productkernelconstructies

Product kernel constructions refer to building kernel functions by taking the product of simpler base kernels defined on individual input components or feature blocks. For an input x partitioned into blocks x = (x1, x2, ..., xK) and kernels ki on each block, a product kernel is k(x, x') = ∏i ki(xi, x'i). This operation preserves positive semidefiniteness: the product of PSD kernels is PSD, so k is a valid kernel. Product kernels enable modeling of interactions across modalities or feature groups without enumerating all cross terms.

In practice, product kernels are used to define covariance structures in Gaussian processes and to construct kernels for structured data. They often arise in multi-view or multi-task learning, where each ki captures information from a different view or task, and the product encodes multiplicative interactions between views. They also appear in kernelized methods as a way to impose multiplicative feature interactions, which can improve expressivity over additive kernels.

Computationally, the kernel matrix under a product kernel can sometimes be exploited via Kronecker or low-rank structures when data are arranged in a grid or block structure. When combining with standard kernels like the RBF or Matérn, product kernels can model complex dependencies but may increase computational burden. They are related to broader kernel construction paradigms, including additive kernels and more general kernel algebra.

See also: kernel methods, reproducing kernel Hilbert space, Gaussian processes, Kronecker product, multi-view learning.