manifold-learning.lle.zip_manifold learning_matlab 流形学习_无监督_机器学习

% LLE ALGORITHM (using K nearest neighbors) % % [Y] = lle(X,K,dmax) % % 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(X,K,d) [D,N] = size(X); fprintf(1,'LLE running on %d points in %d dimensions\n',N,D); % STEP1: COMPUTE PAIRWISE DISTANCES & FIND NEIGHBORS fprintf(1,'-->Finding %d nearest neighbours.\n',K); X2 = sum(X.^2,1); distance = repmat(X2,N,1)+repmat(X2',1,N)-2*X'*X; [sorted,index] = 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(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) = C\ones(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.\n'); % 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(:,2:d+1)'*sqrt(N); % bottom evect is [1,1,1,1...] with eval 0 fprintf(1,'Done.\n'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % other possible regularizers for K>D % C = C + tol*diag(diag(C)); % regularlization % C = C + eye(K,K)*tol*trace(C)*K; % regularlization