Description of slkpca

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slkpca

PURPOSE

SLPCA Learns a Kernel PCA model from training samples

SYNOPSIS

function [A, evs] = slkpca(K0, varargin)

DESCRIPTION

SLPCA Learns a Kernel PCA model from training samples

 $ Syntax $
   - A = slkpca(K0)
   - A = slkpca(K0, ...)
   - [A, evs] = slkpca(K0, ...)

 $ Arguments $
   - X:        the training sample matrix
   - A:        the projection coefficient matrix 
   - evs:      the egienvalues of all n0-1 dimensions 
               (including preserved and discarded)

 $ Description $
   - A = slpca(K0) learns a Kernel PCA model from the samples X by 
     default settings.

   - A = slpca(K0, ...) learns a Kernel PCA model from the samples X 
     according to the properties specified:
     \*
     \t   Table 1. The properties of Kernel PCA learning
     \h    name       &    description                               \\
          'preserve'  & Determine how many components are preserved, it is
                        given in following form: {sch, ...}, which is used
                        as parameters in sldim_by_eigval.       \\
          'weights'   & The 1 x n row vector of sample weights.  
                        If the weights are not specified, then it 
                        considers that all samples have equal weights. 
                        default = [].   \\
        'centralized' & Whether the K0 has been centralized. default =
                        false. If the K0 is not centralized, it will 
                        first centralize it before subsequent steps. \\
     \*
  
 $ Remarks $
   -# The coefficient vectors are normalized so that a^T * K * a = 1.

 $ History $
   - Created by Dahua Lin on May 02, 2006
   - Modified by Dahua Lin on Sep 10, 2006
       - use slmulvec and slmulrowcols to increase efficiency
         in the weighted cases

CROSS-REFERENCE INFORMATION

This function calls:

• slmulrowcols SLMULROWCOLS Multiplies the vectors to all rows and all columns
• slmulvec SLMULVEC multiplies a vector to columns or rows of a matrix
• slsymeig SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
• slcenkernel SLCENKERNEL Compute the centralized kernel matrix
• sldim_by_eigval SLDIM_BY_EIGVAL Determines the dimension of principal subspace by eigenvalues
• slparseprops SLPARSEPROPS Parses input parameters

This function is called by:

SOURCE CODE

0001 function [A, evs] = slkpca(K0, varargin)
0002 %SLPCA Learns a Kernel PCA model from training samples
0003 %
0004 % $ Syntax $
0005 %   - A = slkpca(K0)
0006 %   - A = slkpca(K0, ...)
0007 %   - [A, evs] = slkpca(K0, ...)
0008 %
0009 % $ Arguments $
0010 %   - X:        the training sample matrix
0011 %   - A:        the projection coefficient matrix
0012 %   - evs:      the egienvalues of all n0-1 dimensions
0013 %               (including preserved and discarded)
0014 %
0015 % $ Description $
0016 %   - A = slpca(K0) learns a Kernel PCA model from the samples X by
0017 %     default settings.
0018 %
0019 %   - A = slpca(K0, ...) learns a Kernel PCA model from the samples X
0020 %     according to the properties specified:
0021 %     \*
0022 %     \t   Table 1. The properties of Kernel PCA learning
0023 %     \h    name       &    description                               \\
0024 %          'preserve'  & Determine how many components are preserved, it is
0025 %                        given in following form: {sch, ...}, which is used
0026 %                        as parameters in sldim_by_eigval.       \\
0027 %          'weights'   & The 1 x n row vector of sample weights.
0028 %                        If the weights are not specified, then it
0029 %                        considers that all samples have equal weights.
0030 %                        default = [].   \\
0031 %        'centralized' & Whether the K0 has been centralized. default =
0032 %                        false. If the K0 is not centralized, it will
0033 %                        first centralize it before subsequent steps. \\
0034 %     \*
0035 %
0036 % $ Remarks $
0037 %   -# The coefficient vectors are normalized so that a^T * K * a = 1.
0038 %
0039 % $ History $
0040 %   - Created by Dahua Lin on May 02, 2006
0041 %   - Modified by Dahua Lin on Sep 10, 2006
0042 %       - use slmulvec and slmulrowcols to increase efficiency
0043 %         in the weighted cases
0044 %
0045 
0046 %% parse and verify input arguments
0047 
0048 % for K0
0049 n0 = size(K0, 1);
0050 if ~isequal(size(K0), [n0 n0])
0051     error('sltoolbox:invaliddims', 'K0 should be a square matrix');
0052 end
0053 
0054 % for options
0055 opts.preserve = {};
0056 opts.weights = [];
0057 opts.centralized = false;
0058 opts = slparseprops(opts, varargin{:});
0059 
0060 if isempty(opts.weights)
0061     isweighted = false;
0062 else
0063     isweighted = true;
0064     if ~isequal(size(opts.weights), [1, n0])
0065         error('sltoolbox:sizmismatch', ...
0066             'The weights should be a 1 x n0 row vector');
0067     end
0068 end
0069 
0070 
0071 %% compute
0072 
0073 %% centralize the gram matrix
0074 if ~opts.centralized
0075     K0 = slcenkernel(K0, [], opts.weights);
0076 end
0077 
0078 %% solve the eigen-problem
0079 
0080 if ~isweighted      % non weighted case
0081     [evs, A] = slsymeig(K0);
0082     evs = evs(1:n0-1);
0083     
0084     k = sldim_by_eigval(evs, opts.preserve{:});
0085     A = A(:, 1:k);
0086     A = slmulvec(A, 1 ./ sqrt(evs(1:k))', 2); % normalize
0087     
0088     evs = evs / n0;
0089     
0090 else                % weighted case
0091     
0092     w = max(opts.weights, 0);
0093     sw = sqrt(w)'; % sw is a column vector
0094     K0w = slmulrowcols(K0, sw', sw);
0095     [evs, A] = slsymeig(K0w);
0096     evs = evs(1:n0-1);
0097     
0098     k = sldim_by_eigval(evs, opts.preserve{:});
0099     A = A(:, 1:k);
0100     A = slmulrowcols(A, 1 ./ sqrt(evs(1:k))', sw); % normalize
0101     
0102     evs = evs(1:n0-1) / sum(w);    
0103 end
0104

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