Polynoomkernel

Polynoomkernel, commonly referred to in English as the polynomial kernel, is a kernel function used in kernelized machine learning methods to measure similarity between input vectors in a way that corresponds to a polynomial feature mapping. It is widely employed in support vector machines and other kernel-based algorithms.

The standard form of the polynomial kernel is K(x, y) = (gamma x^T y + coef0)^d, where gamma > 0 is a scaling factor, coef0 is an independent constant, and d is a positive integer representing the degree of the polynomial. Variants include the homogeneous kernel with coef0 = 0, giving K(x, y) = (gamma x^T y)^d, and the non-homogeneous form with coef0 > 0. When gamma = 1 and coef0 = 0, K(x, y) reduces to (x^T y)^d. The kernel is valid (positive semidefinite) for gamma > 0, ensuring it can be used in kernel methods that require a PSD kernel.

Interpretation and feature mapping: the polynomial kernel corresponds to mapping inputs into a feature space consisting of all monomials of the input coordinates up to degree d. The dimensionality of this feature space grows rapidly with the input dimension and the degree, enabling the model to capture interactions between features without an explicit feature construction.

Use and limitations: polynoomkernel is favored for its simplicity and ability to model nonlinear relationships through feature interactions. It can become computationally intensive for large datasets, and high degrees can lead to overfitting. Proper scaling of inputs and careful selection of gamma, coef0, and degree are important for good performance.