%?LLE?ALGORITHM?（using?K?nearest?neighbors）
%
%?[Y]?=?lle（XKdmax）
%
%?X?=?data?as?D?x?N?matrix?（D?=?dimensionality?N?=?#points）
%?K?=?number?of?neighbors
%?dmax?=?max?embedding?dimensionality
%?Y?=?embedding?as?dmax?x?N?matrix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function?[Y]?=?lle（XKd）

[DN]?=?size（X）;
fprintf（1‘LLE?running?on?%d?points?in?%d?dimensions\n‘ND）;


%?STEP1:?COMPUTE?PAIRWISE?DISTANCES?&?FIND?NEIGHBORS?
fprintf（1‘-->Finding?%d?nearest?neighbours.\n‘K）;

X2?=?sum（X.^21）;
distance?=?repmat（X2N1）+repmat（X2‘1N）-2*X‘*X;

[sortedindex]?=?sort（distance）;
neighborhood?=?index（2:（1+K）:）;



%?STEP2:?SOLVE?FOR?RECONSTRUCTION?WEIGHTS
fprintf（1‘-->Solving?for?reconstruction?weights.\n‘）;

if（K>D）?
??fprintf（1‘???[note:?K>D;?regularization?will?be?used]\n‘）;?
??tol=1e-3;?%?regularlizer?in?case?constrained?fits?are?ill?conditioned
else
??tol=0;
end

W?=?zeros（KN）;
for?ii=1:N
???z?=?X（:neighborhood（:ii））-repmat（X（:ii）1K）;?%?shift?ith?pt?to?origin
???C?=?z‘*z;????????????????????????????????????????%?local?covariance
???C?=?C?+?eye（KK）*tol*trace（C）;???????????????????%?regularlization?（K>D）
???W（:ii）?=?C\ones（K1）;???????????????????????????%?solve?Cw=1
???W（:ii）?=?W（:ii）/sum（W（:ii））;??????????????????%?enforce?sum（w）=1
end;


%?STEP?3:?COMPUTE?embedDING?FROM?EIGENVECTS?OF?COST?MATRIX?M=（I-W）‘（I-W）
fprintf（1‘-->Computing?embedding.\n‘）;

%?M=eye（NN）;?%?use?a?sparse?matrix?with?storage?for?4KN?nonzero?elements
M?=?sparse（1:N1:Nones（1N）NN4*K*N）;?
for?ii=1:N
???w?=?W（:ii）;
???jj?=?neighborhood（:ii）;
???M（iijj）?=?M（iijj）?-?w‘;
???M（jjii）?=?M（jjii）?-?w;
???M（jjjj）?=?M（jjjj）?+?w*w‘;
end;

%?CALCULATION?OF?embedDING
options.disp?=?0;?options.isreal?=?1;?options.issym?=?1;?
[Yeigenvals]?=?eigs（Md+10options）;
Y?=?Y（:2:d+1）‘*sqrt（N）;?%?bottom?evect?is?[1111...]?with?eval?0


fprintf（1‘Done.\n‘）;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%?other?possible?regularizers?for?K>D
%???C?=?C?+?tol*diag（diag（C））;???????????????????????%?regularlization
%???C?=?C?+?eye（KK）*tol*trace（C）*K;?????????????????%?regularlization

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----

?????文件???????2140??2014-02-03?10:09??LLE\lle.m

?????文件???????1305??2014-02-03?10:09??LLE\scurve.m

?????文件???????1213??2014-02-03?10:09??LLE\swissroll.m

?????目錄??????????0??2014-02-03?10:11??LLE

-----------?---------??----------?-----??----

?????????????????4658????????????????????4