign of S·X. The elliptical perceptron is introduced as a special case of the spherical perceptron with six weights and six inputs, capable of learning any planar conic (ellipse, parabola, hyperbola) as a decision boundary.

Experimental validation is limited to two‑dimensional data. Synthetic datasets are generated that are naturally separable by a specific conic. The elliptical perceptron is trained using back‑propagation, and the resulting weight vectors are reported in Table 1, demonstrating successful separation for each conic type. No comparison with traditional linear SVMs, kernel methods, or higher‑dimensional tests are provided.

In the conclusion, the authors claim that the Clifford‑based hyperconic perceptron is dimension‑agnostic and can be applied to high‑dimensional problems where decision surfaces of quadratic form are advantageous. They suggest future work on scaling the method, improving computational efficiency of the τ and i mappings, and benchmarking against existing nonlinear classifiers.

Overall, the contribution lies in a mathematically elegant unification of incidence geometry, projective duality, and Clifford algebra to formulate a perceptron that learns quadratic decision surfaces. The paper is rigorous in its algebraic derivations but lacks extensive empirical evaluation, especially in higher dimensions and against baseline methods. The practical impact will depend on demonstrating computational benefits and classification performance in real‑world high‑dimensional datasets.