Full Record

Author | McWhorter, Samuel P. |

Title | Fundamental Issues in Support Vector Machines |

URL | https://digital.library.unt.edu/ark:/67531/metadc500155/ |

Publication Date | 2014 |

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 | 2020-08-05 |

Date Indexed | 2020-08-09 |

Sample Search Hits | Sample Images | Cited Works

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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 ∈
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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…

…selecting a family of *kernel* functions, computing a minimum-distance vector between
the well-separated convex hulls of the categories in a related Hilbert space, then using this
vector to construct a separating function as a sum of linear combinations of…

…*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…