LaplacianScore计算特征得分，对特征进行选择

1、

clear;
load iris.dat
fea = iris(:,1:4);%最后一列是label，这里只需要特征，我们把特征拿出来单独成一个矩阵。每一行表示一个数据
fea = NormalizeFea(fea);% %Normalize each data vector to have L2-norm equal to 1 

W = constructW(fea);
Y = LaplacianScore(fea, W);%Y: Vector of (1-LaplacianScore) for each feature,The features with larger y are more important.
[dump,idx] = sort(-Y);

2、

function fea = NormalizeFea(fea,row)
% if row == 1, normalize each row of fea to have unit norm;
% if row == 0, normalize each column of fea to have unit norm;
%
%   version 3.0 --Jan/2012 
%   version 2.0 --Jan/2012 
%   version 1.0 --Oct/2003 
%
%   Written by Deng Cai (dengcai AT gmail.com)
%

if ~exist('row','var')
    row = 1;
end

if row
    nSmp = size(fea,1);
    feaNorm = max(1e-14,full(sum(fea.^2,2)));
    fea = spdiags(feaNorm.^-.5,0,nSmp,nSmp)*fea;
else
    nSmp = size(fea,2);
    feaNorm = max(1e-14,full(sum(fea.^2,1))');
    fea = fea*spdiags(feaNorm.^-.5,0,nSmp,nSmp);
end
            
return;
if row
    [nSmp, mFea] = size(fea);
    if issparse(fea)
        fea2 = fea';
        feaNorm = mynorm(fea2,1);
        for i = 1:nSmp
            fea2(:,i) = fea2(:,i) ./ max(1e-10,feaNorm(i));
        end
        fea = fea2';
    else
        feaNorm = sum(fea.^2,2).^.5;
        fea = fea./feaNorm(:,ones(1,mFea));
    end
else
    [mFea, nSmp] = size(fea);
    if issparse(fea)
        feaNorm = mynorm(fea,1);
        for i = 1:nSmp
            fea(:,i) = fea(:,i) ./ max(1e-10,feaNorm(i));
        end
    else
        feaNorm = sum(fea.^2,1).^.5;
        fea = fea./feaNorm(ones(1,mFea),:);
    end
end
           

3、

function [Y] = LaplacianScore(X, W)
%    Usage:
%    [Y] = LaplacianScore(X, W)
%
%    X: Rows of vectors of data points
%    W: The affinity matrix.
%    Y: Vector of (1-LaplacianScore) for each feature.
%      The features with larger y are more important.
%
%    Examples:
%
%       fea = rand(50,70);
%       options = [];
%       options.Metric = 'Cosine';
%       options.NeighborMode = 'KNN';
%       options.k = 5;
%       options.WeightMode = 'Cosine';
%       W = constructW(fea,options);
%
%       LaplacianScore = LaplacianScore(fea,W);
%       [junk, index] = sort(-LaplacianScore);
%       
%       newfea = fea(:,index);
%       %the features in newfea will be sorted based on their importance.
%
%    Type "LaplacianScore" for a self-demo.
%
% See also constructW
%
%Reference:
%
%   Xiaofei He, Deng Cai and Partha Niyogi, "Laplacian Score for Feature Selection".
%   Advances in Neural Information Processing Systems 18 (NIPS 2005),
%   Vancouver, Canada, 2005.   
%
%   Deng Cai, 2004/08
if nargin == 0, selfdemo; return; end

[nSmp,nFea] = size(X);

if size(W,1) ~= nSmp
    error('W is error');
end

D = full(sum(W,2));
L = W;

allone = ones(nSmp,1);


tmp1 = D'*X;

D = sparse(1:nSmp,1:nSmp,D,nSmp,nSmp);

DPrime = sum((X'*D)'.*X)-tmp1.*tmp1/sum(diag(D));
LPrime = sum((X'*L)'.*X)-tmp1.*tmp1/sum(diag(D));

DPrime(find(DPrime < 1e-12)) = 10000;

Y = LPrime./DPrime;
Y = Y';
Y = full(Y);

4、

function D = EuDist2(fea_a,fea_b,bSqrt)
%EUDIST2 Efficiently Compute the Euclidean Distance Matrix by Exploring the
%Matlab matrix operations.
%
%   D = EuDist(fea_a,fea_b)
%   fea_a:    nSample_a * nFeature
%   fea_b:    nSample_b * nFeature
%   D:      nSample_a * nSample_a
%       or  nSample_a * nSample_b
%
%    Examples:
%
%       a = rand(500,10);
%       b = rand(1000,10);
%
%       A = EuDist2(a); % A: 500*500
%       D = EuDist2(a,b); % D: 500*1000
%
%   version 2.1 --November/2011
%   version 2.0 --May/2009
%   version 1.0 --November/2005
%
%   Written by Deng Cai (dengcai AT gmail.com)


if ~exist('bSqrt','var')
    bSqrt = 1;
end

if (~exist('fea_b','var')) || isempty(fea_b)
    aa = sum(fea_a.*fea_a,2);
    ab = fea_a*fea_a';
    
    if issparse(aa)
        aa = full(aa);
    end
    
    D = bsxfun(@plus,aa,aa') - 2*ab;
    D(D<0) = 0;
    if bSqrt
        D = sqrt(D);
    end
    D = max(D,D');
else
    aa = sum(fea_a.*fea_a,2);
    bb = sum(fea_b.*fea_b,2);
    ab = fea_a*fea_b';

    if issparse(aa)
        aa = full(aa);
        bb = full(bb);
    end

    D = bsxfun(@plus,aa,bb') - 2*ab;
    D(D<0) = 0;
    if bSqrt
        D = sqrt(D);
    end
end

5、

function W = constructW(fea,options)
%    Usage:
%    W = constructW(fea,options)
%
%    fea: Rows of vectors of data points. Each row is x_i
%   options: Struct value in Matlab. The fields in options that can be set:
%                  
%           NeighborMode -  Indicates how to construct the graph. Choices
%                           are: [Default 'KNN']
%                'KNN'            -  k = 0
%                                       Complete graph
%                                    k > 0
%                                      Put an edge between two nodes if and
%                                      only if they are among the k nearst
%                                      neighbors of each other. You are
%                                      required to provide the parameter k in
%                                      the options. Default k=5.
%               'Supervised'      -  k = 0
%                                       Put an edge between two nodes if and
%                                       only if they belong to same class. 
%                                    k > 0
%                                       Put an edge between two nodes if
%                                       they belong to same class and they
%                                       are among the k nearst neighbors of
%                                       each other. 
%                                    Default: k=0
%                                   You are required to provide the label
%                                   information gnd in the options.
%                                              
%           WeightMode   -  Indicates how to assign weights for each edge
%                           in the graph. Choices are:
%               'Binary'       - 0-1 weighting. Every edge receiveds weight
%                                of 1. 
%               'HeatKernel'   - If nodes i and j are connected, put weight
%                                W_ij = exp(-norm(x_i - x_j)/2t^2). You are 
%                                required to provide the parameter t. [Default One]
%               'Cosine'       - If nodes i and j are connected, put weight
%                                cosine(x_i,x_j). 
%               
%            k         -   The parameter needed under 'KNN' NeighborMode.
%                          Default will be 5.
%            gnd       -   The parameter needed under 'Supervised'
%                          NeighborMode.  Colunm vector of the label
%                          information for each data point.
%            bLDA      -   0 or 1. Only effective under 'Supervised'
%                          NeighborMode. If 1, the graph will be constructed
%                          to make LPP exactly same as LDA. Default will be
%                          0. 
%            t         -   The parameter needed under 'HeatKernel'
%                          WeightMode. Default will be 1
%         bNormalized  -   0 or 1. Only effective under 'Cosine' WeightMode.
%                          Indicates whether the fea are already be
%                          normalized to 1. Default will be 0
%      bSelfConnected  -   0 or 1. Indicates whether W(i,i) == 1. Default 0
%                          if 'Supervised' NeighborMode & bLDA == 1,
%                          bSelfConnected will always be 1. Default 0.
%            bTrueKNN  -   0 or 1. If 1, will construct a truly kNN graph
%                          (Not symmetric!). Default will be 0. Only valid
%                          for 'KNN' NeighborMode
%
%
%    Examples:
%
%       fea = rand(50,15);
%       options = [];
%       options.NeighborMode = 'KNN';
%       options.k = 5;
%       options.WeightMode = 'HeatKernel';
%       options.t = 1;
%       W = constructW(fea,options);
%       
%       
%       fea = rand(50,15);
%       gnd = [ones(10,1);ones(15,1)*2;ones(10,1)*3;ones(15,1)*4];
%       options = [];
%       options.NeighborMode = 'Supervised';
%       options.gnd = gnd;
%       options.WeightMode = 'HeatKernel';
%       options.t = 1;
%       W = constructW(fea,options);
%       
%       
%       fea = rand(50,15);
%       gnd = [ones(10,1);ones(15,1)*2;ones(10,1)*3;ones(15,1)*4];
%       options = [];
%       options.NeighborMode = 'Supervised';
%       options.gnd = gnd;
%       options.bLDA = 1;
%       W = constructW(fea,options);      
%       
%
%    For more details about the different ways to construct the W, please
%    refer:
%       Deng Cai, Xiaofei He and Jiawei Han, "Document Clustering Using
%       Locality Preserving Indexing" IEEE TKDE, Dec. 2005.
%    
%
%    Written by Deng Cai (dengcai2 AT cs.uiuc.edu), April/2004, Feb/2006,
%                                             May/2007
% 

bSpeed  = 1;

if (~exist('options','var'))
   options = [];
end

if isfield(options,'Metric')
    warning('This function has been changed and the Metric is no longer be supported');
end


if ~isfield(options,'bNormalized')
    options.bNormalized = 0;
end

%=================================================
if ~isfield(options,'NeighborMode')
    options.NeighborMode = 'KNN';
end

switch lower(options.NeighborMode)
    case {lower('KNN')}  %For simplicity, we include the data point itself in the kNN
        if ~isfield(options,'k')
            options.k = 5;
        end
    case {lower('Supervised')}
        if ~isfield(options,'bLDA')
            options.bLDA = 0;
        end
        if options.bLDA
            options.bSelfConnected = 1;
        end
        if ~isfield(options,'k')
            options.k = 0;
        end
        if ~isfield(options,'gnd')
            error('Label(gnd) should be provided under ''Supervised'' NeighborMode!');
        end
        if ~isempty(fea) && length(options.gnd) ~= size(fea,1)
            error('gnd doesn''t match with fea!');
        end
    otherwise
        error('NeighborMode does not exist!');
end

%=================================================

if ~isfield(options,'WeightMode')
    options.WeightMode = 'HeatKernel';
end

bBinary = 0;
bCosine = 0;
switch lower(options.WeightMode)
    case {lower('Binary')}
        bBinary = 1; 
    case {lower('HeatKernel')}
        if ~isfield(options,'t')
            nSmp = size(fea,1);
            if nSmp > 3000
                D = EuDist2(fea(randsample(nSmp,3000),:));
            else
                D = EuDist2(fea);
            end
            options.t = mean(mean(D));
        end
    case {lower('Cosine')}
        bCosine = 1;
    otherwise
        error('WeightMode does not exist!');
end

%=================================================

if ~isfield(options,'bSelfConnected')
    options.bSelfConnected = 0;
end

%=================================================

if isfield(options,'gnd') 
    nSmp = length(options.gnd);
else
    nSmp = size(fea,1);
end
maxM = 62500000; %500M
BlockSize = floor(maxM/(nSmp*3));


if strcmpi(options.NeighborMode,'Supervised')
    Label = unique(options.gnd);
    nLabel = length(Label);
    if options.bLDA
        G = zeros(nSmp,nSmp);
        for idx=1:nLabel
            classIdx = options.gnd==Label(idx);
            G(classIdx,classIdx) = 1/sum(classIdx);
        end
        W = sparse(G);
        return;
    end
    
    switch lower(options.WeightMode)
        case {lower('Binary')}
            if options.k > 0
                G = zeros(nSmp*(options.k+1),3);
                idNow = 0;
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    D = EuDist2(fea(classIdx,:),[],0);
                    [dump idx] = sort(D,2); % sort each row
                    clear D dump;
                    idx = idx(:,1:options.k+1);
                    
                    nSmpClass = length(classIdx)*(options.k+1);
                    G(idNow+1:nSmpClass+idNow,1) = repmat(classIdx,[options.k+1,1]);
                    G(idNow+1:nSmpClass+idNow,2) = classIdx(idx(:));
                    G(idNow+1:nSmpClass+idNow,3) = 1;
                    idNow = idNow+nSmpClass;
                    clear idx
                end
                G = sparse(G(:,1),G(:,2),G(:,3),nSmp,nSmp);
                G = max(G,G');
            else
                G = zeros(nSmp,nSmp);
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    G(classIdx,classIdx) = 1;
                end
            end
            
            if ~options.bSelfConnected
                for i=1:size(G,1)
                    G(i,i) = 0;
                end
            end
            
            W = sparse(G);
        case {lower('HeatKernel')}
            if options.k > 0
                G = zeros(nSmp*(options.k+1),3);
                idNow = 0;
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    D = EuDist2(fea(classIdx,:),[],0);
                    [dump idx] = sort(D,2); % sort each row
                    clear D;
                    idx = idx(:,1:options.k+1);
                    dump = dump(:,1:options.k+1);
                    dump = exp(-dump/(2*options.t^2));
                    
                    nSmpClass = length(classIdx)*(options.k+1);
                    G(idNow+1:nSmpClass+idNow,1) = repmat(classIdx,[options.k+1,1]);
                    G(idNow+1:nSmpClass+idNow,2) = classIdx(idx(:));
                    G(idNow+1:nSmpClass+idNow,3) = dump(:);
                    idNow = idNow+nSmpClass;
                    clear dump idx
                end
                G = sparse(G(:,1),G(:,2),G(:,3),nSmp,nSmp);
            else
                G = zeros(nSmp,nSmp);
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    D = EuDist2(fea(classIdx,:),[],0);
                    D = exp(-D/(2*options.t^2));
                    G(classIdx,classIdx) = D;
                end
            end
            
            if ~options.bSelfConnected
                for i=1:size(G,1)
                    G(i,i) = 0;
                end
            end

            W = sparse(max(G,G'));
        case {lower('Cosine')}
            if ~options.bNormalized
                fea = NormalizeFea(fea);
            end

            if options.k > 0
                G = zeros(nSmp*(options.k+1),3);
                idNow = 0;
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    D = fea(classIdx,:)*fea(classIdx,:)';
                    [dump idx] = sort(-D,2); % sort each row
                    clear D;
                    idx = idx(:,1:options.k+1);
                    dump = -dump(:,1:options.k+1);
                    
                    nSmpClass = length(classIdx)*(options.k+1);
                    G(idNow+1:nSmpClass+idNow,1) = repmat(classIdx,[options.k+1,1]);
                    G(idNow+1:nSmpClass+idNow,2) = classIdx(idx(:));
                    G(idNow+1:nSmpClass+idNow,3) = dump(:);
                    idNow = idNow+nSmpClass;
                    clear dump idx
                end
                G = sparse(G(:,1),G(:,2),G(:,3),nSmp,nSmp);
            else
                G = zeros(nSmp,nSmp);
                for i=1:nLabel
                    classIdx = find(options.gnd==Label(i));
                    G(classIdx,classIdx) = fea(classIdx,:)*fea(classIdx,:)';
                end
            end

            if ~options.bSelfConnected
                for i=1:size(G,1)
                    G(i,i) = 0;
                end
            end

            W = sparse(max(G,G'));
        otherwise
            error('WeightMode does not exist!');
    end
    return;
end


if bCosine && ~options.bNormalized
    Normfea = NormalizeFea(fea);
end

if strcmpi(options.NeighborMode,'KNN') && (options.k > 0)
    if ~(bCosine && options.bNormalized)
        G = zeros(nSmp*(options.k+1),3);
        for i = 1:ceil(nSmp/BlockSize)
            if i == ceil(nSmp/BlockSize)
                smpIdx = (i-1)*BlockSize+1:nSmp;
                dist = EuDist2(fea(smpIdx,:),fea,0);

                if bSpeed
                    nSmpNow = length(smpIdx);
                    dump = zeros(nSmpNow,options.k+1);
                    idx = dump;
                    for j = 1:options.k+1
                        [dump(:,j),idx(:,j)] = min(dist,[],2);
                        temp = (idx(:,j)-1)*nSmpNow+[1:nSmpNow]';
                        dist(temp) = 1e100;
                    end
                else
                    [dump idx] = sort(dist,2); % sort each row
                    idx = idx(:,1:options.k+1);
                    dump = dump(:,1:options.k+1);
                end
                
                if ~bBinary
                    if bCosine
                        dist = Normfea(smpIdx,:)*Normfea';
                        dist = full(dist);
                        linidx = [1:size(idx,1)]';
                        dump = dist(sub2ind(size(dist),linidx(:,ones(1,size(idx,2))),idx));
                    else
                        dump = exp(-dump/(2*options.t^2));
                    end
                end
                
                G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),1) = repmat(smpIdx',[options.k+1,1]);
                G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),2) = idx(:);
                if ~bBinary
                    G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),3) = dump(:);
                else
                    G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),3) = 1;
                end
            else
                smpIdx = (i-1)*BlockSize+1:i*BlockSize;
            
                dist = EuDist2(fea(smpIdx,:),fea,0);
                
                if bSpeed
                    nSmpNow = length(smpIdx);
                    dump = zeros(nSmpNow,options.k+1);
                    idx = dump;
                    for j = 1:options.k+1
                        [dump(:,j),idx(:,j)] = min(dist,[],2);
                        temp = (idx(:,j)-1)*nSmpNow+[1:nSmpNow]';
                        dist(temp) = 1e100;
                    end
                else
                    [dump idx] = sort(dist,2); % sort each row
                    idx = idx(:,1:options.k+1);
                    dump = dump(:,1:options.k+1);
                end
                
                if ~bBinary
                    if bCosine
                        dist = Normfea(smpIdx,:)*Normfea';
                        dist = full(dist);
                        linidx = [1:size(idx,1)]';
                        dump = dist(sub2ind(size(dist),linidx(:,ones(1,size(idx,2))),idx));
                    else
                        dump = exp(-dump/(2*options.t^2));
                    end
                end
                
                G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),1) = repmat(smpIdx',[options.k+1,1]);
                G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),2) = idx(:);
                if ~bBinary
                    G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),3) = dump(:);
                else
                    G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),3) = 1;
                end
            end
        end

        W = sparse(G(:,1),G(:,2),G(:,3),nSmp,nSmp);
    else
        G = zeros(nSmp*(options.k+1),3);
        for i = 1:ceil(nSmp/BlockSize)
            if i == ceil(nSmp/BlockSize)
                smpIdx = (i-1)*BlockSize+1:nSmp;
                dist = fea(smpIdx,:)*fea';
                dist = full(dist);

                if bSpeed
                    nSmpNow = length(smpIdx);
                    dump = zeros(nSmpNow,options.k+1);
                    idx = dump;
                    for j = 1:options.k+1
                        [dump(:,j),idx(:,j)] = max(dist,[],2);
                        temp = (idx(:,j)-1)*nSmpNow+[1:nSmpNow]';
                        dist(temp) = 0;
                    end
                else
                    [dump idx] = sort(-dist,2); % sort each row
                    idx = idx(:,1:options.k+1);
                    dump = -dump(:,1:options.k+1);
                end

                G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),1) = repmat(smpIdx',[options.k+1,1]);
                G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),2) = idx(:);
                G((i-1)*BlockSize*(options.k+1)+1:nSmp*(options.k+1),3) = dump(:);
            else
                smpIdx = (i-1)*BlockSize+1:i*BlockSize;
                dist = fea(smpIdx,:)*fea';
                dist = full(dist);
                
                if bSpeed
                    nSmpNow = length(smpIdx);
                    dump = zeros(nSmpNow,options.k+1);
                    idx = dump;
                    for j = 1:options.k+1
                        [dump(:,j),idx(:,j)] = max(dist,[],2);
                        temp = (idx(:,j)-1)*nSmpNow+[1:nSmpNow]';
                        dist(temp) = 0;
                    end
                else
                    [dump idx] = sort(-dist,2); % sort each row
                    idx = idx(:,1:options.k+1);
                    dump = -dump(:,1:options.k+1);
                end

                G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),1) = repmat(smpIdx',[options.k+1,1]);
                G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),2) = idx(:);
                G((i-1)*BlockSize*(options.k+1)+1:i*BlockSize*(options.k+1),3) = dump(:);
            end
        end

        W = sparse(G(:,1),G(:,2),G(:,3),nSmp,nSmp);
    end
    
    if bBinary
        W(logical(W)) = 1;
    end
    
    if isfield(options,'bSemiSupervised') && options.bSemiSupervised
        tmpgnd = options.gnd(options.semiSplit);
        
        Label = unique(tmpgnd);
        nLabel = length(Label);
        G = zeros(sum(options.semiSplit),sum(options.semiSplit));
        for idx=1:nLabel
            classIdx = tmpgnd==Label(idx);
            G(classIdx,classIdx) = 1;
        end
        Wsup = sparse(G);
        if ~isfield(options,'SameCategoryWeight')
            options.SameCategoryWeight = 1;
        end
        W(options.semiSplit,options.semiSplit) = (Wsup>0)*options.SameCategoryWeight;
    end
    
    if ~options.bSelfConnected
        W = W - diag(diag(W));
    end

    if isfield(options,'bTrueKNN') && options.bTrueKNN
        
    else
        W = max(W,W');
    end
    
    return;
end


% strcmpi(options.NeighborMode,'KNN') & (options.k == 0)
% Complete Graph

switch lower(options.WeightMode)
    case {lower('Binary')}
        error('Binary weight can not be used for complete graph!');
    case {lower('HeatKernel')}
        W = EuDist2(fea,[],0);
        W = exp(-W/(2*options.t^2));
    case {lower('Cosine')}
        W = full(Normfea*Normfea');
    otherwise
        error('WeightMode does not exist!');
end

if ~options.bSelfConnected
    for i=1:size(W,1)
        W(i,i) = 0;
    end
end

W = max(W,W');