% SLLE ALGORITHM (using K nearest neighbors) % % [Y] = lle(X,K,dmax,a) % % 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 % a=增量因子 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [Y] = lle(X,K,d,a) [D,N] = size(X); fprintf(1,'SLLE running on %d points in %d dimensions ',N,D); % STEP1: COMPUTE PAIRWISE DISTANCES & FIND NEIGHBORS fprintf(1,'-->Finding %d nearest neighbours. ',K); X2 = sum(X.^2,1); distance = repmat(X2,N,1)+repmat(X2',1,N)-2*X'*X; B=ones(N); R=N/(d+1); for i=1:d+1; B(1+R*(i-1):R*i,1+R*(i-1):R*i)=zeros(R); end; distance1=distance+a*max(max(distance))*B; [sorted,index] = sort(distance1); neighborhood = index(2:(1+K),:); % STEP2: SOLVE FOR RECONSTRUCTION WEIGHTS fprintf(1,'-->Solving for reconstruction weights. '); if(K>D) fprintf(1,' [note: K>D; regularization will be used] '); tol=1e-3; % regularlizer in case constrained fits are ill conditioned else tol=0; end; tol=1e-3; W = zeros(K,N); for ii=1:N z = X(:,neighborhood(:,ii))-repmat(X(:,ii),1,K); % shift ith pt to origin C = z'*z; % local covariance C = C + eye(K,K)*tol*trace(C); % regularlization (K>D) W(:,ii) = Cones(K,1); % 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. '); % M=eye(N,N); % use a sparse matrix with storage for 4KN nonzero elements M = sparse(1:N,1:N,ones(1,N),N,N,4*K*N); for ii=1:N w = W(:,ii); jj = neighborhood(:,ii); M(ii,jj) = M(ii,jj) - w'; M(jj,ii) = M(jj,ii) - w; M(jj,jj) = M(jj,jj) + w*w'; end; % CALCULATION OF EMBEDDING options.disp = 0; options.isreal = 1; options.issym = 1; [Y,eigenvals] = eigs(M,d+1,0,options); Y = Y(:,1:d)'*sqrt(N); % bottom evect is [1,1,1,1...] with eval 0 fprintf(1,'Done. '); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % other possible regularizers for K>D % C = C + tol*diag(diag(C)); % regularlization % C = C + eye(K,K)*tol*trace(C)*K; % regularlization
测试用例（瑞士卷，貌似挺好吃的）： clear all,clc N = 2000; K = 12; d = 3; a=0; % Plot true manfold tt0 = (3*pi/2)*(1+2*[0:0.02:1]); hh = [0:0.125:1]*30; xx = (tt0.*cos(tt0))'*ones(size(hh)); yy = ones(size(tt0))'*hh; zz = (tt0.*sin(tt0))'*ones(size(hh)); cc = tt0'*ones(size(hh)); subplot(1,3,1); cla; surf(xx,yy,zz,cc); view([12 20]); grid off; axis off; hold on; lnx=-5*[3,3,3;3,-4,3]; lny=[0,0,0;32,0,0]; lnz=-5*[3,3,3;3,3,-3]; lnh=line(lnx,lny,lnz); set(lnh,'Color',[1,1,1],'LineWidth',2,'LineStyle','-','Clipping','off'); axis([-15,20,0,32,-15,15]); %generate sample data tt = (3*pi/2)*(1+2*rand(1,N)); height = 21*rand(1,N); X = [tt.*cos(tt); height; tt.*sin(tt)]; %scatter plot of sampled data subplot(1,3,2); cla; scatter3(X(1,:),X(2,:),X(3,:),12,tt,'+'); view([12 20]); grid off; axis off; hold on; lnh=line(lnx,lny,lnz); set(lnh,'Color',[1,1,1],'LineWidth',2,'LineStyle','-','Clipping','off'); axis([-15,20,0,32,-15,15]); drawnow; %run LLE algorithm Y=lle(X,K,d); %scatterplot of embedding subplot(1,3,3); cla; scatter(Y(1,:),Y(2,:),12,tt,'+'); grid off; set(gca,'XTick',[]); set(gca,'YTick',[]);