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Title Fundamental Issues in Support Vector Machines
URL
Publication Date
University/Publisher University of North Texas
Abstract This dissertation considers certain issues in support vector machines (SVMs), including a description of their construction, aspects of certain exponential kernels used in some SVMs, and a presentation of an algorithm that computes the necessary elements of their operation with proof of convergence. In its first section, this dissertation provides a reasonably complete description of SVMs and their theoretical basis, along with a few motivating examples and counterexamples. This section may be used as an accessible, stand-alone introduction to the subject of SVMs for the advanced undergraduate. Its second section provides a proof of the positive-definiteness of a certain useful function here called E and dened as follows: Let V be a complex inner product space. Let N be a function that maps a vector from V to its norm. Let p be a real number between 0 and 2 inclusive and for any in V , let ( be N() raised to the p-th power. Finally, let a be a positive real number. Then E() is exp(()). Although the result is not new (other proofs are known but involve deep properties of stochastic processes) this proof is accessible to advanced undergraduates with a decent grasp of linear algebra. Its final section presents an algorithm by Dr. Kallman (preprint), based on earlier Russian work by B.F. Mitchell, V.F Demyanov, and V.N. Malozemov, and proves its convergence. The section also discusses briefly architectural features of the algorithm expected to result in practical speed increases.
Subjects/Keywords Support vector machines; radial basis function kernel; exponential kernel; elementary proof; Support vector machines.; Algorithms.
Contributors Kallman, Robert R.; Brozovic, Douglas; Brand, Neal E.
Language en
Rights Public ; McWhorter, Samuel P. ; Copyright ; Copyright is held by the author, unless otherwise noted. All rights Reserved.
Country of Publication us
Format iii, 48 pages
Record ID info:ark/67531/metadc500155
Other Identifiers ark: ark:/67531/metadc500155
Repository unt
Date Retrieved
Date Indexed 2020-08-09

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…since c ∈ P , H = {w | w ∈ V ∧ hw, vj i + qj = qj } = {w | w ∈ V ∧ hw, vj i = 0} so H is the kernel of the map lj : V → R given by lj (w) = hw, vj i. Since hvj , vj i = 6 0, lj (vj ) 6= 0, so vj ∈ / H. This is true…

…letter at any size, with any acceptable shape of glyph, possibly in the presence of other characters, in some variety of possible orientations. 1.6. Kernel Functions Let F, Φ, V , C1 , C2 , v, f be as in our discussion in the previous section. In that…

…separating function goes, specifying a unified function with certain properties has the same expressive power as separately specifying a Φ and a Hilbert space V , as we will now show. (The fundamental notion here, the “kernel trick”, is a well-known…

…Φ(x2 )i, In this way, we can see that in the construction of support vector machines, selection of a kernel is essentially equivalent to selection of a map Φ from a feature space to a particular Hilbert space. An aside: Although computing…

…operation is perhaps best understood geometrically through real Hilbert spaces, though their implementation is almost always by means of limits of sums of positivedefinite kernel functions from a feature space to R. Construction of this function proceeds by…

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kernel functions with one variable free, and one evaluated at a particular element of the feature space as in 21 equation 2. A practical method of finding such a vector is discussed in section 3. 22 CHAPTER 2 A USEFUL KERNEL 2.1. Introduction Kernels…

…are important in support vector machine operation. Kernels are constructed from positive-definite functions, usually from Rn for some natural number n. A typical example of a kernel function is exp(− kxk2 ), where x varies over Rn for some…

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