Local Tangent Space Alignment LTSA

The local information for a data point x, is calculated by finding the q largest eignvectors of the correction matrix W, of the neighbourhood around Xj such that 

LTSA seeks to minimise a cost function that minimises the distances between points in the low-dimensional space and the tangent space. As shown in [16], the solution to this minimisation problem is formed by the d smallest eigenvectors of an alignment matrix F. The alignment matrix is found by iteratively summing over all local information matrices  

Linear [1,4] Top eigenvectors K = K Xi,Xj) where k is an appropriately chosen kernel function 

Isomap [2] Top eigenvectors K = -IHt(S)H where r(S) is the squared distance matrix tmd H is the centering matrix 

Eigenmaps [14] Bottom eigenvectors K = HFtH where F is the pseudo-inverse of the Laplacian matrix 

---