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Title Fundamental Issues in Support Vector Machines
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…

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…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…