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Additional info for Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (Adaptive Computation and Machine Learning series)
To turn it into a Hilbert space (over ), one completes it in f f . This is done by adding the norm corresponding to the dot product, f : the limit points of sequences that are convergent in that norm (see Appendix B). 2. This is illustrated by the following quotation from an excellent machine learning textbook published in the seventies (p. 174 in ): “The familiar functions of mathematical physics are eigenfunctions of symmetric kernels, and their use is often suggested for the construction of potential functions.
4). 15) is often referred to as the quadratic form induced by K. 5 ((Positive Deﬁnite) Kernel) Let be a nonempty set. A function k on which for all m and all x 1 xm gives rise to a positive deﬁnite Gram matrix is called a positive deﬁnite (pd) kernel. Often, we shall refer to it simply as a kernel. 6 (Terminology) The term kernel stems from the ﬁrst use of this type of function in the ﬁeld of integral operators as studied by Hilbert and others [243, 359, 112]. 16) is called the kernel of Tk . In the literature, a number of different terms are used for positive deﬁnite kernels, such as reproducing kernel, Mercer kernel, admissible kernel, Support Vector kernel, nonnegative deﬁnite kernel, and covariance function.
16) is called the kernel of Tk . In the literature, a number of different terms are used for positive deﬁnite kernels, such as reproducing kernel, Mercer kernel, admissible kernel, Support Vector kernel, nonnegative deﬁnite kernel, and covariance function. One might argue that the term positive deﬁnite kernel is slightly misleading. In matrix theory, the term deﬁnite is cm 0. 15) only occurs if c1 1. The bar in c¯ j denotes complex conjugation; for real numbers, it has no effect. 2 The Representation of Similarities in Linear Spaces 31 Simply using the term positive kernel, on the other hand, could be mistaken as referring to a kernel whose values are positive.