Kernels

Kernels is a term used across disciplines to denote a core concept that captures essential structure in a system, function, or transformation. This article outlines several common meanings in computing, mathematics, statistics, and signal processing.

In computing, the kernel is the central component of an operating system. It manages resources such as memory, processes, and I/O, and provides interfaces between software and hardware. Kernels handle task scheduling, memory protection, and device drivers, and respond to interrupts. There are different design philosophies, including monolithic kernels that bundle many services in one large block of code and microkernels that keep the kernel small and push services into user space. Examples include the Linux kernel and the Windows NT kernel.

In mathematics, the kernel of a function or linear transformation is the set of inputs that map to zero (or to the identity element, in an algebraic setting). For a linear transformation T from a vector space V to W, the kernel is {v in V | T(v) = 0}. The kernel is always a subspace, and its dimension relates to the rank of T via the rank-nullity theorem. In group theory, the kernel of a homomorphism is the set of elements mapped to the identity, forming a normal subgroup.

In statistics and machine learning, kernel functions measure similarity and enable kernel methods. Through the kernel trick, operations can be performed in a high-dimensional feature space without explicit mapping. Common kernels include linear, polynomial, and Gaussian (RBF). Kernel density estimation uses a kernel as a smoothing function to estimate a probability density function, with bandwidth controlling smoothness.

In image and signal processing, a kernel (or convolution kernel) is a small matrix used to apply filters to images or signals, producing effects such as blurring, sharpening, or edge detection.