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Kernel
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 boundary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kernel (operating System)
A kernel is a computer program at the core of a computer's operating system that always has complete control over everything in the system. The kernel is also responsible for preventing and mitigating conflicts between different processes. 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 use of common resources, such as CPU, cache, 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 usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. Definition Let \mathcal X be a nonempty set, sometimes referred to as the index set. A symmetric function K: \mathcal X \times \mathcal X \to \mathbb is called a positive-definite (p.d.) kernel on \mathcal X if holds for all x_1, \dots, x_n\in \mathcal X, n\in \mathbb, c_1, \dots, c_n \in \mathbb. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Reproducing Kernel Hilbert Space
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is an RKHS if the point-evaluation functional L_x:H\to\mathbb, L_x(f)=f(x), is continuous for every x\in X. Equivalently, H is an RKHS if there exists a function K_x \in H such that, for all f \in H,\langle f, K_x \rangle = f(x).The function K_x is then called the ''reproducing kernel'', and it reproduces the value of f at x via the inner product. An immediate consequence of this property is that convergence in norm implies uniform convergence on any subset of X on which \, K_x\, is bounded. However, the converse does not necessarily hold. Often the set X carries a topology, and \, K_x\, depends continuously on x\in X, in which case: convergence in norm implies uniform convergence on compact subsets of X. It is not entirely straightforwar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kernelization
In computer science, a kernelization is a technique for designing efficient algorithms that achieve their efficiency by a preprocessing stage in which inputs to the algorithm are replaced by a smaller input, called a "kernel". The result of solving the problem on the kernel should either be the same as on the original input, or it should be easy to transform the output on the kernel to the desired output for the original problem. Kernelization is often achieved by applying a set of reduction rules that cut away parts of the instance that are easy to handle. In parameterized complexity theory, it is often possible to prove that a kernel with guaranteed bounds on the size of a kernel (as a function of some parameter associated to the problem) can be found in polynomial time. When this is possible, it results in a fixed-parameter tractable algorithm whose running time is the sum of the (polynomial time) kernelization step and the (non-polynomial but bounded by the parameter) time to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 kernel (algebra), 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 Equaliser (mathematics), 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 theory, 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 (mathematics), equaliser of ''f'' and the zero morphism from ''X'' to ''Y''. In symbols: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kernel (algebra)
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kernel (linear Algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: \ker(L) = \left\ = L^(\mathbf). Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kernel (image Processing)
In image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di .... Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image, the kernel is that function. Details The general expression of a convolution is g_ = \omega * f_ = \sum_^a , where g(x,y) is the filtered image, f(x,y) is the original image, \omega is the filter kernel. Every element of the filter kernel is considered by -a \leq i \leq a and -b \leq j \leq b. Depending on the element values, a kernel can cause a wide range of effects: The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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). These methods involve using linear classifiers to solve nonlinear problems. 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 exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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). These methods involve using linear classifiers to solve nonlinear problems. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Apricot Kernel
An apricot kernel is the apricot seed located within the fruit endocarp, which forms a hard shell around the seed called the pyrena (stone or pit). The kernel contains amygdalin, a poisonous compound, in concentrations that vary between cultivars. Together with the related synthetic compound laetrile, amygdalin has been marketed as an alternative cancer treatment. However, studies have found the compounds to be ineffective for treating cancer. __TOC__ Use The kernel is an economically significant byproduct of fruit processing and the extracted oil and resulting press cake have value. Apricot kernel oil gives Disaronno and some other types of amaretto their almond-like flavor. They are also used in Amaretti di Saronno. In Mandarin Chinese, the term () can refer to either apricot kernels or almonds. Two varieties of apricot kernels are used in Chinese cuisines; a more bitter northern variety and a sweeter southern one. In Cantonese cuisine the two are often mixed, whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |