By Bernhard Schlkopf, Alexander J. Smola
In the Nineteen Nineties, a brand new kind of studying set of rules was once built, in accordance with effects from statistical studying conception: the help Vector desktop (SVM). This gave upward thrust to a brand new category of theoretically based studying machines that use a primary proposal of SVMs -- -kernels--for a few studying projects. Kernel machines offer a modular framework that may be tailored to assorted projects and domain names by means of the alternative of the kernel functionality and the bottom set of rules. they're changing neural networks in a number of fields, together with engineering, info retrieval, and bioinformatics.
Learning with Kernels offers an advent to SVMs and comparable kernel tools. even if the publication starts off with the fundamentals, it's also the most recent study. It offers the entire innovations essential to allow a reader built with a few uncomplicated mathematical wisdom to go into the realm of computer studying utilizing theoretically well-founded but easy-to-use kernel algorithms and to appreciate and observe the robust algorithms which have been built over the past few years.
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Extra info for Learning with kernels : support vector machines, regularization, optimization, and beyond
Example text
The algorithm uct based algorithm operating on vectorial data Φ(x 1 ) obtained by replacing k by k˜ then is exactly the same dot product based algorithm, ˜ m ). ˜ 1) Φ(x only that it operates on Φ(x The best known application of the kernel trick is in the case where k is the dot product in the input domain (cf. 5). The trick is not limited to that case, however: k and k˜ can both be nonlinear kernels. , the data set might have to lie in the positive orthant. We shall later see that certain kernels induce feature maps which enforce such properties for the mapped data (cf.
This is due to the fact that ellipses can be written as linear equations in the entries of (z1 z2 z3 ). Therefore, in feature space, the problem reduces to that of estimating a hyperplane from the mapped data points. 13)), the dot product in the three-dimensional space can be computed without computing Φ2 . Later in the book, we shall describe algorithms for constructing hyperplanes which are based on dot products (Chapter 7). 2 The Representation of Similarities in Linear Spaces In what follows, we will look at things the other way round, and start with the kernel rather than with the feature map.
6 Support Vector Regression Let us turn to a problem slightly more general than pattern recognition. Rather 1 , regression estimation is concerned with estithan dealing with outputs y mating real-valued functions. To generalize the SV algorithm to the regression case, an analog of the soft margin is constructed in the space of the target values y (note that we now have 11. 40) by C m rather than C, as done in Chapter 7 below. 8 In SV regression, a tube with radius is fitted to the data. 47). 8, see Chapters 3 and 9) .