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Highlights from
Statistical Learning Toolbox

from Statistical Learning Toolbox by Dahua Lin
Functions for statistical learning, pattern recognition and computer vision, covering many topics.

Description of slkernel
Home > sltoolbox > kernel > slkernel.m

slkernel

PURPOSE ^

SLKERNEL Computes the kernel for samples

SYNOPSIS ^

function K = slkernel(varargin)

DESCRIPTION ^

SLKERNEL Computes the kernel for samples

 $ Syntax $
   - K = slkernel(X0, kernel_type, ...)
   - K = slkernel(X0, X, kernel_type, ...)

 $ Description $
   - K = slkernel(X0, kernel_type, ...) Computes the Gram matrix for
     the samples in matrix X0 using the kernel specified by kernel_type. 
     kernel_type can be name of a built-in kernel type, or the name, 
     handle of an user-specified function. The user-specified function
     should take in two d x n matrix containing n pairs of vectors, and
     output a 1 x n vector of their kernel values. The available builtin
     kernels are given as follows:
     \*
     \t   Table 1.  Built-in Kernel Types          \\
     \h     name     &    description              \\
           'lin'     &  Linear Kernel: 
                        x1' * x2, 
                        with no parameters         \\
           'gauss'   &  Gaussian RBF Kernel:
                        exp(- ||x1 - x2||^2 / (2 * sigma^2)), 
                        with one parameter sigma   \\ 
           'poly'    &  Polynomial Kernel:
                        ((x1' * x2) + a)^k,
                        with two parameters: k and a \\
           'sigmoid' &  Sigmoidal Kernel:
                        tanh(k * (x1' * x2) + a),    
                        with two parameters: k and a \\
           'invquad' &  Inverse Quadratic Kernel:
                        1 / sqrt(||x1 - x2||^2 + a), 
                        with one parameter: a        \\
     \* 

   - K = slkernel(X0, X, kernel_type, p1, p2, ...) Computes the empirical
     kernel mapping for samples X with respect to data set X0. Suppose
     there are n0 samples in X0, n samples in X, then K will be an n0 * n
     matrix. Each column in K corresponds to a column in X.

 $ History $
   - Created by Dahua Lin on Jul 13rd, 2005
   - Modified by Dahua Lin on May 2nd, 2005
       - base on sltoolbox v4
       - re-organize the code structure
       - add the ability of user-specified kernel functions.

CROSS-REFERENCE INFORMATION ^

This function calls:
  • slmetric_pw SLMETRIC_PW Compute the metric between column vectors pairwisely
  • slpweval SLPWEVAL Perform pairwise computation
This function is called by:
  • slkernelfea SLKERNELFEA Extracts kernelized mapped features

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function K = slkernel(varargin)
0002 %SLKERNEL Computes the kernel for samples
0003 %
0004 % $ Syntax $
0005 %   - K = slkernel(X0, kernel_type, ...)
0006 %   - K = slkernel(X0, X, kernel_type, ...)
0007 %
0008 % $ Description $
0009 %   - K = slkernel(X0, kernel_type, ...) Computes the Gram matrix for
0010 %     the samples in matrix X0 using the kernel specified by kernel_type.
0011 %     kernel_type can be name of a built-in kernel type, or the name,
0012 %     handle of an user-specified function. The user-specified function
0013 %     should take in two d x n matrix containing n pairs of vectors, and
0014 %     output a 1 x n vector of their kernel values. The available builtin
0015 %     kernels are given as follows:
0016 %     \*
0017 %     \t   Table 1.  Built-in Kernel Types          \\
0018 %     \h     name     &    description              \\
0019 %           'lin'     &  Linear Kernel:
0020 %                        x1' * x2,
0021 %                        with no parameters         \\
0022 %           'gauss'   &  Gaussian RBF Kernel:
0023 %                        exp(- ||x1 - x2||^2 / (2 * sigma^2)),
0024 %                        with one parameter sigma   \\
0025 %           'poly'    &  Polynomial Kernel:
0026 %                        ((x1' * x2) + a)^k,
0027 %                        with two parameters: k and a \\
0028 %           'sigmoid' &  Sigmoidal Kernel:
0029 %                        tanh(k * (x1' * x2) + a),
0030 %                        with two parameters: k and a \\
0031 %           'invquad' &  Inverse Quadratic Kernel:
0032 %                        1 / sqrt(||x1 - x2||^2 + a),
0033 %                        with one parameter: a        \\
0034 %     \*
0035 %
0036 %   - K = slkernel(X0, X, kernel_type, p1, p2, ...) Computes the empirical
0037 %     kernel mapping for samples X with respect to data set X0. Suppose
0038 %     there are n0 samples in X0, n samples in X, then K will be an n0 * n
0039 %     matrix. Each column in K corresponds to a column in X.
0040 %
0041 % $ History $
0042 %   - Created by Dahua Lin on Jul 13rd, 2005
0043 %   - Modified by Dahua Lin on May 2nd, 2005
0044 %       - base on sltoolbox v4
0045 %       - re-organize the code structure
0046 %       - add the ability of user-specified kernel functions.
0047 %
0048 
0049 %% parse and verify input arguments
0050 
0051 % for X0
0052 if ~isnumeric(varargin{1})
0053     error('sltoolbox:invalidarg', ...
0054         'The first argument should be an numeric matrix');
0055 end
0056 X0 = varargin{1};
0057 if ndims(X0) ~= 2
0058     error('sltoolbox:invaliddims', ...
0059         'The X0 should be a 2D matrix');
0060 end
0061 
0062 % for X
0063 if isnumeric(varargin{2})
0064     X = varargin{2};    
0065     if ndims(X) ~= 2
0066         error('sltoolbox:invaliddims', ...
0067             'X should be a 2D matrix');
0068     end
0069     ipkt = 3;       % argument index of kernel type
0070 else
0071     X = X0;
0072     ipkt = 2;        
0073 end
0074 
0075 % for kernel type
0076 if nargin < ipkt || isempty(varargin{ipkt})
0077     error('sltoolbox:invalidarg', ...
0078         'kernel type is not specified');
0079 end
0080 kernel_type = varargin{ipkt};
0081 
0082 % for extra parameters
0083 if nargin == ipkt       % no extra parameters
0084     params = {};
0085 else
0086     params = varargin(ipkt+1:end);
0087 end
0088 
0089 
0090 %% Determine the computation routine and delegate to it
0091 
0092 % determine for the built-in ones
0093 
0094 bik = false;
0095 if ischar(kernel_type)
0096     switch kernel_type
0097         case 'lin'
0098             bik = true;   % bik means built-in kernel
0099             fh_kernel = @lin_kernel;
0100         case 'gauss'
0101             bik = true;
0102             fh_kernel = @gauss_kernel;
0103         case 'poly'
0104             bik = true;
0105             fh_kernel = @poly_kernel;
0106         case 'sigmoid'
0107             bik = true;
0108             fh_kernel = @sigmoid_kernel;
0109         case 'invquad'
0110             bik = true;
0111             fh_kernel = @invquad_kernel;
0112     end
0113 end
0114 
0115 % delegate
0116 if bik
0117     K = fh_kernel(X0, X, params{:});
0118 else
0119     K = slpweval(X0, X, kernel_type, params{:});
0120 end
0121 
0122 
0123 %% The built-in kernel functions
0124 
0125 % Linear Kernel
0126 function K = lin_kernel(X0, X)
0127 
0128 K = X0' * X;
0129 
0130 
0131 % Gaussian Radius Base Function Kernel
0132 function K = gauss_kernel(X0, X, sigma)
0133 
0134 D2 = slmetric_pw(X0, X, 'sqdist');
0135 K = exp(- D2 / (2 * sigma * sigma));
0136 
0137 
0138 % Polynomial Kernel
0139 function K = poly_kernel(X0, X, k, a)
0140 
0141 K = (X0' * X + a).^k;
0142 
0143 
0144 % Sigmoidal Kernel
0145 function K = sigmoid_kernel(X0, X, k, a)
0146 
0147 K = tanh(k * (X0' * X) + a);
0148 
0149 
0150 % Inverse Quadratic Kernel
0151 function K = invquad_kernel(X0, X, a)
0152 
0153 D2 = slmetric_pw(X0, X, 'sqdist');
0154 K = 1 ./ sqrt(D2 + a);
0155 
0156
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