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Statistical Learning Toolbox

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

Description of sllocaltanspace
Home > sltoolbox > manifold > sllocaltanspace.m

sllocaltanspace

PURPOSE ^

SLLOCALTANSPACE Solves the local tangent spaces

SYNOPSIS ^

function [LM, LP, LS] = sllocaltanspace(X0, G, dl)

DESCRIPTION ^

SLLOCALTANSPACE Solves the local tangent spaces

 $ Syntax $
   - [LM, LP] = sllocaltanspace(X0, G, dl)
   - [LM, LP, LS] = sllocaltanspace(...)

 $ Arguments $
   - X0:       The referenced sample matrix (d0 x n0)
   - G:        The neighborhood graph (n0 x n)
   - dl:       The dimension of local tangent spaces
               (should be strictly less than d0 and n0)
   - LM:       The local means (d0 x n)
   - LP:       The local tangent space basis (d0 x dl x n)
   - LS:       The local spectrum (dl x n)

 $ Description $
   - [LM, LP] = sllocaltanspace(X0, G, dl) solves the local tangent
     spaces based on the neighborhood graph G. Suppose G is n0 x n,
     (n0 source points and n target points), then it solves the local
     tangent spaces at n target points with the space constructed with
     their neighbors in X0. If G is valued, the values in G are the 
     weights of the samples in constructing local tangent space.

   - [LM, LP, LS] = sllocaltanspace(...) additionally outputs the
     eigen-spectrum of the local spaces. 

 $ Remarks $
   - The local dimensions are sorted in descending order of the 
     corresponding eigenvalues. In the case of local rank < dl,
     for the last dl - rank dimensions, the eigenvalues are set to
     zeros, and the eigenvectors are set to zero vectors.

 $ History $
   - Created by Dahua Lin, on Sep 13rd, 2006

CROSS-REFERENCE INFORMATION ^

This function calls:
  • sladdvec SLADDVEC adds a vector to columns or rows of a matrix
  • slmulvec SLMULVEC multiplies a vector to columns or rows of a matrix
  • slnormalize SLNORMALIZE Normalize the sub-arrays
  • slsymeig SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
  • sladjmat SLADJMAT Constructs the adjacency matrix representation of a graph
  • slgraphinfo SLGRAPHINFO Extracts basic information of a given graph representation
  • slmean SLMEAN Compute the mean vector of samples
This function is called by:
  • slltsa SLLTSA Performs Local Tangent Space Alignment Learning

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function [LM, LP, LS] = sllocaltanspace(X0, G, dl) 
0002 %SLLOCALTANSPACE Solves the local tangent spaces
0003 %
0004 % $ Syntax $
0005 %   - [LM, LP] = sllocaltanspace(X0, G, dl)
0006 %   - [LM, LP, LS] = sllocaltanspace(...)
0007 %
0008 % $ Arguments $
0009 %   - X0:       The referenced sample matrix (d0 x n0)
0010 %   - G:        The neighborhood graph (n0 x n)
0011 %   - dl:       The dimension of local tangent spaces
0012 %               (should be strictly less than d0 and n0)
0013 %   - LM:       The local means (d0 x n)
0014 %   - LP:       The local tangent space basis (d0 x dl x n)
0015 %   - LS:       The local spectrum (dl x n)
0016 %
0017 % $ Description $
0018 %   - [LM, LP] = sllocaltanspace(X0, G, dl) solves the local tangent
0019 %     spaces based on the neighborhood graph G. Suppose G is n0 x n,
0020 %     (n0 source points and n target points), then it solves the local
0021 %     tangent spaces at n target points with the space constructed with
0022 %     their neighbors in X0. If G is valued, the values in G are the
0023 %     weights of the samples in constructing local tangent space.
0024 %
0025 %   - [LM, LP, LS] = sllocaltanspace(...) additionally outputs the
0026 %     eigen-spectrum of the local spaces.
0027 %
0028 % $ Remarks $
0029 %   - The local dimensions are sorted in descending order of the
0030 %     corresponding eigenvalues. In the case of local rank < dl,
0031 %     for the last dl - rank dimensions, the eigenvalues are set to
0032 %     zeros, and the eigenvectors are set to zero vectors.
0033 %
0034 % $ History $
0035 %   - Created by Dahua Lin, on Sep 13rd, 2006
0036 %
0037 
0038 %% parse and verify input arguments
0039 
0040 if ~isnumeric(X0) || ndims(X0) ~= 2
0041     error('sltoolbox:invalidarg', ...
0042         'The sample matrix X0 should be a 2D numeric matrix');
0043 end
0044 [d0, n0] = size(X0);
0045 
0046 gi = slgraphinfo(G);
0047 if gi.n ~= n0
0048     error('sltoolbox:invalidarg', ...
0049         'The graph is not consistent with the sample number');
0050 end
0051 if ~strcmp(gi.form, 'adjmat')
0052     G = sladjmat(G, 'sparse', true);
0053 end
0054 n = gi.nt;
0055 
0056 if dl >= d0 || dl >= n0
0057     error('sltoolbox:invalidarg', ...
0058         'The local dimension dl should be strictly less than d0 and n0');
0059 end
0060 
0061 if nargout >= 3
0062     want_sp = true;
0063 else
0064     want_sp = false;
0065 end
0066 
0067 if isnumeric(G)
0068     use_weights = true;
0069 else
0070     use_weights = false;
0071 end
0072 
0073 %% main skeleton
0074 
0075 % prepare storage
0076 LM = zeros(d0, n);
0077 LP = zeros(d0, dl, n);
0078 if want_sp
0079     LS = zeros(dl, n);
0080 end
0081 
0082 % do computation
0083 for i = 1 : n    
0084     localinds = find(G(:,i));
0085     if use_weights
0086         localw = G(localinds, i)';
0087     else
0088         localw = [];
0089     end
0090     localX = X0(:, localinds);
0091     
0092     [cm, cp, csp] = solvelocalspace(localX, localw, d0, dl);
0093     
0094     LM(:,i) = cm;
0095     LP(:,:,i) = cp;
0096     if want_sp
0097         LS(:,i) = csp;
0098     end
0099 end
0100 
0101 
0102 %% core routine to compute local tangent spaces
0103 
0104 function [vmean, P, spectrum] = solvelocalspace(X, w, d0, dl)
0105 
0106 n = size(X, 2);
0107 dm = min([dl, d0, n]);
0108 
0109 % preprocess samples: centralize and weight
0110 vmean = slmean(X, w, true);
0111 X = sladdvec(X, -vmean, 1);
0112 if ~isempty(w)
0113     X = slmulvec(X, w, 2);
0114 end
0115 
0116 % solve eigen-problem
0117 if d0 <= n / 2
0118     C = X * X';
0119     [spectrum, P] = slsymeig(C, dm); 
0120 else
0121     C = X' * X;
0122     [spectrum, P] = slsymeig(C, dm);
0123     P = slnormalize(X * P);
0124 end
0125 
0126 % truncate to rank
0127 rk = sum(spectrum > n * eps(spectrum(1)));
0128 if rk < dm
0129     spectrum = spectrum(1:rk);
0130     P = P(:, 1:rk);    
0131 end
0132 spectrum = spectrum / n;
0133 
0134 % complement to dl
0135 if rk < dl
0136     spectrum = [spectrum; zeros(dl-rk, 1)];
0137     P = [P, zeros(d0, dl-rk)];
0138 end
0139
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