This article is about machine learning. For polynomial kernels in complexity theory, see Kernelization.
Illustration of the mapping . On the left a set of samples in the input space, on the right the same samples in the feature space where the polynomial kernel (for some values of the parameters and ) is the inner product. The hyperplane learned in feature space by an SVM is an ellipse in the input space.
In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models.
Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features.[1]
^Yoav Goldberg and Michael Elhadad (2008). splitSVM: Fast, Space-Efficient, non-Heuristic, Polynomial Kernel Computation for NLP Applications. Proc. ACL-08: HLT.
machine learning, the polynomialkernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents...
splines). A final alternative is to use kernelized models such as support vector regression with a polynomialkernel. If residuals have unequal variance,...
learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular,...
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is...
is the sum of the (polynomial time) kernelization step and the (non-polynomial but bounded by the parameter) time to solve the kernel. Indeed, every problem...
usually used for SVM. In situ adaptive tabulation Kernel machines Fisher kernel Platt scaling Polynomialkernel Predictive analytics Regularization perspectives...
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)} and...
two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u...
distribution) combined with popular embedding kernels k {\displaystyle k} (e.g. the Gaussian kernel or polynomialkernel), or can be accurately empirically estimated...
A kernel smoother is a statistical technique to estimate a real valued function f : R p → R {\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} } as the weighted...
Schölkopf (2006). "A unifying view of Wiener and Volterra theory and polynomialkernel regression". Neural Computation. 18 (12): 3097–3118. doi:10.1162/neco...
of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more...
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition...
root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number...
z , {\displaystyle K(x,z)=x^{\mathsf {T}}z,} the polynomialkernel, inducing the space of polynomial functions of order d {\displaystyle d} : K ( x ,...
admit a polynomial-sized approximate kernelization scheme (PSAKS): for any ε > 0 {\displaystyle \varepsilon >0} it is possible to compute a polynomial-sized...
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets...
the Mehler kernel for Hermite polynomials, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above...
properties of tensor sketches, particularly focused on applications to polynomialkernels. In this context, the sketch is required not only to preserve the...
by reduction to binary tasks. It is a type of kernel machine that uses an inhomogeneous polynomialkernel. Hastie, Trevor; Tibshirani, Robert; Friedman...
Schölkopf, B. (2006). "A unifying view of Wiener and Volterra theory and polynomialkernel regression". Neural Computation. 18 (12): 3097–3118. doi:10.1162/neco...
on 2013-08-11. Ninh, Pham; Pagh, Rasmus (2013). Fast and scalable polynomialkernels via explicit feature maps. SIGKDD international conference on Knowledge...
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