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The function : is often referred to as a kernel or a kernel function. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or integral . Certain problems in machine learning have more structure than an arbitrary weighting function k {\displaystyle k} .
Thus, in a sufficiently rich hypothesis space—or equivalently, for an appropriately chosen kernel—the SVM classifier will converge to the simplest function (in terms of ) that correctly classifies the data. This extends the geometric interpretation of SVM—for linear classification, the empirical risk is minimized by any function whose ...
Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. [2] The feature space of the kernel has an infinite number of dimensions; for =, its expansion using the multinomial theorem is: [3]
For degree-d polynomials, the polynomial kernel is defined as [2](,) = (+)where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial.
Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis.
[2] [3] [4] This has enabled detailed comparisons between SVM and other forms of Tikhonov regularization, and theoretical grounding for why it is beneficial to use SVM's loss function, the hinge loss. [5]
Positive-definite kernels, through their equivalence with reproducing kernel Hilbert spaces (RKHS), are particularly important in the field of statistical learning theory because of the celebrated representer theorem which states that every minimizer function in an RKHS can be written as a linear combination of the kernel function evaluated at ...
k-NN classifier with a {0-1} loss function. [2] Support Vector Machine (SVM) classification with a bounded kernel and where the regularizer is a norm in a Reproducing Kernel Hilbert Space. A large regularization constant leads to good stability. [4] Soft margin SVM classification. [4] Regularized Least Squares regression. [4]