Polynoomkern

Polynoomkern is the Dutch term for polynomial kernel, an important class of kernel functions in support vector machine and kernel ridge regression methods. It is defined as k(x, y) = (γ⟨x, y⟩ + r)ᵈ, where γ and r are scaling parameters, ⟨·, ·⟩ denotes the inner product, and d is the degree of the polynomial. The polynomial kernel maps input vectors into a feature space that includes all monomials up to degree d, enabling learning of non‑linear relationships while retaining computational efficiency through the kernel trick.

The kernel was introduced in the mid‑1990s as part of the broader development of the support vector machine by Cortes, Vapnik, and later extended by Scholkopf, Smola, and Steinwart, who formalised its theoretical properties. In practice, common choices include the linear kernel (d = 1), the quadratic kernel (d = 2), and higher‑order polynomials for complex data. Parameter selection is often performed by cross‑validation; setting γ to 1/number_of_features is a common default.

Polynomials kernels are universal approximators and can represent a wide variety of functions. However, high degrees can lead to over‑fitting and numerical instability. Thus, they are frequently combined with regularisation parameters in the optimisation problem. In many machine‑learning libraries, the polynomial kernel is implemented as a default option alongside Gaussian (RBF) and sigmoid kernels.