mySVD.m

function [U, S, V] = mySVD(X,ReducedDim)
%mySVD    Accelerated singular value decomposition.
%   [U,S,V] = mySVD(X) produces a diagonal matrix S, of the  
%   dimension as the rank of X and with nonnegative diagonal elements in
%   decreasing order, and unitary matrices U and V so that
%   X = U*S*V'.
%
%   [U,S,V] = mySVD(X,ReducedDim) produces a diagonal matrix S, of the  
%   dimension as ReducedDim and with nonnegative diagonal elements in
%   decreasing order, and unitary matrices U and V so that
%   Xhat = U*S*V' is the best approximation (with respect to F norm) of X
%   among all the matrices with rank no larger than ReducedDim.
%
%   Based on the size of X, mySVD computes the eigvectors of X*X^T or X^T*X
%   first, and then convert them to the eigenvectors of the other.  
%
%   See also SVD.
%
%   version 2.0 --Feb/2009 
%   version 1.0 --April/2004 
%
%   Written by Deng Cai (dengcai AT gmail.com)
%                                                   

MAX_MATRIX_SIZE = 10000; % You can change this number according your machine computational power
EIGVECTOR_RATIO = 0.1; % You can change this number according your machine computational power


if ~exist('ReducedDim','var')
    ReducedDim = 0;
end

[nSmp, mFea] = size(X);
if mFea/nSmp > 1.0713
    ddata = X*X';
    ddata = max(ddata,ddata');
    
    dimMatrix = size(ddata,1);
    if (ReducedDim > 0) && (dimMatrix > MAX_MATRIX_SIZE) && (ReducedDim < dimMatrix*EIGVECTOR_RATIO)
        option = struct('disp',0);
        [U, eigvalue] = eigs(ddata,ReducedDim,'la',option);
        eigvalue = diag(eigvalue);
    else
        if issparse(ddata)
            ddata = full(ddata);
        end
        
        [U, eigvalue] = eig(ddata);
        eigvalue = diag(eigvalue);
        [dump, index] = sort(-eigvalue);
        eigvalue = eigvalue(index);
        U = U(:, index);
    end
    clear ddata;
    
    maxEigValue = max(abs(eigvalue));
    eigIdx = find(abs(eigvalue)/maxEigValue < 1e-10);
    eigvalue(eigIdx) = [];
    U(:,eigIdx) = [];
    
    if (ReducedDim > 0) && (ReducedDim < length(eigvalue))
        eigvalue = eigvalue(1:ReducedDim);
        U = U(:,1:ReducedDim);
    end
    
    eigvalue_Half = eigvalue.^.5;
    S =  spdiags(eigvalue_Half,0,length(eigvalue_Half),length(eigvalue_Half));

    if nargout >= 3
        eigvalue_MinusHalf = eigvalue_Half.^-1;
        V = X'*(U.*repmat(eigvalue_MinusHalf',size(U,1),1));
    end
else
    ddata = X'*X;
    ddata = max(ddata,ddata');
    
    dimMatrix = size(ddata,1);
    if (ReducedDim > 0) && (dimMatrix > MAX_MATRIX_SIZE) && (ReducedDim < dimMatrix*EIGVECTOR_RATIO)
        option = struct('disp',0);
        [V, eigvalue] = eigs(ddata,ReducedDim,'la',option);
        eigvalue = diag(eigvalue);
    else
        if issparse(ddata)
            ddata = full(ddata);
        end
        
        [V, eigvalue] = eig(ddata);
        eigvalue = diag(eigvalue);
        
        [dump, index] = sort(-eigvalue);
        eigvalue = eigvalue(index);
        V = V(:, index);
    end
    clear ddata;
    
    maxEigValue = max(abs(eigvalue));
    eigIdx = find(abs(eigvalue)/maxEigValue < 1e-10);
    eigvalue(eigIdx) = [];
    V(:,eigIdx) = [];
    
    if (ReducedDim > 0) && (ReducedDim < length(eigvalue))
        eigvalue = eigvalue(1:ReducedDim);
        V = V(:,1:ReducedDim);
    end
    
    eigvalue_Half = eigvalue.^.5;
    S =  spdiags(eigvalue_Half,0,length(eigvalue_Half),length(eigvalue_Half));
    
    eigvalue_MinusHalf = eigvalue_Half.^-1;
    U = X*(V.*repmat(eigvalue_MinusHalf',size(V,1),1));
end