Description of slcenkernel

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slcenkernel

PURPOSE

SLCENKERNEL Compute the centralized kernel matrix

SYNOPSIS

function KC = slcenkernel(K0, K, w)

DESCRIPTION

SLCENKERNEL Compute the centralized kernel matrix

 $ Syntax $
   - KC = slcenkernel(K0)
   - KC = slcenkernel(K0, [], w)
   - KC = slcenkernel(K0, K)
   - KC = slcenkernel(K0, K, w)

 $ Arguments $
   - K0:       the gram matrix of the referenced samples
   - K:        the kernel matrix for target samples
   - w:        the weights for the referenced samples
   - KC:       the centralized kernel matrix.

 $ Description $
   - KC = slcenkernel(K0) compute the centralized kernel matrix from
     the original kernel gram matrix K0.
 
   - KC = slcenkernel(K, [], w) compute the centralized kernel gram
     matrix from the original kernel gram matrix K0. The mean feature
     is obtained with the weights for referenced samples, given by w.

   - KC = slcenkernel(K0, K) compute the centralized kernel matrix
     for target samples, with the original gram matrix for referenced
     samples K0 and the kernel matrix for target samples w.r.t the 
     referenced samples K given.

   - KC = slcenkernel(K0, K, w) compute the centralied kernel matrix
     for target samples, with feature mean computed in a weighted 
     manner.

 $ Remarks $
   -# For original kernel matrix K, it is defined as 
      K(i, j) = <phi(i), phi(j)>, given that phi(i) and phi(j) are
      the feature map of the samples x(i) and x(j) respectively.
      Then the centralized kernel matrix is defined as
      KC(i, j) = <phi(i) - mean_phi, phi(j) - mean_phi>, where
      mean_phi is the mean of all referenced feature maps. If w
      is specified mean_phi is given by weighted mean. 
      Kernel centralization plays an important role in many 
      kernelized algorithms such as Kernel PCA.

   -# Suppose the mean feature map is defined by 
      mean_phi = sum_i w_i phi(i).
      Then the centralized kernel gram matrix can be written as
      KC = K - 1 * (w^T * K) - (K * w) * 1^T + 1 * (w^T * K * w) * 1^T.
      It can be easily shown that the mean of columns (rows) of KC is
      a zero vector. Thus, centralize the centralized kernel matrix
      would keep the input unchanged. 

   -# Instead of applying the formula given above, the function 
      implements a more efficient computational routine by reducing
      the redundant computations.

 $ History $
   - Created by Dahua Lin on May 2nd, 2006
   - Modified by Dahua Lin on Sep 10, 2006
       - use sladdrowcols to replace sladd to increase efficiency

CROSS-REFERENCE INFORMATION

This function calls:

• sladdrowcols SLADDROWCOLS Adds the vectors to all rows and all columns

This function is called by:

• slgda SLGDA Performs Baudat's Generalized Discriminant Analysis
• slkernelfea SLKERNELFEA Extracts kernelized mapped features
• slkpca SLPCA Learns a Kernel PCA model from training samples

SOURCE CODE

0001 function KC = slcenkernel(K0, K, w)
0002 %SLCENKERNEL Compute the centralized kernel matrix
0003 %
0004 % $ Syntax $
0005 %   - KC = slcenkernel(K0)
0006 %   - KC = slcenkernel(K0, [], w)
0007 %   - KC = slcenkernel(K0, K)
0008 %   - KC = slcenkernel(K0, K, w)
0009 %
0010 % $ Arguments $
0011 %   - K0:       the gram matrix of the referenced samples
0012 %   - K:        the kernel matrix for target samples
0013 %   - w:        the weights for the referenced samples
0014 %   - KC:       the centralized kernel matrix.
0015 %
0016 % $ Description $
0017 %   - KC = slcenkernel(K0) compute the centralized kernel matrix from
0018 %     the original kernel gram matrix K0.
0019 %
0020 %   - KC = slcenkernel(K, [], w) compute the centralized kernel gram
0021 %     matrix from the original kernel gram matrix K0. The mean feature
0022 %     is obtained with the weights for referenced samples, given by w.
0023 %
0024 %   - KC = slcenkernel(K0, K) compute the centralized kernel matrix
0025 %     for target samples, with the original gram matrix for referenced
0026 %     samples K0 and the kernel matrix for target samples w.r.t the
0027 %     referenced samples K given.
0028 %
0029 %   - KC = slcenkernel(K0, K, w) compute the centralied kernel matrix
0030 %     for target samples, with feature mean computed in a weighted
0031 %     manner.
0032 %
0033 % $ Remarks $
0034 %   -# For original kernel matrix K, it is defined as
0035 %      K(i, j) = <phi(i), phi(j)>, given that phi(i) and phi(j) are
0036 %      the feature map of the samples x(i) and x(j) respectively.
0037 %      Then the centralized kernel matrix is defined as
0038 %      KC(i, j) = <phi(i) - mean_phi, phi(j) - mean_phi>, where
0039 %      mean_phi is the mean of all referenced feature maps. If w
0040 %      is specified mean_phi is given by weighted mean.
0041 %      Kernel centralization plays an important role in many
0042 %      kernelized algorithms such as Kernel PCA.
0043 %
0044 %   -# Suppose the mean feature map is defined by
0045 %      mean_phi = sum_i w_i phi(i).
0046 %      Then the centralized kernel gram matrix can be written as
0047 %      KC = K - 1 * (w^T * K) - (K * w) * 1^T + 1 * (w^T * K * w) * 1^T.
0048 %      It can be easily shown that the mean of columns (rows) of KC is
0049 %      a zero vector. Thus, centralize the centralized kernel matrix
0050 %      would keep the input unchanged.
0051 %
0052 %   -# Instead of applying the formula given above, the function
0053 %      implements a more efficient computational routine by reducing
0054 %      the redundant computations.
0055 %
0056 % $ History $
0057 %   - Created by Dahua Lin on May 2nd, 2006
0058 %   - Modified by Dahua Lin on Sep 10, 2006
0059 %       - use sladdrowcols to replace sladd to increase efficiency
0060 %
0061 
0062 %% parse and verify input arguments
0063 
0064 % for K0
0065 n0 = size(K0, 1);
0066 if ndims(K0) ~= 2 || size(K0, 2) ~= n0;
0067     error('sltoolbox:invaliddims', ...
0068         'K0 should be a 2D square matrix');
0069 end
0070 
0071 % for K
0072 if nargin < 2 || isempty(K)
0073     K = K0;
0074 else
0075     if ndims(K) ~= 2
0076         error('sltoolbox:invaliddims', ...
0077             'K should be a 2D matrix');
0078     end
0079     if size(K, 1) ~= n0
0080         error('sltoolbox:sizmismatch', ...
0081             'Size inconsistency between K0 and K');
0082     end
0083 end
0084     
0085 % for w
0086 if nargin < 3 || isempty(w)
0087     isweighted = false;
0088 else
0089     if ~isequal(size(w), [1, n0])
0090         error('sltoolbox:sizmismatch', ...
0091             'Size inconsistency between K0 and w');
0092     end    
0093     isweighted = true;
0094 end
0095 
0096 
0097 %% compute
0098 
0099 % Steps:
0100 % 1. compute v1: mean row vector of K (1 x n)
0101 % 2. compute v2: mean column vector of K0 (n x 1)
0102 % 3. compute s3: mean value of of all elements of K0 (1 x 1)
0103 % 4. KC = K - expand(v1) - expand(v2) + s3
0104 
0105 if ~isweighted  % non-weighted case
0106 
0107     v1 = sum(K, 1) * (1 / n0);
0108     v2 = sum(K0, 2) * (1 / n0);
0109     s3 = sum(v2) * (1 / n0);
0110     
0111 else            % weighted case
0112     
0113     w = w / sum(w);     % normalize the weights
0114     
0115     v1 = w * K;
0116     v2 = K0 * w';
0117     s3 = w * v2;
0118     
0119 end
0120 
0121 KC = sladdrowcols(K, -v1, -v2+s3);
0122 
0123

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