sv_kernel.tex

一种新颖的SVM算法

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%/*
% * File:        sv_kernel.tex
% * Purpose:     Explanation of kernels
% *
% * Author:      Mark O. Stitson
% * Created:     00/00/97
% * Updated:     29/01/98
% *
% * $Id: sv_kernel.tex,v 1.9.2.2 1998/02/12 17:21:45 markst Exp $
% * 
% * Copyright (c) 1998  RHBNC London - All rights reserved
% * THIS IS PROPRIETARY SOURCE CODE of RHBNC London
% */

\author{M. O. Stitson}
\title{RHBNC SVM Kernels}

\begin{document}
\parindent=0pt
\parskip=7pt

\maketitle

\section{SV Kernels}
This is a list of the kernel functions in the RHUL SV Machine:
\begin{itemize}
\item 1. The simple dot product:

$$K(x,y)=x\cdot y$$

\item 2. The simple polynomial kernel:

$$K(x,y)=((x\cdot y)+1)^d$$

where $d$ is user defined.

Taken from \cite{vapnik95}.

\item 3. Vovk's real polynomial:

$$K(x,y)=\frac{1-(x\cdot y)^d}{1-(x\cdot y)}$$

where $d$ is user defined and where $-1<(x\cdot y)<1$.

From private communications with V. Vovk.

\item 4. Vovk's real infinite polynomial:

$$K(x,y)=\frac{1}{1-(x\cdot y)}$$

where $-1<(x\cdot y)<1$.

From private communications with V. Vovk.

\item 5. Radial Basis function:

$$\exp(-\gamma |x-y|^2)$$

where $\gamma$ is user defined.

Taken from \cite{vapnik95}.

\item 6. Two layer neural network:

$$\tanh(\frac{b(x \cdot y)}{1}-c)$$

where $b$ and $c$ are user defined.

Taken from \cite{vapnik95}.

\item 7. Linear splines with an infinite number of points:

For the one-dimensional case:

$$1+x_ix_j+x_ix_j\min(x_i,x_j)-\frac{x_i+x_j}{2}(\min(x_ix_j))^2+\frac{(\min(x_i,x_j))^3}{3}$$

For the multi-dimensional case $K(x,y)=\prod_{k=1}^{n}K_k(x^k,y^k)$

Taken from \cite{vladgolo}.

\item 8. Full polynomial kernel:

$$\big(\frac{x\cdot y}{a}+b\big)^d$$

where $a$, $b$ and $d$ are user defined.

From \cite{vapnik95} and generalized.

\item 9. Regularized Fourier (weaker mode regularization)

For the one-dimensional case:

$$\frac{\pi}{2\gamma}\frac{\cosh\frac{\pi-|x_i-x_j|}{\gamma}}
{\sinh\frac{\pi}{\gamma}}$$

where $0\leq |x_i-x_j| \leq 2\pi$ and $\gamma$ is user defined.

For the multi-dimensional case $K(x,y)=\prod_{k=1}^{n}K_k(x^k,y^k)$

From \cite{vladgolo} and \cite{vlad98}.

\item 10. Semi Local Kernel

$$[(x_i\cdot x_j)+1]^d\exp({-||x_i-x_j||^2\sigma^2})$$

where $d$ and $\sigma$ are user defined and weight between global and
local approximation.

From private communications with V. Vapnik.

\item 11. Regularized Fourier (stronger mode regularization)

For the one-dimensional case:

$$\frac{1-\gamma^2}{2(1-2\gamma\cos(x_i-x_j)+\gamma^2)}$$

where $0\leq |x_i-x_j| \leq 2\pi$ and $\gamma$ is user defined.

For the multi-dimensional case $K(x,y)=\prod_{k=1}^{n}K_k(x^k,y^k)$

From \cite{vladgolo} and \cite{vlad98}.

\item 17. Anova 1

$$K(x,y)=(\sum_{k=1}^{n}\exp(-\gamma(x^k-y^k)^2))^{d}$$

where the degree $d$ and $\gamma$ are user defined.

From private communications with V. Vapnik.

\item 18. Generic Kernel 1

This is a kernel intended for experiments, just modify the appropriate
function in kernel\_generic\_1\_c.C. You can use the parameters a\_val,
b\_val, c\_val, d\_val and e\_val.

\item 19. Generic Kernel 2

This is a kernel intended for experiments, just modify the appropriate
function in kernel\_generic\_2\_c.C. You can use the parameters a\_val,
b\_val, c\_val, d\_val and e\_val.

\end{itemize}

\include{bibliography}

\end{document}