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Kernels
Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learning * Kernelization, a technique for designing efficient algorithms ** Kernel, a routine that is executed in a vectorized loop, for example in general-purpose computing on graphics processing units *KERNAL, the Commodore operating system Mathematics Objects * Kernel (algebra), a general concept that includes: ** Kernel (linear algebra) or null space, a set of vectors mapped to the zero vector ** Kernel (category theory), a generalization of the kernel of a homomorphism ** Kernel (set theory), an equivalence relation: partition by image under a function ** Difference kernel, a binary equalizer: the kernel of the difference of two functions Functions * Kernel (geometry), the set of points within a polygon from which the whole polygon bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Kernel
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas. This article will discuss some of the historical and current developments of the theory of positive-definite kernels, starting with the general idea and properties before considering practical applications. Definition Let \mathcal X be a nonempty set, sometimes referred to as the index set. A symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (statistics)
The term kernel is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics. Bayesian statistics In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (operating System)
The kernel is a computer program at the core of a computer's operating system and generally has complete control over everything in the system. It is the portion of the operating system code that is always resident in memory and facilitates interactions between hardware and software components. A full kernel controls all hardware resources (e.g. I/O, memory, cryptography) via device drivers, arbitrates conflicts between processes concerning such resources, and optimizes the utilization of common resources e.g. CPU & cache usage, file systems, and network sockets. On most systems, the kernel is one of the first programs loaded on startup (after the bootloader). It handles the rest of startup as well as memory, peripherals, and input/output (I/O) requests from software, translating them into data-processing instructions for the central processing unit. The critical code of the kernel is usually loaded into a separate area of memory, which is protected from access by application ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transition Kernel
In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels are the Markov kernels. Definition Let (S, \mathcal S) , (T, \mathcal T) be two measurable spaces. A function : \kappa \colon S \times \mathcal T \to , +\infty is called a (transition) kernel from S to T if the following two conditions hold: *For any fixed B \in \mathcal T , the mapping :: s \mapsto \kappa(s,B) :is \mathcal S/ \mathcal B( , +\infty-measurable; *For every fixed s \in S , the mapping :: B \mapsto \kappa(s, B) :is a measure on (T, \mathcal T). Classification of transition kernels Transition kernels are usually classified by the measures they define. Those measures are defined as : \kappa_s \colon \mathcal T \to , + \infty with : \kappa_s(B)=\kappa(s,B) for all B \in \mathcal T and all s \in S . With thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Kernel
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Formal definition Let (X,\mathcal A) and (Y,\mathcal B) be measurable spaces. A ''Markov kernel'' with source (X,\mathcal A) and target (Y,\mathcal B) is a map \kappa : \mathcal B \times X \to ,1/math> with the following properties: # For every (fixed) B \in \mathcal B, the map x \mapsto \kappa(B, x) is \mathcal A-measurable # For every (fixed) x \in X, the map B \mapsto \kappa(B, x) is a probability measure on (Y, \mathcal B) In other words it associates to each point x \in X a probability measure \kappa(dy, x): B \mapsto \kappa(B, x) on (Y,\mathcal B) such that, for every measurable set B\in\mathcal B, the map x\mapsto \kappa(B, x) is measurable with respect to the \sigma-algebra \mathcal A. Examples Simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Trick
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified ''feature map'': in contrast, kernel methods require only a user-specified ''kernel'', i.e., a similarity function over all pairs of data points computed using Inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the Representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing. Kernel methods owe their name to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General-purpose Computing On Graphics Processing Units
General-purpose computing on graphics processing units (GPGPU, or less often GPGP) is the use of a graphics processing unit (GPU), which typically handles computation only for computer graphics, to perform computation in applications traditionally handled by the central processing unit (CPU). The use of multiple video cards in one computer, or large numbers of graphics chips, further parallelizes the already parallel nature of graphics processing. Essentially, a GPGPU pipeline is a kind of parallel processing between one or more GPUs and CPUs that analyzes data as if it were in image or other graphic form. While GPUs operate at lower frequencies, they typically have many times the number of cores. Thus, GPUs can process far more pictures and graphical data per second than a traditional CPU. Migrating data into graphical form and then using the GPU to scan and analyze it can create a large speedup. GPGPU pipelines were developed at the beginning of the 21st century for graphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.See and . For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Method
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified ''feature map'': in contrast, kernel methods require only a user-specified ''kernel'', i.e., a similarity function over all pairs of data points computed using Inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the Representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing. Kernel methods owe their name to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (category Theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' that yields zero when composed with (followed by) ''f''. Note that kernel pairs and difference kernels (also known as binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article. Definition Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary morphism in C, then a kernel of ''f'' is an equaliser of ''f'' and the zero morphism from ''X'' to ''Y''. In symbols: :ker(''f'') = eq(''f'', 0''XY'') To be more explicit, the following universal pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corn Kernel
Corn kernels are the fruits of corn (called maize in many countries). Maize is a grain, and the kernels are used in cooking as a vegetable or a source of starch. The kernel comprise endosperm, germ, pericarp, and tip cap. One ear of corn contains roughly 800 kernels in 16 rows. Corn kernels are readily available in bulk throughout maize-producing areas. They have a number of uses, including food and biofuel. Corn consists of the husk and the silk, often mistaken for the husk. Description Corn kernels are the fruits of maize. Maize is a grain, and the kernels are used in cooking as a vegetable or a source of starch. The kernels can be of various colors: blackish, bluish-gray, purple, green, red, white and yellow. One ear of corn contains roughly 800 kernels in 16 rows. One hundred bushels of corn can contain upwards of 7,280,000 kernels. Transportation and packaging of dried clean corn kernels to non-producing areas adds to the cost. Parts The kernel of maize consists of a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
