Description of slkernelembed

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slkernelembed

PURPOSE

SLKERNELEMBED Finds an embedding space to preserve inner products

SYNOPSIS

function [X, spectrum] = slkernelembed(K, d, w)

DESCRIPTION

SLKERNELEMBED Finds an embedding space to preserve inner products

 $ Syntax $
   - X = slkernelembed(K, d)
   - X = slkernelembed(K, d, w)
   - [X, info] = slkernelembed(...)

 $ Arguments $
   - K:        The pairwise inner product matrix (kernel matrix): n x n
   - d:        The dimension of the embedded space 
   - w:        The weights of the samples: 1 x n
   - X:        The embeded sample coordinates: d x n
   - spectrum: The eigenvalues of the embeded space (dx1)

 $ Description $
   - X = slkernelembed(K, d, w) finds an embedding space in which the
     the inner product structure of the original sample set is best
     preserved in terms of square error of reconstructed kernel matrix.

   - X = slkernelembed(K, d, w) runs the algorithm on weighted samples.

   - [X, spectrum] = slkernelembed(...) additionally outputs the
     eigen-spectrum of the embedding space, which is a column vector
     of eigenvalues associated with all dimensions.

 $ Remarks $
   - The embeded dimension d should be less than the number of samples n.

   - The implementation is based on EVD decomposition.

 $ History $
   - Created by Dahua Lin, on Sep 8, 2006

CROSS-REFERENCE INFORMATION

This function calls:

• slsymeig SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
• raise_lackinput RAISE_LACKINPUT Raises an error indicating lack of input argument

This function is called by:

• slcmds SLMDS Performs Classical Multidimensional scaling

SOURCE CODE

0001 function [X, spectrum] = slkernelembed(K, d, w)
0002 %SLKERNELEMBED Finds an embedding space to preserve inner products
0003 %
0004 % $ Syntax $
0005 %   - X = slkernelembed(K, d)
0006 %   - X = slkernelembed(K, d, w)
0007 %   - [X, info] = slkernelembed(...)
0008 %
0009 % $ Arguments $
0010 %   - K:        The pairwise inner product matrix (kernel matrix): n x n
0011 %   - d:        The dimension of the embedded space
0012 %   - w:        The weights of the samples: 1 x n
0013 %   - X:        The embeded sample coordinates: d x n
0014 %   - spectrum: The eigenvalues of the embeded space (dx1)
0015 %
0016 % $ Description $
0017 %   - X = slkernelembed(K, d, w) finds an embedding space in which the
0018 %     the inner product structure of the original sample set is best
0019 %     preserved in terms of square error of reconstructed kernel matrix.
0020 %
0021 %   - X = slkernelembed(K, d, w) runs the algorithm on weighted samples.
0022 %
0023 %   - [X, spectrum] = slkernelembed(...) additionally outputs the
0024 %     eigen-spectrum of the embedding space, which is a column vector
0025 %     of eigenvalues associated with all dimensions.
0026 %
0027 % $ Remarks $
0028 %   - The embeded dimension d should be less than the number of samples n.
0029 %
0030 %   - The implementation is based on EVD decomposition.
0031 %
0032 % $ History $
0033 %   - Created by Dahua Lin, on Sep 8, 2006
0034 %
0035 
0036 %% parse and verify input arguments
0037 
0038 if nargin < 2
0039     raise_lackinput('slkernelembed', 2);
0040 end
0041 
0042 if ndims(K) ~= 2 || size(K, 1) ~= size(K, 2)
0043     error('sltoolbox:invalidarg', ...
0044         'The K should be a square matrix');
0045 end
0046 n = size(K, 1);
0047 
0048 if d >= n
0049     error('sltoolbox:exceedbound', ...
0050         'The dimension d should be less than the number of samples n');
0051 end
0052 
0053 if nargin < 3
0054     w = [];
0055 else
0056     if ~isempty(w)
0057         if ~isequal(size(w), [1, n])
0058             error('sltoolbox:sizmismatch', ...
0059                 'If w is specified, it should be an 1 x n row vector');
0060         end
0061     end
0062 end
0063 
0064 
0065 %% compute
0066 
0067 % enforce symmetry
0068 K = 0.5 * (K + K');
0069 
0070 % assign weights
0071 if ~isempty(w)
0072     for i = 1 : n
0073         K(i,:) = K(i,:) * w(i);
0074     end
0075     for i = 1 : n
0076         K(:,i) = K(:,i) * w(i);
0077     end
0078 end
0079 
0080 % decompose
0081 [spectrum, X] = slsymeig(K, d);
0082 
0083 spectrum = max(spectrum, 0);
0084 s = sqrt(spectrum);
0085 for i = 1 : d
0086     X(:,i) = X(:,i) * s(i);
0087 end
0088 X = X';
0089 
0090 % de-weights
0091 if ~isempty(w)
0092     for i = 1 : n
0093         X(:,i) = X(:,i) / w(i);
0094     end
0095 end
0096     
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